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background-position: left; width: 550px; min-height: 105px; _height: 105px; background-repeat: no-repeat; '> <div class='titlewrapper' style='background: transparent'> <h1 class='title' style='background: transparent; border-width: 0px'> <a href='http://www.righto.com/'> Ken Shirriff's blog </a> </h1> </div> <div class='descriptionwrapper'> <p class='description'><span>Computer history, restoring vintage computers, IC reverse engineering, and whatever</span></p> </div> </div> </div></div> </div> </div> <div class='header-cap-bottom cap-bottom'> <div class='cap-left'></div> <div class='cap-right'></div> </div> </div> </header> <div class='tabs-outer'> <div class='tabs-cap-top cap-top'> <div class='cap-left'></div> <div class='cap-right'></div> </div> <div class='fauxborder-left tabs-fauxborder-left'> <div class='fauxborder-right tabs-fauxborder-right'></div> <div class='region-inner tabs-inner'> <div class='tabs no-items section' id='crosscol'></div> <div class='tabs no-items section' id='crosscol-overflow'></div> </div> </div> <div class='tabs-cap-bottom cap-bottom'> <div class='cap-left'></div> <div class='cap-right'></div> </div> </div> <div class='main-outer'> <div class='main-cap-top cap-top'> <div class='cap-left'></div> <div class='cap-right'></div> </div> <div class='fauxborder-left main-fauxborder-left'> <div class='fauxborder-right main-fauxborder-right'></div> <div class='region-inner main-inner'> <div class='columns fauxcolumns'> <div class='fauxcolumn-outer fauxcolumn-center-outer'> <div class='cap-top'> <div class='cap-left'></div> <div class='cap-right'></div> </div> <div class='fauxborder-left'> <div class='fauxborder-right'></div> <div class='fauxcolumn-inner'> </div> </div> <div class='cap-bottom'> <div class='cap-left'></div> <div class='cap-right'></div> </div> </div> <div class='fauxcolumn-outer fauxcolumn-left-outer'> <div class='cap-top'> <div class='cap-left'></div> <div class='cap-right'></div> </div> <div class='fauxborder-left'> <div class='fauxborder-right'></div> <div class='fauxcolumn-inner'> </div> </div> <div class='cap-bottom'> <div class='cap-left'></div> <div class='cap-right'></div> </div> </div> <div class='fauxcolumn-outer fauxcolumn-right-outer'> <div class='cap-top'> <div class='cap-left'></div> <div class='cap-right'></div> </div> <div class='fauxborder-left'> <div class='fauxborder-right'></div> <div class='fauxcolumn-inner'> </div> </div> <div class='cap-bottom'> <div class='cap-left'></div> <div class='cap-right'></div> </div> </div>  <div class='columns-inner'> <div class='column-center-outer'> <div class='column-center-inner'> <div class='main section' id='main'><div class='widget Blog' data-version='1' id='Blog1'> <div class='blog-posts hfeed'> <div class="date-outer"> <div class="date-posts"> <div class='post-outer'> <div class='post hentry' itemprop='blogPost' itemscope='itemscope' itemtype='http://schema.org/BlogPosting'> <meta content='//static.righto.com/images/ARM1/2-chip_labeled-w600.png' itemprop='image_url'/> <meta content='6264947694886887540' itemprop='blogId'/> <meta content='9190561993947712649' itemprop='postId'/> <a name='9190561993947712649'></a> <h3 class='post-title entry-title' itemprop='name'> <a href='http://www.righto.com/2016/01/conditional-instructions-in-arm1.html'>Conditional instructions in the ARM1 processor, reverse engineered</a> </h3> <div class='post-header'> <div class='post-header-line-1'></div> </div> <div class='post-body entry-content' id='post-body-9190561993947712649' itemprop='description articleBody'> <meta property="og:image" content="//static.righto.com/images/ARM1/mux_detail.png"> By carefully examining the layout of the ARM1 processor, it can be reverse engineered. This article describes the interesting circuit used for conditional instructions: this circuit is marked in red on the die photo below. Unlike most processors, the ARM executes every instruction conditionally. Each instruction specifies a condition and is only executed if the condition is satisfied. For every instruction, the condition circuit reads the condition from the instruction register (blue), evaluates the condition flags (purple), and informs the control logic (yellow) if the instruction should be executed or skipped. <p> <a href="//static.righto.com/images/ARM1/3-chip_labeled.png"><img alt="The ARM1 processor chip showing the condition evaluation circuit (red) and the main components it interacts with. Original photo courtesy of Computer History Museum." class="hilite" height="546" src="//static.righto.com/images/ARM1/2-chip_labeled-w600.png" title="The ARM1 processor chip showing the condition evaluation circuit (red) and the main components it interacts with. Original photo courtesy of Computer History Museum." width="600"> </a> <p> <div class="cite"> The ARM1 processor chip showing the condition evaluation circuit (red) and the main components it interacts with. Original photo courtesy of <a href="http://www.computerhistory.org/revolution/digital-logic/12/286">Computer History Museum</a>. </div> <p> Why care about the ARM1 chip? It is the <a href="http://spectrum.ieee.org/semiconductors/processors/25-microchips-that-shook-the-world">highly-influential</a> ancestor of the extremely popular ARM processor. The ARM1 processor got off to a slow start in 1985 but now ARM processors are now sold by the tens of billions; your smart phone probably runs on ARM. This article is part of my series on reverse engineering the ARM1; start with my <a href="http://www.righto.com/2015/12/reverse-engineering-arm1-ancestor-of.html">first article</a> for an overview of the chip. <h2>What are conditional instructions?</h2> A key part of any computer is the ability of a program to change what it is doing based on various conditions. Most computers provide conditional branch instructions, which cause execution to jump to a different part of the program based on various condition flags. For example, consider the code <code>if (x == 0) { do_something }</code>. Compiled to assembly code, this first tests the value of variable x and sets the Zero flag if x is 0. Next, a conditional branch instruction jump over the <code>do_something</code> code if the Zero flag is not set. <p> The ARM processor takes conditionals much further than other processors: <i>every</i> instruction becomes a conditional instruction. Every instruction includes one of 16 conditions and the instruction is only executed if the condition is true; otherwise the instruction is skipped. (This is also known as <a href="https://en.wikipedia.org/wiki/Branch_predication">predication</a>.) The motivation is to avoid inefficient jumping around in the code. <p> The <a href="http://bitsavers.trailing-edge.com/pdf/acorn/VTI_ARM_Databook_1990.pdf">ARM manual excerpt</a> below shows how four bits in each 32-bit instruction specify one of 16 conditions. Most of the conditions are straightforward, checking if values are equal, negative, higher, and so forth. Most instructions will use the "always" condition, which simply means the instruction always executes. The opposite "never" condition is not highly useful - an instruction with that condition never executes - but it can be used for a NOP, patching code, or adjusting timing of an instruction sequence. <p> <a href="//static.righto.com/images/ARM1/condition_bits.png"><img alt="Every instruction in the ARM processor has one of 16 conditions specified. The instruction is executed only if the condition is satisfied." class="hilite" height="352" src="//static.righto.com/images/ARM1/condition_bits-w600.png" title="Every instruction in the ARM processor has one of 16 conditions specified. The instruction is executed only if the condition is satisfied." width="600"> </a> <p> <div class="cite"> Every instruction in the ARM processor has one of 16 conditions specified. The instruction is executed only if the condition is satisfied. </div> <p> Studying the different conditions reveals much of how the condition circuit works. It is based on four condition flags. The zero (Z) flag is set if a value is zero. The negative (N) flag is set if a value is negative. The carry (C) flag is set if there is a carry or borrow from addition or subtraction. The overflow (V) flag is set if there is an overflow during signed arithmetic (<a href="http://www.righto.com/2012/12/the-6502-overflow-flag-explained.html">details</a>). <p> The top three bits of the instruction select one of eight conditions, as highlighted in yellow. The fourth bit selects the condition or its opposite (blue). If the fourth bit is 0, the condition must be true; if the fourth bit is 1, the condition must be false. <h2>Implementation of the circuit</h2> The implementation of the conditional logic circuit matches the above description. First, the eight conditions are generated from the four flags. One of the conditions is selected based on the three instruction bits. If the fourth instruction bit is set, the condition is flipped. The result is 1 if the condition is satisfied, and 0 if the condition is not satisfied. One unexpected part of the circuit is that an undefined instruction or and interrupt causes the condition to be cleared, preventing execution of the instruction. The resulting condition signal output is connected to a control part of the chip, where it causes the instruction to be executed or not, as desired. <p> <a href="//static.righto.com/images/ARM1/overall_diagram.png"><img alt="The condition code evaluation circuit from the ARM1 processor." class="hilite" height="588" src="//static.righto.com/images/ARM1/overall_diagram-w500.png" title="The condition code evaluation circuit from the ARM1 processor." width="500"> </a> <p> <div class="cite"> The condition code evaluation circuit from the ARM1 processor. </div> <p> The diagram above shows the condition code circuit of the chip as it appears in the <a href="http://visual6502.org/sim/varm/armgl.html">simulator</a>; this is a zoomed-in version of the red rectangle indicated on the die earlier. The chip consists of multiple layers, indicated by different colors. Transistors appear as red or blue regions. NMOS transistors are red; they turn on with a 1 input and can pull their output low. PMOS transistors (blue) are complementary; they turn on with a 0 input and can pull their output high. Physically above the transistors is the polysilicon wiring layer (green). When polysilicon crosses a transistor it forms the gate (yellow) that controls the transistor. Finally, two layers of metal wiring (gray) are above the polysilicon. <p> The circuit is arranged in columns. The first column of transistors forms the logic gates to generate the conditions from the flag values. The next column is the multiplexer, a circuit that takes the eight input conditions and selects one. The rightmost column contains 8 NAND gates that decode the three instruction bits into 8 control lines. Each line is fed into the multiplexer to select the corresponding condition. At the right is the wiring for the 3 instruction bits and their complements. A few miscellaneous gates are at the bottom of the multiplexer and decoder columns. These include inverters to complement the instruction bits. <h2>The condition generation gates</h2> <p> The diagram below zooms in on the left third of the circuit above. This part of the circuit uses standard CMOS logic gates to computes the conditions from the flags. Each gate is built from NMOS (red) and PMOS (blue) transistors in a horizontal strip. Comparing the text description of conditions from the manual with the logic shows how the conditions are generated. For instance, the HI (unsigned higher) condition requires flags "C set and Z clear". The top three gates generate this condition. The GE (greater than or equal) condition is more complex, requiring flags "N set and V set, or N clear and V clear". The next two gates compute this value. (Due to the way CMOS gates are constructed, an OR-NAND gate is constructed as a single gate.) Likewise, the other conditions are generated. The AL (always) condition is simply a 1, and doesn't require any circuitry. The conditions are fed into the multiplexer, which will be discussed below. <p> The output coming back from the multiplexer is the selected condition, labeled "cond" below. The NAND and OR-NAND gates flip the condition if instruction register bit 28 (ireg28) is set. This implements the eight opposite conditions. The result is labeled "ok", indicating the overall condition is satisfied. The final three gates block instruction execution for an interrupt or undefined instruction. <p> <a href="//static.righto.com/images/ARM1/gates.png"><img alt="Gates in the ARM1 processor generate the various conditionals from the flag values." class="hilite" height="744" src="//static.righto.com/images/ARM1/gates-w500.png" title="Gates in the ARM1 processor generate the various conditionals from the flag values." width="500"> </a> <p> <div class="cite"> Gates in the ARM1 processor generate the various conditionals from the flag values. </div> <p> One thing I'd like to emphasize about the ARM1 is that its layout is very orderly and non-optimized. While it may appear chaotic, the gates are arranged by combining relatively fixed blocks ("standard cells") and wiring them together. Each gate forms a strip and the gates are stacked together in columns. The polysilicon and metal layers connect the gates as necessary. <p> The layout of the ARM1 chip is a consequence of the VLSI Technology chip design software used to create it. The resulting layout is simple, but doesn't use space very efficiently. Since the ARM1 uses very few transistors for its time, the designers weren't worried about optimizing the layout. In contrast, earlier chips such as the Z-80 were hand-drawn, with each transistor and wire carefully shaped to use the minimum space possible. The diagram below shows a small part of the Z-80 processor layout, showing the extremely irregular but dense arrangement of the chip. The transistors are not arranged in rows as in the ARM1 above, but fit together to use all the available space. <p> <a href="//static.righto.com/images/ARM1/z80.png"><img alt="A detail of the Z-80 processor layout, showing the complex hand-drawn layout. Each transistor and wire is carefully shaped to minimize the chip's size." class="hilite" height="291" src="//static.righto.com/images/ARM1/z80-w250.png" title="A detail of the Z-80 processor layout, showing the complex hand-drawn layout. Each transistor and wire is carefully shaped to minimize the chip's size." width="250"> </a> <p> <div class="cite"> A detail of the Z-80 processor layout, showing the complex hand-drawn layout. Each transistor and wire is carefully shaped to minimize the chip's size. </div> <p> <h2>The multiplexer and decoders</h2> Selecting the desired condition out of the eight possibilities is the job of a circuit called the multiplexer. The multiplexer takes 8 inputs (the conditions) and 8 control signals (based on the instruction) and selects the desired condition. To the right of the multiplexer, 8 NAND gates generate the 8 control signals by decoding the three instruction bits. Each gate simply looks at three bit values and outputs a 0 if the bits select that condition. For instance, if the first two bits are 0 and the third is 1, the gate for condition 1 outputs a 0, selecting that condition in the multiplexer. The animation below shows the circuit as the instruction bits cycle through the eight conditions. You can see the activated condition moving downwards through the circuit. <p> <a href="http://imgur.com/Hsadokk"><img alt="Animation of the multiplexer in the ARM1 condition code evaluation circuit." height="622" src="http://i.imgur.com/Hsadokk.gif" title="Animation of the multiplexer in the ARM1 condition code evaluation circuit." width="300"> </a> <div> Animation of the multiplexer in the ARM1 condition code evaluation circuit. </div> <p> While a multiplexer can be built from standard logic gates, the ARM1 multiplexer is built from a different type of circuitry called <a href="https://en.wikipedia.org/wiki/Transmission_gate">transmission gates</a> (which the ARM1 also uses in its <a href="http://www.righto.com/2016/01/counting-bits-in-hardware-reverse.html">bit counter</a>). A multiplexer built from transmission gates is more compact and faster than one built from standard logic (NAND gates). One feature of CMOS is that by combining an NMOS transistor and a PMOS transistor in parallel, a transmission gate switch can be built. Feeding 1 into the NMOS gate and 0 into the PMOS gate turns on both transistors and they pass their input through. With the opposite gate values, both transistors turn off and the switch opens. The multiplexer is built from 8 of these CMOS switches. Each condition input feeds into one switch, and the switch outputs are connected together. One switch is turned on at a time, selecting the corresponding input as the output value. <p> The diagram below shows the schematic of the multiplexer as well as its physical layout on the chip. Only the first three segments of the eight are shown; the remainder are similar. Each input is connected to two transistors forming a CMOS switch. Because the NMOS and PMOS gates require opposite signals, the multiplexer has an inverter for each control signal. Each inverter also consists of two transistors, but wired differently from the switch. <p> <a href="//static.righto.com/images/ARM1/mux_schematic.png"><img alt="Schematic of the multiplexer inside the ARM1 processor's condition code evaluation circuit." height="441" src="//static.righto.com/images/ARM1/mux_schematic-w450.png" style="vertical-align: top; margin-top: 20px" title="Schematic of the multiplexer inside the ARM1 processor's condition code evaluation circuit." width="450"> </a> <a href="//static.righto.com/images/ARM1/mux_diagram.png"><img alt="Diagram of the multiplexer inside the ARM1 processor's condition code evaluation circuit." height="406" src="//static.righto.com/images/ARM1/mux_diagram-w600.png" style="vertical-align:top" title="Diagram of the multiplexer inside the ARM1 processor's condition code evaluation circuit." width="600"> </a> <p> <div class="cite"> Schematic and diagram of the multiplexer inside the ARM1 processor's condition code evaluation circuit. </div> <p> Working together the decode circuit, inverters, and CMOS switches form the multiplexer that selects the desired condition from the eight choices. The logic described earlier allows this condition to be flipped, for a total of 16 possible conditions. <h2>Conclusion</h2> One unusual feature of the ARM instruction set is that every instruction has a condition associated with it and is only executed if the condition is true. The ARM1 chip is simple enough that the condition circuitry on the chip can be examined and understood at the transistor and gate level. Now that you've seen the internals of the condition logic, you can use the <a href="http://visual6502.org/sim/varm/armgl.html">Visual ARM1 simulator</a> to see the circuit in action. While the ARM1 may seem like a historical artifact of the 1980s, ARM processors power most smartphones, so there's probably a similar circuit controlling your phone right now. <p> Thanks to the Visual 6502 team for providing the simulator and ARM1 chip layout data. If you're interested in ARM1 internals, see <a href="http://www.righto.com/search/label/arm">my full set of ARM posts</a> and <a href="http://daveshacks.blogspot.co.uk/2015/12/inside-alu-of-armv1-first-arm.html">Dave Mugridge's series of posts</a>. <div style='clear: both;'></div> </div> <div class='post-footer'> <div class='post-footer-line post-footer-line-1'><span class='post-comment-link'> <a class='comment-link' href='https://www.blogger.com/comment/fullpage/post/6264947694886887540/9190561993947712649' onclick=''> No comments: </a> </span> <span class='post-icons'> <span class='item-action'> <a href='https://www.blogger.com/email-post/6264947694886887540/9190561993947712649' title='Email Post'> <img alt='' class='icon-action' height='13' src='http://img1.blogblog.com/img/icon18_email.gif' width='18'/> </a> </span> <span class='item-control blog-admin pid-1138732533'> <a href='https://www.blogger.com/post-edit.g?blogID=6264947694886887540&postID=9190561993947712649&from=pencil' title='Edit Post'> <img alt='' class='icon-action' height='18' src='https://resources.blogblog.com/img/icon18_edit_allbkg.gif' width='18'/> </a> </span> </span> <span class='post-backlinks post-comment-link'> </span> <div class='post-share-buttons goog-inline-block'> <a class='goog-inline-block share-button sb-email' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=9190561993947712649&target=email' target='_blank' title='Email This'><span class='share-button-link-text'>Email This</span></a><a class='goog-inline-block share-button sb-blog' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=9190561993947712649&target=blog' onclick='window.open(this.href, "_blank", "height=270,width=475"); return false;' target='_blank' title='BlogThis!'><span class='share-button-link-text'>BlogThis!</span></a><a class='goog-inline-block share-button sb-twitter' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=9190561993947712649&target=twitter' target='_blank' title='Share to X'><span class='share-button-link-text'>Share to X</span></a><a class='goog-inline-block share-button sb-facebook' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=9190561993947712649&target=facebook' onclick='window.open(this.href, "_blank", "height=430,width=640"); return false;' target='_blank' title='Share to Facebook'><span class='share-button-link-text'>Share to Facebook</span></a><a class='goog-inline-block share-button sb-pinterest' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=9190561993947712649&target=pinterest' target='_blank' title='Share to Pinterest'><span class='share-button-link-text'>Share to Pinterest</span></a> </div> </div> <div class='post-footer-line post-footer-line-2'><span class='post-labels'> Labels: <a href='http://www.righto.com/search/label/arm' rel='tag'>arm</a>, <a href='http://www.righto.com/search/label/reverse-engineering' rel='tag'>reverse-engineering</a> </span> </div> <div class='post-footer-line post-footer-line-3'></div> </div> </div> </div> </div></div> <div class="date-outer"> <div class="date-posts"> <div class='post-outer'> <div class='post hentry' itemprop='blogPost' itemscope='itemscope' itemtype='http://schema.org/BlogPosting'> <meta content='//static.righto.com/images/ARM1/full_encoder_labeled-w800.png' itemprop='image_url'/> <meta content='6264947694886887540' itemprop='blogId'/> <meta content='775448634558980905' itemprop='postId'/> <a name='775448634558980905'></a> <h3 class='post-title entry-title' itemprop='name'> <a href='http://www.righto.com/2016/01/more-arm1-processor-reverse-engineering.html'>More ARM1 processor reverse engineering: the priority encoder</a> </h3> <div class='post-header'> <div class='post-header-line-1'></div> </div> <div class='post-body entry-content' id='post-body-775448634558980905' itemprop='description articleBody'> <style type="text/css"> img.hilite {z-index:10;} </style> <meta property="og:image" content="//static.righto.com/images/ARM1/full_encoder_labeled-w400.png" /> In this article, I reverse-engineer the priority encoder in the ARM1 processor. By examining the chip layout provided by the <a href="http://blog.visual6502.org/2015/11/the-visual-arm1.html">Visual ARM1 project</a>, I have determined how this circuit works and created a schematic. <p> The ARM1 chip is the ancestor of the extremely popular ARM processors used in most smart phones. The ARM1 is a good choice for reverse engineering since it was designed in 1985 and its simple RISC silicon circuits are easier to understand than modern processors. This article jumps into the chip details; if you want an overview of the ARM1 internals, start with my first article on <a href="http://www.righto.com/2015/12/reverse-engineering-arm1-ancestor-of.html">reverse engineering the ARM1</a>. <p> The priority encoder takes a 16-bit binary field, finds the bits that are set and outputs the 4-bit binary positions of these bits in sequence. For example, if the input field is 1000000000001011, successive outputs will be 0, 1, 3, and 15. (Bits are scanned starting with bit 0, the rightmost bit.) The priority encoder gets its name because it selects bits by priority (rightmost first) and encodes the result into binary. <p> The diagram below shows the layout of the priority encoder on the chip. It is implemented as 16 bit slices, one for each bit, arranged left to right ("backwards"). Slice 2 is highlighted in red; slices 5 through 13 have been cut out to make the image fit. The 16 input bits arrive through the data bus on the bottom and each bit enters a slice through one of the bit input lines (green). If the bit is currently the highest priority, the output encoder at the top of the slice generates the 4-bit binary value on the output bus. The pullups pull the output bus lines to the high state. Finally, the drivers amplify the output signals and send them to other parts of the chip.<a class="ref" href="#ref1">[1]</a> <p> <a href="//static.righto.com/images/ARM1/full_encoder_labeled.png"><img alt="The priority encoder circuit in the ARM1 consists of 16 slices, one for each bit. One slice is highlighted in red." class="hilite" height="448" src="//static.righto.com/images/ARM1/full_encoder_labeled-w800.png" title="The priority encoder circuit in the ARM1 consists of 16 slices, one for each bit. One slice is highlighted in red." width="800"> </a> <p> <div class="cite"> The priority encoder circuit in the ARM1 consists of 16 slices, one for each bit. One slice is highlighted in red. Slices 5 to 13 are omitted. </div> <p> The priority encoder is a key part of the ARM processor's block data transfer instructions, which efficiently copy data between on-chip registers and memory storage.<a class="ref" href="#ref2">[2]</a> These instructions can transfer any subset of ARM's 16 registers in a single instruction. The desired registers are specified by setting the corresponding bits in a 16-bit field in the instruction. The role of the priority encoder is to scan this field and determine which register to transfer during each step of the operation. <h2>Implementation of the priority encoder</h2> The schematic below shows one of the 16 slices in the priority encoder. The input bit from the bus, <i>bus_bit</i> enters at the bottom. The green bit select block determines if the bit is currently the high-priority bit. If so, <i>bit_selected</i> becomes 1. The output encoder (blue) puts the binary value associated with the selected bit onto the bus. Finally, the bit used latch (red) marks the bit as used, blocking it and allowing the next bit in sequence to be active in the next cycle. The two-phase clock signals <i>Φ1</i> and <i>Φ2</i> cause the priority encoder to move from bit to bit. <p> <a href="//static.righto.com/images/ARM1/schematic_labeled_cropped.png"><img alt="Schematic of the priority encoder in the ARM1 processor, showing one slice." class="hilite" height="680" src="//static.righto.com/images/ARM1/schematic_labeled_cropped-w800.png" title="Schematic of the priority encoder in the ARM1 processor, showing one slice." width="800"> </a> <p> <div class="cite"> Schematic of the priority encoder in the ARM1 processor, showing one slice. </div> <p> The bit selection logic (green) is fairly straightforward. The input <i>clear_to_left</i> is 1 if all the bits to the left are clear. If all the bits to the left are clear and the current bit is set, then this bit is selected by the priority encoder. This also blocks <i>clear_to_left</i> from being passed to the next slice. Otherwise, <i>clear_to_left</i> is passed along. Thus, as it passes through the circuit <i>clear_to_left</i> will be 1 until a bit is encountered, and then 0 from that point. If the final <i>clear_to_left</i> output is 1, then all bits are clear and encoding is done. The logic for <i>clear_to_right</i> is similar, allowing the highest-priority bit to be selected from the right instead. Normally the initial <i>clear_to_left</i> input is 0, and the initial <i>clear_to_right</i> bit is 1, enabling the left scan and disabling the right scan. <p> The bit used latch (red) keeps track of which bits have already been output. It is what allows the priority encoder to move from bit to bit each clock cycle. The two <a href="https://en.wikipedia.org/wiki/Transmission_gate">transmission gates</a> (indicated with the four-triangle symbol) are clocked alternately so the <i>bit_selected</i> signal will move through the circuit after two half-clocks. Two NAND gates are connected as an <a href="https://en.wikipedia.org/wiki/Flip-flop_(electronics)#SR_NAND_latch">SR latch</a> to store this signal. Once a slice has selected a bit, the latch remembers that the bit has been used and blocks <i>bus_bit</i> from flowing into the bit select circuit. This allows the next bit in sequence to be selected. The bit used circuit also has a <i>clear</i> signal that resets the latch for a new instruction. <p> The bus pullup circuit (purple) and the output encoder (blue) work together to output the binary value corresponding to the selected bit. They use <a href="https://en.wikipedia.org/wiki/Dynamic_logic_(digital_electronics)">dynamic logic</a> rather than standard gates to reduce the circuit size. This logic depends on the clock and the capacitance of the output bus lines to generate the right values. In phase 2 of the clock, the bus pullup transistors pull the output bus lines high. Then, in phase 1, the output encoder in the active slice pulls the appropriate lines low so the bus will have the correct value. The schematic above shows the encoder for slice 6: the transistors attached to lines 8 and 1 pull them low, leaving 4 and 2 high; the resulting binary 0110 is 6. One set of pullup transistors supports the whole priority encoder, while each slice has its own output encoder transistors. <p> The output bus lines pass through drivers to boost the current; the signal on the output bus is relatively weak since it is generated by dynamic logic. The output flows to the register select circuit to select the appropriate register for the data transfer. See Dave Mugridge's article on <a href="https://daveshacks.blogspot.com/2015/12/inside-armv1-register-bank-register.html">ARM1 register selection</a> for details on how registers are selected. <p> <h2>Discussion</h2> The block data move transfer instructions in the ARM1 require two special functional units: the priority encoder and the bit counter (which I reverse-engineered <a href="http://www.righto.com/2016/01/counting-bits-in-hardware-reverse.html">earlier</a>). These two circuits are highlighted in red in the ARM1 die photo below. Supporting block data transfers added significant complexity to the chip (about 3% by area), but the chip designers felt the performance gain from block transfers was worth it. <p> <a href="//static.righto.com/images/ARM1/chip_labeled_bit.png"><img alt="The ARM1 processor chip with major functional groups labeled. The bit counter and priority encoder used for the LDM/STM instructions are highlighted in red. These take up about 3% of the chip's area." class="hilite" height="554" src="//static.righto.com/images/ARM1/chip_labeled_bit-w500.png" title="The ARM1 processor chip with major functional groups labeled. The bit counter and priority encoder used for the LDM/STM instructions are highlighted in red. These take up about 3% of the chip's area." width="500"> </a> <p> <div class="cite"> The ARM1 processor chip with major functional groups labeled. The bit counter and priority encoder used for the LDM/STM instructions are highlighted in red. These take up about 3% of the chip's area.ARM1 die photo courtesy of <a href="http://www.computerhistory.org/revolution/digital-logic/12/286">Computer History Museum</a>. </div> <p> One interesting thing about the priority encoder's design is alternating slices have inverted logic: NAND gates become NOR gates and vice versa. The reason is to avoid inverters between stages. You'll note on the schematic that the <i>clear_to_right</i> and <i>clear_to_left</i> outputs are inverted. The obvious design would add inverters to fix the polarity. However, this would add an extra gate delay in each stage, which is significant when the signal has to ripple through 16 stages. By "flipping" alternate stages, this delay is avoided. The trick of alternating stages to avoid inverters is used in other chips. For example, the <a href="http://www.righto.com/2013/03/register-file-8085.html">8085's incrementer</a> and the 6502's ALU. <p> One surprise with the ARM1 priority encoder is it supports both low-to-high priority and high-to-low priority, but high-to-low priority is disabled and not used. That is, the rightmost <i>clear_to_right</i> is wired to 1, so the rightmost bit circuitry will never be active. The explanation for this unused circuitry is interesting. <p> When using the block data operations to push and pull registers on the stack, you'd expect to push R0, R1, R2, etc and then pop in the reverse order R2, R1, R0.<a class="ref" href="#ref3">[3]</a> To handle this, the priority encoder needs to provide the registers in either order, and the address incrementer needs to increment or decrement addresses depending on whether you're pushing or popping, and the chip includes this circuitry. However, there's a flaw that wasn't discovered until midway through the design of the ARM1. Register 15 (the program counter) must always be updated last, or else you can't recover from a fault during the instruction because you've lost the address.<a class="ref" href="#ref4">[4]</a> <p> The solution used in the ARM1 is to always read or write registers starting with the lowest register and the lowest address. In other words, to pop R2, R1, R0, the ARM1 jumps into the middle of the stack and pops R0, R1, R2 in the reverse order. It sounds crazy but it works. (The bit counter determines how many words to shift the starting position.) The consequence of this redesign was that the circuitry to decrement addresses and priority encode in reverse order is never used. This circuity was removed from the ARM2. <h2>Conclusion</h2> The priority encoder is a large functional unit in the ARM1 chip, used for the block data transfer instructions. By looking at one of the 16 slices in the encoder, the circuit can be reverse-engineered and understood. While largely built from standard logic gates, the circuit also uses transmission gates and dynamic logic for efficiency. One surprise is the priority encoder contains unused logic allowing it to work in either direction. This wasted circuitry is left over from a design change during the development of the ARM1. <p> Now that you've seen the internals of the priority encoder, you can use the <a href="http://visual6502.org/sim/varm/armgl.html">Visual ARM1 simulator</a> to see the circuit in action.<a class="ref" href="#ref5">[5]</a> <h2>Notes and references</h2> <a name="ref1"></a>[1] The drivers also invert and buffer clock signals that are used by the priority encoder. <p> <a name="ref2"></a>[2] ARM's block data transfer instructions are called STM (Store Multiple) and LDM (Load Multiple), storing and loading multiple registers with one instruction. These instructions can be used for copying data or for stack push/pop, saving registers in a subroutine call or interrupt handler. Note that these instructions are not implemented in microcode, but in hardware that steps through the registers and memory. These instruction are explained in detail on the <a href="http://www.heyrick.co.uk/armwiki/STM">ARMwiki</a>. <p> <a name="ref3"></a>[3] The block data transfer instructions work for general register copying, not just pushing and popping to a stack. It's simpler to explain the instructions in terms of a stack, though. <p> <a name="ref4"></a>[4] If an instruction encounters a memory fault (e.g. a virtual memory page is missing), you want to take an interrupt, fix the problem (e.g. load in the page), and then restart the instruction. However, if you update registers high-to-low, R15 (the program counter) will be updated first. If a fault happens during the instruction, the address of the instruction (R15) is lost, and restarting the instruction is a problem. <p> One solution would be to push registers high-to-low and pop low-to-high so R15 is always updated last. Apparently the ARM designers wanted the low register at the low address, even if the stack grows upwards, so popping R15 least wouldn't work. Another alternative is to have a "shadow" program counter that can restore the program counter during a fault. The ARM1 designers considered this alternative too complex. For details, see page 248 of "VLSI RISC Architecture and Organization", by Stephen Furber, one of the ARM1's designers. <p> <a name="ref5"></a>[5] Thanks to the Visual 6502 team for providing the simulator and ARM1 chip layout data. If you're interested in ARM1 internals, also see <a href="http://daveshacks.blogspot.co.uk/2015/12/inside-alu-of-armv1-first-arm.html">Dave Mugridge's series of posts</a>. <p> <div style='clear: both;'></div> </div> <div class='post-footer'> <div class='post-footer-line post-footer-line-1'><span class='post-comment-link'> <a class='comment-link' href='https://www.blogger.com/comment/fullpage/post/6264947694886887540/775448634558980905' onclick=''> No comments: </a> </span> <span class='post-icons'> <span class='item-action'> <a href='https://www.blogger.com/email-post/6264947694886887540/775448634558980905' title='Email Post'> <img alt='' class='icon-action' height='13' src='http://img1.blogblog.com/img/icon18_email.gif' width='18'/> </a> </span> <span class='item-control blog-admin pid-1138732533'> <a href='https://www.blogger.com/post-edit.g?blogID=6264947694886887540&postID=775448634558980905&from=pencil' title='Edit Post'> <img alt='' class='icon-action' height='18' src='https://resources.blogblog.com/img/icon18_edit_allbkg.gif' width='18'/> </a> </span> </span> <span class='post-backlinks post-comment-link'> </span> <div class='post-share-buttons goog-inline-block'> <a class='goog-inline-block share-button sb-email' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=775448634558980905&target=email' target='_blank' title='Email This'><span class='share-button-link-text'>Email This</span></a><a class='goog-inline-block share-button sb-blog' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=775448634558980905&target=blog' onclick='window.open(this.href, "_blank", "height=270,width=475"); return false;' target='_blank' title='BlogThis!'><span class='share-button-link-text'>BlogThis!</span></a><a class='goog-inline-block share-button sb-twitter' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=775448634558980905&target=twitter' target='_blank' title='Share to X'><span class='share-button-link-text'>Share to X</span></a><a class='goog-inline-block share-button sb-facebook' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=775448634558980905&target=facebook' onclick='window.open(this.href, "_blank", "height=430,width=640"); return false;' target='_blank' title='Share to Facebook'><span class='share-button-link-text'>Share to Facebook</span></a><a class='goog-inline-block share-button sb-pinterest' href='https://www.blogger.com/share-post.g?blogID=6264947694886887540&postID=775448634558980905&target=pinterest' target='_blank' title='Share to Pinterest'><span class='share-button-link-text'>Share to Pinterest</span></a> </div> </div> <div class='post-footer-line post-footer-line-2'><span class='post-labels'> Labels: <a href='http://www.righto.com/search/label/arm' rel='tag'>arm</a>, <a href='http://www.righto.com/search/label/reverse-engineering' rel='tag'>reverse-engineering</a> </span> </div> <div class='post-footer-line post-footer-line-3'></div> </div> </div> </div> </div></div> <div class="date-outer"> <div class="date-posts"> <div class='post-outer'> <div class='post hentry' itemprop='blogPost' itemscope='itemscope' itemtype='http://schema.org/BlogPosting'> <meta content='//static.righto.com/images/ARM1/chip_labeled_bit-w600.png' itemprop='image_url'/> <meta content='6264947694886887540' itemprop='blogId'/> <meta content='5240240538862637372' itemprop='postId'/> <a name='5240240538862637372'></a> <h3 class='post-title entry-title' itemprop='name'> <a href='http://www.righto.com/2016/01/counting-bits-in-hardware-reverse.html'>Counting bits in hardware: reverse engineering the silicon in the ARM1 processor</a> </h3> <div class='post-header'> <div class='post-header-line-1'></div> </div> <div class='post-body entry-content' id='post-body-5240240538862637372' itemprop='description articleBody'> <meta property="og:image" content="//static.righto.com/images/ARM1/adder.png" /> How can you count bits in hardware? In this article, I reverse-engineer the circuit used by the ARM1 processor to count the number of set bits in a 16-bit field, showing how individual transistors form multiplexers, which are combined into adders, and finally form the bit counter. The ARM1 is the ancestor of the processor in most cell phones, so you may have a descendent of this circuit in your pocket. <p> ARM is now the world's most popular instruction set but it has humble beginnings. The original ARM1 processor was designed in 1985 by a UK company called Acorn Computer for the <a href="https://en.wikipedia.org/wiki/BBC_Micro">BBC Micro</a> home/educational computer. A few years later Apple needed a low-power, high-performance processor for its ill-fated <a href="https://en.wikipedia.org/wiki/Apple_Newton">Newton</a> handheld system and chose ARM.<a class="ref" href="#ref1">[1]</a> In 1990, Acorn Computers, Apple, and chip manufacturer VLSI Technology formed the company <a href="https://en.wikipedia.org/wiki/ARM_Holdings#Founding">Advanced RISC Machines</a> to continue ARM development. ARM became very popular for low power applications (such as phones) and now more than 50 billion ARM processors have been manufactured. <p> One way ARM processors increase performance is through block data transfer instructions, which efficiently copy data between on-chip registers and memory storage.<a class="ref" href="#ref2">[2]</a> These instructions can transfer any subset of ARM's 16 registers in a single instruction. The desired registers are specified by setting the corresponding bits in a 16-bit field in the instruction. To implement the block transfer instructions, the ARM requires two specialized circuits. The first circuit, the bit counter, counts the number of bits set in the register select field to determine how many registers are being transferred.<a class="ref" href="#ref3">[3]</a> The second circuit, the priority encoder, scans the register select field and finds the next set bit, indicating which register to load/store next. <p> <a href="//static.righto.com/images/ARM1/chip_labeled_bit.png"><img alt="The ARM1 processor chip with major functional groups labeled. The bit counter and priority encoder used for the LDM/STM instructions are highlighted in red. These take up about 3% of the chip's area." class="hilite" height="665" src="//static.righto.com/images/ARM1/chip_labeled_bit-w600.png" title="The ARM1 processor chip with major functional groups labeled. The bit counter and priority encoder used for the LDM/STM instructions are highlighted in red. These take up about 3% of the chip's area." width="600"> </a> <p> <div class="cite"> The ARM1 processor chip with major functional groups labeled. The bit counter and priority encoder used for the LDM/STM instructions are highlighted in red. These take up about 3% of the chip's area.ARM1 die photo courtesy of <a href="http://www.computerhistory.org/revolution/digital-logic/12/286">Computer History Museum</a>. </div> These two circuits are highlighted in red in the ARM1 die photo above. As you can see, the circuits take up a significant fraction of the chip (about 3%), but the chip designers felt the performance gain from block transfers was worth the increase in chip size and complexity. This article explains the bit counter, and I plan to describe the priority encoder later. <p> Zooming in on the bit counter reveals the circuit below. It looks like a jumble of lines, but by examining it carefully, you can get an understanding of what is going on. The remainder of the article explains how a special type of circuitry called pass transistor logic is used to build a multiplexer — a circuit that selects one of its two inputs. The multiplexers are used to form logic gates, which are then combined to form a full adder, which adds three bits. Finally, the adders are combined to create the bit counting circuit. If you're not familiar with digital logic or the ARM processor, you might want to start with my earlier article on <a href="http://www.righto.com/2015/12/reverse-engineering-arm1-ancestor-of.html">reverse-engineering the ARM1</a> for an overview. <p> <a href="//static.righto.com/images/ARM1/sim_cropped.png"><img alt="The bit counter circuit from the ARM1 processor chip. This circuit counts the number of registers selected by the LDM/STM instructions." class="hilite" height="435" src="//static.righto.com/images/ARM1/sim_cropped-w600.png" title="The bit counter circuit from the ARM1 processor chip. This circuit counts the number of registers selected by the LDM/STM instructions." width="600"> </a> <p> <div class="cite"> The bit counter circuit from the ARM1 processor chip. This circuit counts the number of registers selected by the LDM/STM instructions. </div> <p> <h2>Pass transistors and transmission gates</h2> The bit counter is built from a type of circuitry called <a href="https://en.wikipedia.org/wiki/Pass_transistor_logic">pass transistor logic</a>. Unlike normal logic gates, pass transistor logic switches the inputs themselves to pass an input directly to the output. Pass transistors are used because sums (i.e. XORs) are inconvenient to generate with standard logic and can be generated more efficiently with pass transistor logic. <p> The ARM1 chip, like most modern chips, is built from a technology called CMOS. The <i>C</i> in <i>CMOS</i> stands for <i>complementary</i> because CMOS circuits are built from two complementary types of transistors. NMOS transistors switch on when the control signal on the gate is high, and can pull the output low. PMOS transistors are opposite; they switch on when the gate's control signal is low, and can pull the output high. Combining an NMOS transistor and a PMOS transistor in parallel forms a transmission gate. If both transistors are on, the input will be passed to the output whether it is low or high. If both transistors are off, the input is blocked. Thus, the circuit acts as a switch that can either pass the input through to the output or block it. <p> The diagram below shows two transistors (circled) connected to form a transmission gate. The upper one is NMOS and the lower one is PMOS. On the right is the symbol for a transmission gate. Note that because the transistors are complementary, they require opposite enable signals. <p> <a href="//static.righto.com/images/ARM1/passtrans.png"><img alt="Schematic symbols for a CMOS pass gate. On the left, the two transistors are shown. On the right is the equivalent pass gate symbol. The circles around the transistors are to make the transistors clear and are not part of the symbol." class="hilite" height="267" src="//static.righto.com/images/ARM1/passtrans-w500.png" title="Schematic symbols for a CMOS pass gate. On the left, the two transistors are shown. On the right is the equivalent pass gate symbol. The circles around the transistors are to make the transistors clear and are not part of the symbol." width="500"> </a> <p> <div class="cite"> Schematic symbols for a CMOS transmission gate. On the left, the two transistors are shown. On the right is the equivalent transmission gate symbol. The circles around the transistors are to make the transistors clear and are not part of the symbol. </div> <h2>The multiplexer</h2> Next, we can look at how transmission gates are used in the chip. The diagram below shows a multiplexer as it appears in the <a href="http://visual6502.org/sim/varm/armgl.html">Visual ARM1</a> simulator. The ARM1 chip is constructed from five layers, which appear as different colors in the simulator. (The layers are harder to distinguish in the real chip.) The bottom layer is the silicon that makes up the transistors of the chip. During manufacturing, regions of the silicon are modified (doped) by applying different impurities. Silicon can be doped positive to form a PMOS transistor (blue) or doped negative for an NMOS transistor (red). Undoped silicon (black) is basically an insulator. Polysilicon wires (green) are deposited on top of the silicon. When polysilicon crosses doped silicon, it forms the gate of a transistor (yellow). Finally, two layers of metal<a class="ref" href="#ref4">[4]</a> (gray) are on top of the polysilicon and provide wiring. Black squares are contacts that form connections between the different layers. <p> <a href="//static.righto.com/images/ARM1/mux_layers.png"><img alt="A pass-gate multiplexer in the ARM1 processor, showing how different layers are displayed in the Visual ARM1 simulator." class="hilite" height="383" src="//static.righto.com/images/ARM1/mux_layers-w350.png" title="A pass-gate multiplexer in the ARM1 processor, showing how different layers are displayed in the Visual ARM1 simulator." width="350"> </a> <p> <div class="cite"> A pass-gate multiplexer in the ARM1 processor, showing how different layers are displayed in the Visual ARM1 simulator. </div> <p> Each multiplexer consists of four transistors: two NMOS (red) and two PMOS (blue); the gate appears in yellow between the two sides of the transistor. These form two transmission gates allowing either the left input or the right input to be connected to the output. If "Select left" is high and "Select right" is low, the two transistors on the left turn on, connecting the left input to the output. Conversely, if "Select right" is high and "Select left" is low, the two transistors on the right turn on, connecting the right input to the output. <p> <a href="//static.righto.com/images/ARM1/mux_circuit.png"><img alt="A pass-gate multiplexer circuit in the ARM1 processor. The left shows the physical construction of the circuit, as it appears in the Visual ARM1 simulator. The corresponding schematic is on the right. If 'Select left' is high, the two transistors on the left will be active, connecting the left input to the output. If 'Select right' is high, the two transistors on the right will connect the right input to the output." class="hilite" height="308" src="//static.righto.com/images/ARM1/mux_circuit-w600.png" title="A pass-gate multiplexer circuit in the ARM1 processor. The left shows the physical construction of the circuit, as it appears in the Visual ARM1 simulator. The corresponding schematic is on the right. If 'Select left' is high, the two transistors on the left will be active, connecting the left input to the output. If 'Select right' is high, the two transistors on the right will connect the right input to the output." width="600"> </a> <p> <div class="cite"> A pass-gate multiplexer circuit in the ARM1 processor. The left shows the physical construction of the circuit, as it appears in the Visual ARM1 simulator. The corresponding schematic is on the right. If 'Select left' is high, the two transistors on the left will be active, connecting the left input to the output. If 'Select right' is high, the two transistors on the right will connect the right input to the output. </div> The symbol for a multiplexer is shown below. If the select line is 1, the input labeled 1 is selected for the output, and conversely for 0. Note that the inverted select line is also required, but isn't explicitly shown in the symbol. This is important, since the inverted select must be generated in the circuit. <p> <a href="//static.righto.com/images/ARM1/mux_symbol.png"><img alt="Symbol for a two-input multiplexer. Based on the select line, one of the inputs goes to the output." class="hilite" height="185" src="//static.righto.com/images/ARM1/mux_symbol-w250.png" title="Symbol for a two-input multiplexer. Based on the select line, one of the inputs goes to the output." width="250"> </a> <p> <div class="cite"> Symbol for a two-input multiplexer. Based on the select line, one of the inputs goes to the output. </div> <h2>Building a full adder from multiplexers</h2> A <a href"=https://en.wikipedia.org/wiki/Adder_(electronics)#Full_adder">full adder</a> is a digital circuit to add two bits along with a carry in, generating a sum output and a carry output. (If you think of the output as a binary sum, the sum output is the low bit and the carry output is the high bit.) Equivalently, the full adder can be thought of as adding three input bits. The full adder is the building block of the ARM1's bit counting circuit. <p> In the ARM1, a full adder is built from multiplexers, along with a few inverters. The diagram below shows how a full adder appears in the simulator. Counting the yellow rectangles, you can see that there are 29 transistors in the circuit. The transistors are connected by metal wires (gray) and polysilicon wires (green). While the layout may appear chaotic, the transistors are arranged in an orderly way: a row of PMOS transistors (blue), two rows of NMOS transistors (red), and a second row of PMOS transistors (blue).<a class="ref" href="#ref5">[5]</a> <p> <a href="//static.righto.com/images/ARM1/1-adder.png"><img alt="A full-adder circuit in the ARM1 processor, as it appears in the Visual ARM1 simulator." class="hilite" height="416" src="//static.righto.com/images/ARM1/adder-w400.png" title="A full-adder circuit in the ARM1 processor, as it appears in the Visual ARM1 simulator." width="400"> </a> <p> <div class="cite"> A full-adder circuit in the ARM1 processor, as it appears in the Visual ARM1 simulator. </div> Arranging components for high density wasn't important to the ARM1 designers, so they built circuits from standard blocks (or cells) using computerized design tools, resulting in the regular layout seen above. On the other hand, the designers of earlier processors such as the 6502 and Z-80 tried to minimize the chip size as much as possible, so the chip layout was highly optimized. Each transistor and wire was hand-drawn to fit as tightly as possible, almost like a jigsaw puzzle. The image below shows part of the Z-80 chip, demonstrating the tightly-packed, irregular layout. The difference between hand-draw, optimized layout and computer-generated layout is striking. <p> <a href="//static.righto.com/images/ARM1/z80.png"><img alt="A detail of the Z-80 processor layout, showing the complex hand-drawn layout. Each transistor and wire is carefully shaped to minimize the chip's size." class="hilite" height="408" src="//static.righto.com/images/ARM1/z80-w350.png" title="A detail of the Z-80 processor layout, showing the complex hand-drawn layout. Each transistor and wire is carefully shaped to minimize the chip's size." width="350"> </a> <p> <div class="cite"> A detail of the Z-80 processor layout, showing the complex hand-drawn layout. Each transistor and wire is carefully shaped to minimize the chip's size. Z-80 data is from the Visual 6502 project. </div> <p> The schematic below shows how the full adder in the ARM1 is built from multiplexers. In the lower left, a multiplexer generates "A XOR B", which is the single-bit sum of A and B. If you try the combinations of A and B, you'll find that the output is 1 if exactly one of the inputs is 1, and otherwise 0. The next multiplexer reverses the A inputs and computes the complement of A XOR B.<a class="ref" href="#ref6">[6]</a> The third multiplexer implements a NAND gate: If B is 1 and A is 1, the output is 0.<a class="ref" href="#ref7">[7]</a> <p> <a href="//static.righto.com/images/ARM1/adder_schematic.png"><img alt="Schematic of a full-adder in the ARM1 processor, showing its construction from multiplexers. Inverters for A and B are not shown." class="hilite" height="426" src="//static.righto.com/images/ARM1/adder_schematic-w500.png" title="Schematic of a full-adder in the ARM1 processor, showing its construction from multiplexers. Inverters for A and B are not shown." width="500"> </a> <p> <div class="cite"> Schematic of a full-adder in the ARM1 processor, showing its construction from multiplexers. Inverters for A and B are not shown. </div> <p> The multiplexers in the upper half compute the sum and carry (i.e. bit 0 and bit 1 of the binary sum), as can be verified by trying the input combinations. You might wonder why inverters are used, rather than generating the desired outputs directly. The reason is to boost the signals, since the outputs of multiplexers are relatively weak. <p> The diagram below indicates the multiplexers and inverters<a class="ref" href="#ref6">[6]</a> that make up a full adder, with the components highlighted. Each multiplexer is built as described earlier, and they are arranged as in the schematic above. The multiplexers are connected together by polysilicon and metal wires. The three inputs are at the bottom and the two outputs are at the top. This adder is the main block used to build the bit counter, and the next section will show how adders are connected together. <p> <a href="//static.righto.com/images/ARM1/adder_labeled.png"><img alt="A full-adder circuit in the ARM1 processor, showing how it is built from pass-gate multiplexers and inverters." class="hilite" height="534" src="//static.righto.com/images/ARM1/adder_labeled-w600.png" title="A full-adder circuit in the ARM1 processor, showing how it is built from pass-gate multiplexers and inverters." width="600"> </a> <p> <div class="cite"> A full-adder circuit in the ARM1 processor, showing how it is built from pass-gate multiplexers and inverters. </div> <h2>Building the bit counter from adders</h2> The bit counter takes 16 bit inputs and generates a 4-bit count as output, using adders as building blocks. The flow chart below shows how it operates, with data flowing from top to bottom. Each box is an adder, with carry (C) and sum (S) outputs. Boxes are colored according to which bit of the sum they are computing: red for the 1's bit, green for the 2's bit, blue for the 4's bit and purple for the 8's bit. Each box passes its sum output down and passes its carry to the left. <p> Overall, the process is similar to long addition if you could just add three digits at a time. You compute partial sums, then add up those sums, and so forth until all the sums are added up. Then the carries need to be added up, along with the sums of those carries, and so forth. If there are carries from those digits, they need to be added up, until finally everything has been added. <p> The first step of counting the bits is to add each triple of bits with a full adder, generating a two bit count (0, 1, or 2). Inconveniently, since the sixteen input bits aren't divisible by 3, one bit is left over and is handled separately. Next, the five partial sums are added by more adders (red). As carries are generated, they also get added (green). Carries from the carries are also added (blue). In the final step, two-input half adders<a class="ref" href="#ref8">[8]</a> compute the sum output; these half adders are simpler than the three-input full adders.<a class="ref" href="#ref9">[9]</a> <p> <p> <a href="//static.righto.com/images/ARM1/sum_diagram.png"><img alt="The bit counter in the ARM1 processor is built from full-adders and half adders. Red corresponds to sum bit 0, green is bit 1, blue is bit 2, and purple is bit 3." class="hilite" height="669" src="//static.righto.com/images/ARM1/sum_diagram-w600.png" title="The bit counter in the ARM1 processor is built from full-adders and half adders. Red corresponds to sum bit 0, green is bit 1, blue is bit 2, and purple is bit 3." width="600"> </a> <p> <div class="cite"> The bit counter in the ARM1 processor is built from full-adders and half adders. Red corresponds to sum bit 0, green is bit 1, blue is bit 2, and purple is bit 3. To simplify the diagram, outputs from the first stage are indicated by letters rather than lines. </div> <p> The diagram below shows how the flow chart above is implemented on the chip to create the entire bit counter circuit. The adders are numbered to match the flow chart. The data bus is at the bottom, connected to the bit counter inputs by 16 polysilicon wires (green). Data flows generally upwards through the circuit, opposite to the flow chart. The five adders at the bottom add triples of input bits, and the remaining adders combine the sums and carries. The four half adders are connected to the output drivers in the upper right. The control circuit enables and disables the output drivers, so the bit count is output to the bus at the right times. <p> <a href="//static.righto.com/images/ARM1/summer_labeled.png"><img alt="The bit counter circuit in the ARM1 processor. The full-adders and half-adders are indicated with numbers. The bits enter and the bottom and the count is output at the upper right." class="hilite" height="438" src="//static.righto.com/images/ARM1/summer_labeled-w600.png" title="The bit counter circuit in the ARM1 processor. The full-adders and half-adders are indicated with numbers. The bits enter and the bottom and the count is output at the upper right." width="600"> </a> <p> <div class="cite"> The bit counter circuit in the ARM1 processor. The full-adders and half-adders are indicated with numbers. The bits enter and the bottom and the count is output at the upper right. </div> <h2>Conclusion</h2> Well, it's been quite a journey from individual transistors to the bit counter, a complex functional block in a real processor. Hopefully this article has taken some of the mystery out of how circuits in a processor are constructed. Now you can try out the <a href="http://visual6502.org/sim/varm/armgl.html">Visual ARM1 simulator</a> and take a look at this circuit in action.<a class="ref" href="#ref10">[10]</a> <h2>Notes and references</h2> <a name="ref1"></a>[1] An interesting <a href="https://queue.acm.org/detail.cfm?id=1716385">interview with Steve Furber</a>, co-designer of the ARM1, explains how ARM achieved low power consumption. Acorn wanted to use a low-cost plastic package for the chip, but it could only handle 1 Watt. The designers didn't have good tools for estimating power consumption, so they were conservative in their design and the final power consumption was way below the target, just 1/10 Watt. In addition, ARM1 had a simple RISC (Reduced Instruction Set Computer) design, which also reduced power consumption: ARM1 had about 25,000 transistors compared to 275,000 in the 80386 which came out the same year. Thus, the low power consumption of ARM that led to its wild success in mobile applications was largely accidental. <p> <a name="ref2"></a>[2] ARM's block data transfer instructions are called STM (Store Multiple) and LDM (Load Multiple), storing and loading multiple registers with one instruction. These instructions don't exactly fit the RISC processor philosophy since they are fairly complex and perform many memory accesses, but the ARM designers took the pragmatic approach and implemented them for efficiency. These instructions can be used for copying data or for stack push/pop, saving registers in a subroutine call or interrupt handler. Note that these instructions are not implemented in microcode, but in hardware that steps through the registers and memory. <p> <a name="ref3"></a>[3] It's not obvious why a bit counter is required at all. You'd think the chip could just store registers until it's done, without knowing the total count. The unexpected answer is that LDM/STM always start with the lowest address working upwards. For example, if you're popping 4 registers off the stack with LDM, you'd expect to start at the top of the stack and work down. Instead, the ARM pulls registers out of the middle of the stack: it starts four words from the top, pops registers in reverse order going up, and then updates the stack pointer to the bottom. The results are exactly the same as popping from the top, just the memory accesses are in the reverse order. (The STM instruction is explained in detail on the <a href="http://www.heyrick.co.uk/armwiki/STM">ARMwiki</a>.) Thus, the bit counter is needed to figure out how far down to jump in memory at the start of the instruction. <p> That raises the question of why would memory accesses always go low to high, even when that seems backwards. The explanation is that you want to update register 15 (the program counter) last, so if there's a fault during the instruction you haven't clobbered the instruction address and can restart. This problem was discovered partway through the ARM1 design, causing the designers to implement the new strategy that block transfers always go from lowest register to highest register and lowest address to highest address. The bit counter was added to support this. Some remnants of the earlier, simpler design are visible in the ARM1. Specifically, the priority encoder can operate either direction, but high-to-low is never used. In addition, the address incrementer can both increment and decrement addresses, but decrement is never used. The unused circuitry was removed from the later ARM2. <p> <a name="ref4"></a>[4] At the time, having two layers of metal in the chip instead of one was a risky technology. However, the ARM1 designers wanted the convenience of two layers, which made routing the chip much simpler. <p> <a name="ref5"></a>[5] A few other things to point out in the multiplexer layout. Note that the second input to each multiplexer matches the first input to the next multiplexer. This lets neighboring multiplexers share inputs, so they can be packed together more closely. Another thing of interest is the transistor sizes. The PMOS transistors are about twice the size of the NMOS transistors in order to provide the same current. The reason is that electrons carry the charge around in NMOS transistors, while "holes" carry the charge in PMOS transistor, and electrons move faster, providing more current (<a href="https://www.ece.umd.edu/class/enee302h.F2005/enee302h-week5.pdf">details</a>). Finally, the transistors in the upper right are larger. These transistor drive the outputs from the multiplexer, so they must provide more current. <p> <a name="ref6"></a>[6] You might wonder why the circuit computes the complement of A XOR B when it isn't used in the schematic. The reason is the multiplexer uses both the select input and the complement of the select input. Thus, the complement is used; it just isn't explicitly shown in the schematic. Likewise an inverter complements B, so it is available for the select lines. <p> <a name="ref7"></a>[7] It is very unusual to implement a NAND gate with a multiplexer. Normally CMOS circuits implement a NAND gate with a standard four transistor circuit. But since the circuit already had multiplexers, adding an additional one was more efficient than the standard NAND gate. <p> <a name="ref8"></a>[8] The half adder is built from standard gates, rather than multiplexers, as shown in the schematic below. The half adder's behavior is different from a standard half adder: it computes A+B+1 instead of A+B. Thus, the output of the four half adders is equivalent to adding binary 1111 to the sum, equivalent to subtracting 1. The output drivers invert this, so the output on the bus is the twos complement of the sum. The outputs are also shifted two bits on the bus, multiplying the value by 4 (since ARM registers are 4 bytes long). For example, if you pop 3 registers the stack will be decremented by 12 bytes. <p> <a href="//static.righto.com/images/ARM1/halfadder.png"><img alt="The half-adder circuit from the ARM1 processor's bit counter. The outputs from this half-adder are different from normal, as it is used to generate a twos-complement negative output." class="hilite" height="311" src="//static.righto.com/images/ARM1/halfadder-w250.png" title="The half-adder circuit from the ARM1 processor's bit counter. The outputs from this half-adder are different from normal, as it is used to generate a twos-complement negative output." width="250"> </a> <p> <div class="cite"> The half-adder circuit from the ARM1 processor's bit counter. The outputs from this half-adder are different from normal, as it is used to generate a twos-complement negative output. </div> <p> <a name="ref9"></a>[9] The circuit complexity of the bit counter is interesting. To sum 16 bits requires 15 adders. In general, summing N bits will require N-1 adders (for N a power of 2). Note that each adder takes 3 lines down to 2, so reducing N lines down to 1output requires N-1 adders. (There are 4 outputs, not 1, but the half adders bump the total back to N-1). The number of adders for each output bit is a power of 2: 1 purple, 2 blue, 4 green, 8 red. Larger sum circuits could be created by combining two smaller ones. For example, two 16-bit counter circuits could be combined to create a 32-bit counter circuit by adding four more full adders to add the results from each half, before the final half-adder layer. The circuit used in the ARM1 isn't quite the recursive design, pushing more adders to the first layer. An important part of the design is to minimize propagation delay; in the ARM1 design, signals go through 6 adders in worst case, slightly better than the purely recursive design. <p> <a name="ref10"></a>[10] Thanks to the Visual 6502 team for providing the simulator and ARM1 chip layout data. 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class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2024/01/'> January </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/'> 2023 </a> <span class='post-count' dir='ltr'>(35)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/12/'> December </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/11/'> November </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/10/'> October </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/09/'> September </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/08/'> August </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/07/'> July </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/05/'> May </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/04/'> April </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/03/'> March </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/02/'> February </a> <span class='post-count' dir='ltr'>(5)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2023/01/'> January </a> <span class='post-count' dir='ltr'>(8)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/'> 2022 </a> <span class='post-count' dir='ltr'>(18)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/11/'> November </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/08/'> August </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/07/'> July </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/06/'> June </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/05/'> May </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/04/'> April </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/03/'> March </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/02/'> February </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2022/01/'> January </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/'> 2021 </a> <span class='post-count' dir='ltr'>(26)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/12/'> December </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/11/'> November </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/09/'> September </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/08/'> August </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/07/'> July </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/06/'> June </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/05/'> May </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/04/'> April </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/03/'> March </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/02/'> February </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2021/01/'> January </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/'> 2020 </a> <span class='post-count' dir='ltr'>(33)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/12/'> December </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/11/'> November </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/10/'> October </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/09/'> September </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/08/'> August </a> <span class='post-count' dir='ltr'>(5)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/07/'> July </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/06/'> June </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/05/'> May </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/04/'> April </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/03/'> March </a> <span class='post-count' dir='ltr'>(5)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2020/01/'> January </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/'> 2019 </a> <span class='post-count' dir='ltr'>(18)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/11/'> November </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/10/'> October </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/09/'> September </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/08/'> August </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/07/'> July </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/04/'> April </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/02/'> February </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2019/01/'> January </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/'> 2018 </a> <span class='post-count' dir='ltr'>(17)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/12/'> December </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/09/'> September </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/08/'> August </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/06/'> June </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/05/'> May </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/04/'> April </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/03/'> March </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/02/'> February </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2018/01/'> January </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/'> 2017 </a> <span class='post-count' dir='ltr'>(21)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/12/'> December </a> <span class='post-count' dir='ltr'>(5)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/11/'> November </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/10/'> October </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/08/'> August </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/07/'> July </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a 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class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2017/01/'> January </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate expanded'> <a class='toggle' href='javascript:void(0)'> <span class='zippy toggle-open'> ▼  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/'> 2016 </a> <span class='post-count' dir='ltr'>(34)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/12/'> December </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/10/'> October </a> <span class='post-count' dir='ltr'>(5)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/09/'> September </a> <span class='post-count' dir='ltr'>(8)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/08/'> August </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/07/'> July </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/06/'> June </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/05/'> May </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/04/'> April </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/03/'> March </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/02/'> February </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate expanded'> <a class='toggle' href='javascript:void(0)'> <span class='zippy toggle-open'> ▼  </span> </a> <a class='post-count-link' href='http://www.righto.com/2016/01/'> January </a> <span class='post-count' dir='ltr'>(3)</span> <ul class='posts'> <li><a href='http://www.righto.com/2016/01/conditional-instructions-in-arm1.html'>Conditional instructions in the ARM1 processor, re...</a></li> <li><a href='http://www.righto.com/2016/01/more-arm1-processor-reverse-engineering.html'>More ARM1 processor reverse engineering: the prior...</a></li> <li><a href='http://www.righto.com/2016/01/counting-bits-in-hardware-reverse.html'>Counting bits in hardware: reverse engineering the...</a></li> </ul> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2015/'> 2015 </a> <span class='post-count' dir='ltr'>(12)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2015/12/'> December </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2015/11/'> November </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2015/10/'> October </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2015/08/'> August </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2015/05/'> May </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2015/03/'> March </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2015/02/'> February </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2014/'> 2014 </a> <span class='post-count' dir='ltr'>(13)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2014/12/'> December </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2014/10/'> October </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2014/09/'> September </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2014/05/'> May </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2014/03/'> March </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2014/02/'> February </a> <span class='post-count' dir='ltr'>(5)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/'> 2013 </a> <span class='post-count' dir='ltr'>(24)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/11/'> November </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/09/'> September </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/08/'> August </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/07/'> July </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/06/'> June </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/04/'> April </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/03/'> March </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/02/'> February </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2013/01/'> January </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2012/'> 2012 </a> <span class='post-count' dir='ltr'>(10)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2012/12/'> December </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2012/11/'> November </a> <span class='post-count' dir='ltr'>(5)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2012/10/'> October </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2012/05/'> May </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2012/03/'> March </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2012/02/'> February </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2011/'> 2011 </a> <span class='post-count' dir='ltr'>(11)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2011/12/'> December </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2011/07/'> July </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2011/05/'> May </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2011/04/'> April </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2011/03/'> March </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2011/02/'> February </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/'> 2010 </a> <span class='post-count' dir='ltr'>(22)</span> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/12/'> December </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/11/'> November </a> <span class='post-count' dir='ltr'>(4)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/10/'> October </a> <span class='post-count' dir='ltr'>(3)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/08/'> August </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/06/'> June </a> <span class='post-count' dir='ltr'>(1)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a class='toggle' href='javascript:void(0)'> <span class='zippy'> ►  </span> </a> <a class='post-count-link' href='http://www.righto.com/2010/05/'> May </a> <span class='post-count' dir='ltr'>(2)</span> </li> </ul> <ul class='hierarchy'> <li class='archivedate collapsed'> <a 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