Recursion Theory and Recursive Functions

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relative w-full h-full object-cover" src="https://media.licdn.com/dms/image/v2/D5612AQHmEjzSPK73LA/article-cover_image-shrink_720_1280/article-cover_image-shrink_720_1280/0/1703378366254?e=2147483647&v=beta&t=7t35H6nhecAQtSpxcltns6hnpFdLAkNpFOKXCinyxF8" fetchpriority="high" data-embed-id="cover-image" alt="Recursion Theory and Recursive Functions"> </div> </div>  </figure> <header> <section class="core-section-container relative my-3"> <h1 class="pulse-title text-display-md text-color-text lg:text-display-lg font-bold w-11/12">Recursion Theory and Recursive Functions</h1> <div class="ellipsis-menu absolute right-0 top-0"> <div class="collapsible-dropdown flex items-center relative hyphens-auto"> <button class="ellipsis-menu__trigger collapsible-dropdown__button btn-md btn-tertiary cursor-pointer !py-[6px] !px-1 flex items-center rounded-[50%] " aria-expanded="false" aria-label="Open menu" data-tracking-control-name="article-ssr-frontend-pulse_ellipsis-menu-trigger" tabindex="0"> <icon 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class="base-card__full-link absolute top-0 right-0 bottom-0 left-0 p-0 z-[2]" href="https://www.linkedin.com/in/sabonchi" data-tracking-control-name="article-ssr-frontend-pulse_publisher-author-card" data-tracking-will-navigate> <span class="sr-only"> Hasanain Alsabonchi </span> </a> <img class="inline-block relative rounded-[50%] w-6 h-6 " data-delayed-url="https://media.licdn.com/dms/image/v2/D5603AQFsCDzWBqBSxQ/profile-displayphoto-shrink_400_400/profile-displayphoto-shrink_400_400/0/1715275569997?e=2147483647&v=beta&t=PIp7t8FBMvhGJzRggkWkIZU_noERJENBCVb4jrSBHS0" data-ghost-classes="bg-color-entity-ghost-background" data-ghost-url="https://static.licdn.com/aero-v1/sc/h/9c8pery4andzj6ohjkjp54ma2" alt="Hasanain Alsabonchi"> <div class="base-main-card__info self-center ml-1 flex-1 relative break-words papabear:min-w-0 mamabear:min-w-0 babybear:w-full ">  <h3 class="base-main-card__title font-sans text-[18px] font-bold text-color-text overflow-hidden "> Hasanain Alsabonchi </h3> <h4 class="base-main-card__subtitle body-text text-color-text overflow-hidden "> Software Engineer | Previously SWE @ NASA | M.S. in Computer Science. </h4>  <div class="body-text text-color-text-low-emphasis base-main-card__metadata"> Published Dec 27, 2023 </div>  </div> <div class="base-main-card__ctas z-[3] self-center ml-3 babybear:ml-1 babybear:self-start"> <a class="base-main-card__cta btn-sm btn-secondary" href="https://www.linkedin.com/signup/cold-join?session_redirect=%2Fpulse%2Frecursion-theory-recursive-functions-hasanain-alsabonchi-eh9bc&trk=article-ssr-frontend-pulse_publisher-author-card" data-tracking-control-name="article-ssr-frontend-pulse_publisher-author-card" data-tracking-will-navigate> + Follow </a> </div> </div> </div> </section> </header> <div data-test-id="article-content-blocks">  <div class="article-main__content" data-test-id="publishing-text-block"> <h3><span class>December 2023, Hasanain Alsabonchi and Julien Chaffraix.</span></h3> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>Recursion theory is a subfield of Computer Science. Almost everything in programming has a mathematical foundation, including sets, functions, arrays, and maps. The mathematical definition of recursion involves defining an object (a function, sequence, or structure) in terms of itself or a part of itself. One such example is factorial. The function </span><span class="italic">f </span><span class>(</span><span class="italic">n</span><span class>−1) represents a subproblem of the original problem </span><span class="italic">f </span><span class>(</span><span class="italic">n</span><span class>), and </span><span class="italic">n</span><span class> - 1 is a step toward reaching the base case. The recursive definition of a recursive function consists of two parts:</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class> <ol> <li><span class>One or more base cases</span></li> <li><span class>The recursive step</span></li> </ol> </span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>A factorial of a number 'N' is the product of all integers between 1 and </span><span class="italic">N</span><span class>. For example, if '</span><span class="italic">N</span><span class>' equals 5, then the factorial of '</span><span class="italic">N</span><span class>' would be 5 </span><span class="italic"> 4 </span><span class> 3 </span><span class="italic"> 2 </span><span class> 1, which equals to 120. A recursive definition can be used to solve a problem mathematically or programmatically. The factorial function 'Fact(n)' can be defined recursively as follows:</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class> <ol> <li><span class>Base Case: </span><span class="italic">f </span><span class>(0) = 1</span></li> <li><span class>Recursive Step: </span><span class="font-[700]">(for n > 0)</span><span class> </span><span class="italic">f </span><span class>(</span><span class="italic">n</span><span class>) = n * </span><span class="italic"> </span><span class>(</span><span class="italic">n</span><span class>−1) </span></li> </ol> </span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class="font-[700]">For instance</span><span class>:</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class="italic">f </span><span class>(5) = 5 * </span><span class="italic">f </span><span class>(4)</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class="italic">f </span><span class>(4) = 4 * </span><span class="italic">f </span><span class>(3)</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class="italic">f </span><span class>(3) = 3 *</span><span class="italic">f </span><span class>(2)</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class="italic">f </span><span class>(2) = 2 *</span><span class="italic">f </span><span class>(1)</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class="italic">f </span><span class>(1) = 1 * </span><span class="italic">f </span><span class>(0) (where </span><span class="italic">f </span><span class>(0) = 1)</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>The Fibonacci sequence is a good example of a recursive relation with two base cases.</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class> <ol> <li><span class>Base Case: </span><span class="italic">f </span><span class>(0) = 0 or </span><span class="italic">f </span><span class>(1) = 1</span></li> <li><span class>Recursive Step: </span><span class="font-[700]">(for n > 1)</span><span class> </span><span class="italic">f </span><span class>(n) = </span><span class="italic">f </span><span class>(n−1) + </span><span class="italic">f </span><span class>(n−2)</span></li> </ol> </span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>For instance, to find the 4th Fibonacci term:</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class="italic">f </span><span class>(0) = 0</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class="italic">f </span><span class>(1) = 1</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class="italic">f </span><span class>(2) = f(0) + </span><span class="italic">f </span><span class>(1) ⇒ 1</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class="italic">f </span><span class>(3) = f(2) +</span><span class="italic">f </span><span class>(1) ⇒ 2</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>The base case serves as a way to break out of the recursion. A recursive function should reach and execute one or more base cases to avoid infinite recursion. For tree problems, you might see 'if (node == null) return,' which is a typical base case. The recursive step, on the other hand, is where a function calls itself with a typically smaller input size/value and utilizes the result of the recursive call to help solve the current problem. It can involve multiple recursive calls, as seen in problems like the Fibonacci sequence.</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>Having explored recursion, let's now examine and contrast recursive algorithms with their iterative counterparts.</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <h3><span class>Recursion vs. Iterative Programming</span></h3> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>Recursion allows solving complex programming tasks concisely. Inversely, an iterative approach is less concise as we repeat a single statement or a block of statements a certain number of times or until a specific condition is met. The Church-Turing thesis proves that what is computable by a recursive function is computable by an iterative one and vice versa. This is because recursion can be implemented using loops and a stack. Recursive algorithms have a space complexity, specifically in non-tail recursion, which is directly related to the maximum depth of the function call stack during execution. However, the iterative approach typically has no associated space complexity unless it adds an extra stack. The following code computes n! (n factorial) using two different approaches</span> </p> </div>  <div class="flex flex-col mt-2" data-test-id="publishing-image-block" rel="ugc"> <figure> <img class="lazy-load block w-full" data-delayed-url="https://media.licdn.com/dms/image/v2/D5612AQEwSdNLkmLcYg/article-inline_image-shrink_400_744/article-inline_image-shrink_400_744/0/1703380176616?e=2147483647&v=beta&t=B2D_M2ZjwNhcZ-5hF8JUJwX0kZ8z4A6yS6IvTQCp-lM">  </figure> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>One important component of recursion is the stack. Let’s look at how the stack is managed.</span> </p> </div> <div class="inline-articles bg-color-surface-new border-hidden rounded-sm p-1 mx-auto my-1.5 mamabear:w-2/3 inline-articles__after"> <h2 class="text-xl font-medium ml-1.5 my-1 text-color-text mamabear:text-center">Recommended by LinkedIn</h2> <div class="flex flex-wrap mx-0.5 babybear:mx-1.5"> <div class="content-author-card flex flex-col max-w-[270px] h-auto rounded-md border-1 border-system-gray-30 border-solid pb-1.5 inline-article mx-auto my-1 border-hidden papabear:w-[219px] babybear:min-w-full mamabear:min-w-full h-auto bg-color-canvas-tint"> <a class="no-underline" data-tracking-control-name="inline-recommended-articles" data-tracking-will-navigate href="https://www.linkedin.com/pulse/data-science-20-andriy-burkov-dixxe"> <img class="content-image w-full h-[150px] bg-cover rounded-t-md object-cover" data-delayed-url="https://media.licdn.com/dms/image/v2/D4E12AQE3XEBCysX3Xg/article-cover_image-shrink_720_1280/article-cover_image-shrink_720_1280/0/1707793787632?e=2147483647&v=beta&t=So5wqqLDVx_C-Mrzj-ysIug43jAkepHshpiSYFDlc_8"> <div class="inline-article-title text-md text-color-text font-bold pl-2 pt-1 pr-1.5">Data Science #20</div> <div class="inline-article-meta px-2 mt-1 font-sans text-xs font-normal text-color-text-low-emphasis h-full"> <span class="inline-article-author pt-0.5"> Andriy Burkov </span> <span class="before:middot pt-0.5 inline-article-published-date"> 10 months ago </span> </div> </a>  </div> <div class="content-author-card flex flex-col max-w-[270px] h-auto rounded-md border-1 border-system-gray-30 border-solid pb-1.5 inline-article mx-auto my-1 border-hidden papabear:w-[219px] babybear:min-w-full mamabear:min-w-full h-auto bg-color-canvas-tint"> <a class="no-underline" data-tracking-control-name="inline-recommended-articles" data-tracking-will-navigate href="https://www.linkedin.com/pulse/introduction-random-walks-brownian-motions-related-vincent-granville"> <img class="content-image w-full h-[150px] bg-cover rounded-t-md object-cover" data-delayed-url="https://media.licdn.com/dms/image/v2/D5612AQFAHpHcTSkBng/article-cover_image-shrink_720_1280/article-cover_image-shrink_720_1280/0/1676307183968?e=2147483647&v=beta&t=zABJ8nO_jzQC1SvQ7PP3wwaGX6TSdPrD2I_AOZCgE-g"> <div class="inline-article-title text-md text-color-text font-bold pl-2 pt-1 pr-1.5">Introduction to Random Walks, Brownian Motions, and…</div> <div class="inline-article-meta px-2 mt-1 font-sans text-xs font-normal text-color-text-low-emphasis h-full"> <span class="inline-article-author pt-0.5"> Vincent Granville </span> <span class="before:middot pt-0.5 inline-article-published-date"> 1 year ago </span> </div> </a>  </div> <div class="content-author-card flex flex-col max-w-[270px] h-auto rounded-md border-1 border-system-gray-30 border-solid pb-1.5 inline-article mx-auto my-1 border-hidden papabear:w-[219px] babybear:min-w-full mamabear:min-w-full h-auto bg-color-canvas-tint"> <a class="no-underline" data-tracking-control-name="inline-recommended-articles" data-tracking-will-navigate href="https://www.linkedin.com/pulse/understanding-bubble-sort-algorithm-amr-saafan"> <img class="content-image w-full h-[150px] bg-cover rounded-t-md object-cover" data-delayed-url="https://media.licdn.com/dms/image/v2/D4D12AQHFc9BqwVjIlg/article-cover_image-shrink_720_1280/article-cover_image-shrink_720_1280/0/1697555238349?e=2147483647&v=beta&t=3IqsHTpHkZodgRfWG8cCiGQ0QzHmZWfNlpxpnX_-VqY"> <div class="inline-article-title text-md text-color-text font-bold pl-2 pt-1 pr-1.5">Understanding the Bubble Sort Algorithm</div> <div class="inline-article-meta px-2 mt-1 font-sans text-xs font-normal text-color-text-low-emphasis h-full"> <span class="inline-article-author pt-0.5"> Amr Saafan </span> <span class="before:middot pt-0.5 inline-article-published-date"> 1 year ago </span> </div> </a>  </div> </div> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <h3><span class>Stack Memory Management </span></h3> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>When an operating system (OS) runs a program, the loader first loads the program’s data and code into the main memory. This enables the CPU to access and execute the program's instructions and work with its data. The stack plays a central role in managing a program's execution. One crucial component of stack memory operations is the Stack Pointer (SP), a CPU register that indicates the top of the stack, The Stack Pointer is automatically adjusted each time a stack operation is executed. The stack memory is pre-allocated at program start because, by definition, it must be contiguous in memory. This also means that the stack memory is fixed throughout the lifetime of the program.</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <h3><span class>Stack Register</span></h3> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>'ebp’ and ‘esp’ are x86 architecture registers used to manage the stack. ‘esp’ is the stack pointer, used to manage the stack. It is adjusted to allocate space for local variables and to pass arguments to functions. ‘ebp’ is the base pointer, keeping track of the start of the current stack frame. It is sometimes used to access parameters and local variables. Now, let's examine the following recursive program.</span> </p> </div>  <div class="flex flex-col mt-2" data-test-id="publishing-image-block" rel="ugc"> <figure> <img class="lazy-load block w-full" data-delayed-url="https://media.licdn.com/dms/image/v2/D5612AQEyp5QHB2GvRQ/article-inline_image-shrink_1000_1488/article-inline_image-shrink_1000_1488/0/1703378796186?e=2147483647&v=beta&t=S6vI5bFtiScSzku8ayW80Oia-TP-mCIRzNaIUl1PDWE">  </figure> </div>  <div class="flex flex-col mt-2" data-test-id="publishing-image-block" rel="ugc"> <figure> <img class="lazy-load block w-full" data-delayed-url="https://media.licdn.com/dms/image/v2/D5612AQEGAkHSMMk99A/article-inline_image-shrink_1000_1488/article-inline_image-shrink_1000_1488/0/1703378831378?e=2147483647&v=beta&t=dakQfLFNHGrKaOJhWaD3Mc8jRC2oeGc-HU-hc8g1XFs"> <figcaption class="mt-[0.8rem] text-center text-sm font-normal text-color-text-low-emphasis" data-test-id="publishing-image-block-caption"> <span class> The diagram shows how the snippet code below is executed inside the stack memory.</span> </figcaption> </figure> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>Inside 'foo()', just before calling 'bar()', we need to pass two arguments to the function. This is accomplished by pushing them onto the stack, as illustrated in Figure 1. The CPU is only aware of the next instruction to execute, stored in a register known as the Instruction Pointer (IP). To return from a function call, we must save the next instruction to be executed after the call. Therefore, this instruction is pushed to the stack in Figure 2, just before calling 'bar()' (the call is achieved by updating IP to the beginning of the function ‘bar()’).</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>Upon returning from 'bar()', the program must revert the frame to the state as it was in Figure 1 (this operation is referred to as "unwinding" the stack). To achieve this, we save the stack base pointer (ebp) in the stack, just before updating it for the new function. When returning, the compiler will generate code that resets the stack pointer (esp) to the current ebp and then restores the previously pushed ebp. Figure 3 represents the code immediately after saving the caller’s ebp.</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>Finally, the two variables, 'x' and 'y' in 'bar()', are pushed to the stack, as shown in Figure 4.</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class> Each programming language has its distinct calling convention and is designed to work with various platforms (CPU architecture + operating system). Understanding the call stack is fundamental when working with function calls and recursion. When a function is called, whether it's a normal function call or a recursive call, the program's execution environment creates a new stack frame (also known as an activation record). This stack frame contains information such as local variables, function parameters, and the return address. Since it operates on the Last In, First Out (LIFO) principle, storing data on the stack is called 'pushing' (using the PUSH instruction in x86 assembly), and restoring data from the stack is called 'popping' (using the POP instruction). These operations are common in assembly or low-level languages.</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <br> </p> </div>  <div class="flex flex-col mt-2" data-test-id="publishing-image-block" rel="ugc"> <figure> <img class="lazy-load block w-full" data-delayed-url="https://media.licdn.com/dms/image/v2/D5612AQG4pIyiURWykg/article-inline_image-shrink_400_744/article-inline_image-shrink_400_744/0/1703379562411?e=2147483647&v=beta&t=zOL5oySA6MKN8pYs87vKow-EVvF2N3EBKfJgTOgPxms"> <figcaption class="mt-[0.8rem] text-center text-sm font-normal text-color-text-low-emphasis" data-test-id="publishing-image-block-caption"> <span class> </span> </figcaption> </figure> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>In the case of recursive functions, each recursive call creates a new stack frame on top of the previous one. The call stack ensures that the program can keep track of its execution flow and return to the appropriate caller after a function completes its task. </span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>When a function encounters a return statement or throws an exception, it exits the current recursive routine with its current inputs. As the function completes its execution, this information is popped from the stack.</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>Tree problems are a type of problems that can be solved neatly using recursion. Let's look at one final example to see how the stack gets updated.</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>The problem we’re looking at is whether a tree is balanced. This is the </span><span class><a href="https://www.linkedin.com/redir/redirect?url=https%3A%2F%2Fleetcode%2Ecom%2Fproblems%2Fbalanced-binary-tree%2F&urlhash=-QUa&trk=article-ssr-frontend-pulse_little-text-block" target="_blank" data-tracking-control-name="article-ssr-frontend-pulse_little-text-block" data-tracking-will-navigate data-test-link>LeetCode exercise</a></span><span class> if you’re curious. A common solution is to recurse on each subtree and track whether each subtree is balanced. If one subtree is unbalanced, we can propagate the information up to the root (which is modeled as the magic constant -1 in the following code snippets):</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>Here is an example of the function calls involved and their return values:</span> </p> </div>  <div class="flex flex-col mt-2" data-test-id="publishing-image-block" rel="ugc"> <figure> <img class="lazy-load block w-full" data-delayed-url="https://media.licdn.com/dms/image/v2/D5612AQGSms-lzhg8rA/article-inline_image-shrink_1000_1488/article-inline_image-shrink_1000_1488/0/1703380030081?e=2147483647&v=beta&t=KGn-9pM4SCTBBK38w5OB-eZT-3Gujm_v0Zj-0lQNG70">  </figure> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>Now if we look at the hardware stack, the following diagram shows the different states that it will go through as we invoke ‘height’. The sequence is similar to the previous example above pushing the return address, then the previous 'ebp' then any arguments, and finally the local variable.</span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>In summary, recursion is a powerful programming technique, often used for tasks like calculating factorials. It differs from iterative approaches but can achieve the same results. However, recursion can be space-inefficient. Stack memory is crucial in program execution, especially in tracking function calls and recursion. Understanding the call stack and its management is essential for programmers. Recursion, while elegant, can be challenging to master.</span> </p> </div>  <hr data-test-id="publishing-divider-block" class="my-[3.2rem] mx-auto h-[1px] border-0 bg-black-a16">  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>¹ Stackoverflow </span><span class><a href="https://www.linkedin.com/redir/redirect?url=https%3A%2F%2Fstackoverflow%2Ecom%2Fquestions%2F931762%2Fcan-every-recursion-be-converted-into-iteration&urlhash=VRXA&trk=article-ssr-frontend-pulse_little-text-block" target="_blank" data-tracking-control-name="article-ssr-frontend-pulse_little-text-block" data-tracking-will-navigate data-test-link>Can every recursion be converted into iteration?</a></span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>² Tail Recursion </span><span class><a href="https://en.wikipedia.org/wiki/Tail_call?trk=article-ssr-frontend-pulse_little-text-block" target="_blank" data-tracking-control-name="article-ssr-frontend-pulse_little-text-block" data-tracking-will-navigate data-test-link>Tail Call</a></span> </p> </div>  <div class="article-main__content" data-test-id="publishing-text-block"> <p> <span class>³ Calling Convention </span><span class><a href="https://en.wikipedia.org/wiki/Calling_convention?trk=article-ssr-frontend-pulse_little-text-block" target="_blank" data-tracking-control-name="article-ssr-frontend-pulse_little-text-block" data-tracking-will-navigate data-test-link>https://en.wikipedia.org/wiki/Calling_convention¥</a></span> </p> </div>  </div>   </article>  <code id="ugcPostUrl" style="display: none"></code> <code id="ugcPost" style="display: none"></code> <code id="disableLinkedinShareRedirectLink" style="display: none"></code> <code id="ugcPostUrn" style="display: none"></code> <code id="signInUrl" style="display: none"></code>  <div class="article-social-activity z-[201] x-social-activity-wrapper xs:sticky bottom-0 flex 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comment-with-action my-1.5"> <div> <a href="https://at.linkedin.com/in/anton-kuscher-859172331?trk=article-ssr-frontend-pulse_x-social-details_comments-action_comment_actor-image" data-tracking-control-name="article-ssr-frontend-pulse_x-social-details_comments-action_comment_actor-image" data-tracking-will-navigate> <img class="inline-block relative rounded-[50%] w-4 h-4 " data-delayed-url="https://media.licdn.com/dms/image/v2/D4D03AQE47ATw3xi-sg/profile-displayphoto-shrink_100_100/profile-displayphoto-shrink_100_100/0/1727900695292?e=2147483647&v=beta&t=dfw8XLCZyXw_Po92kfkHLgy6Oqp5020nJcLFvDl3XVM" data-ghost-classes="bg-color-entity-ghost-background" data-ghost-url="https://static.licdn.com/aero-v1/sc/h/9c8pery4andzj6ohjkjp54ma2" alt="Anton Kuscher"> </a> </div> <div class="w-full min-w-0"> <div class="comment__body ml-0.5 mb-1 pt-1 pr-1.5 pb-1 pl-1.5 rounded-[8px] rounded-tl-none relative"> <div class="comment__header flex flex-col"> <a class="text-sm link-styled no-underline leading-open comment__author truncate pr-6" href="https://at.linkedin.com/in/anton-kuscher-859172331?trk=article-ssr-frontend-pulse_x-social-details_comments-action_comment_actor-name" data-tracking-control-name="article-ssr-frontend-pulse_x-social-details_comments-action_comment_actor-name" data-tracking-will-navigate> Anton Kuscher </a> <p class="!text-xs text-color-text-low-emphasis leading-[1.33333] mb-0.5 truncate comment__author-headline">Student at Fachhochschule Technikum Wien</p> <span class="text-xs text-color-text-low-emphasis comment__duration-since absolute right-6 top-1 inline-block"> 2mo </span> <div class="ellipsis-menu absolute right-0 top-0 !mr-0.5 babybear:!-top-0.5" data-test-id="comment__ellipsis-menu" data-feed-action="show-comment-dropdown"> <div class="collapsible-dropdown flex items-center relative hyphens-auto"> <button class="ellipsis-menu__trigger collapsible-dropdown__button btn-md btn-tertiary cursor-pointer !py-[6px] !px-1 flex items-center rounded-[50%] ellipsis-menu__trigger--small w-[32px] h-[32px] babybear:rotate-90 mamabear:!h-4 mamabear:!min-h-[32px] papabear:!h-4 papabear:!min-h-[32px]" aria-expanded="false" aria-label="Open menu" data-tracking-control-name="article-ssr-frontend-pulse_x-social-details_comments-action_comment_ellipsis-menu-trigger" tabindex="0"> <icon class="ellipsis-menu__trigger-icon m-0 p-0 centered-icon" data-delayed-url="https://static.licdn.com/aero-v1/sc/h/c26vw8uiog92lvrqov98hvbxr"></icon> </button> <ul class="collapsible-dropdown__list hidden container-raised absolute w-auto overflow-y-auto flex-col items-stretch z-1 bottom-auto top-[100%]" role="menu" tabindex="-1"> <li class="ellipsis-menu__item border-t-1 border-solid border-color-border-low-emphasis first-of-type:border-none flex"> <a href="/uas/login?session_redirect=https%3A%2F%2Fwww.linkedin.com%2Fpulse%2Frecursion-theory-recursive-functions-hasanain-alsabonchi-eh9bc&trk=article-ssr-frontend-pulse_x-social-details_comments-action_comment_ellipsis-menu-semaphore-sign-in-redirect&guestReportContentType=COMMENT&_f=guest-reporting" data-tracking-control-name="article-ssr-frontend-pulse_x-social-details_comments-action_comment_ellipsis-menu-semaphore-sign-in-redirect" data-tracking-will-navigate data-item-type="semaphore" data-semaphore-content-type="COMMENT" data-semaphore-content-urn="urn:li:comment:(urn:li:ugcPost:7145866253088284672,7247343289803165698)" data-semaphore-tracking-prefix="article-ssr-frontend-pulse_x-social-details_comments-action_comment_ellipsis-menu-semaphore" data-is-logged-in="false" data-modal="semaphore__toggle" class="semaphore__toggle visited:text-color-text-secondary ellipsis-menu__semaphore ellipsis-menu__item-button flex items-center w-full p-1 cursor-pointer font-sans text-sm font-bold link-styled focus:link-styled link:no-underline active:bg-color-background-container-tint focus:bg-color-background-container-tint hover:bg-color-background-container-tint outline-offset-[-2px]">  <icon class="ellipsis-menu__item-icon text-color-text h-[24px] w-[24px] mr-1" data-delayed-url="https://static.licdn.com/aero-v1/sc/h/iq0x9q37wj214o129ai1yjut"> </icon> Report this comment </a>  </li>  </ul>  </div> </div> </div> <div class="attributed-text-segment-list__container relative mt-1 mb-1.5 babybear:mt-0 babybear:mb-0.5"> <p class="attributed-text-segment-list__content text-color-text !text-sm whitespace-pre-wrap break-words comment__text babybear:mt-0.5" dir="ltr">You might want to correct the recursive function to calculate factorials. The if statement needs to check for n<=0 instead of n<0. Otherwise it will return 0 for every positive number. :D</p>  </div>   </div> <div class="comment__actions flex flex-row flex-wrap items-center ml-2" aria-label="Comment Controls"> <a href="https://www.linkedin.com/signup/cold-join?session_redirect=%2Fpulse%2Frecursion-theory-recursive-functions-hasanain-alsabonchi-eh9bc&trk=article-ssr-frontend-pulse_x-social-details_comments-action_comment_like" class="comment__action inline-block font-sans text-xs text-color-text-low-emphasis cursor-pointer" data-tracking-control-name="article-ssr-frontend-pulse_x-social-details_comments-action_comment_like" data-tracking-will-navigate> <div class="flex flex-row items-center "> <icon class="action-btn__icon w-2 mr-0.5" data-svg-class-name="action-btn__icon--svg" data-delayed-url="https://static.licdn.com/aero-v1/sc/h/70k0g8kmgdfjjymflqqzipzxj"> </icon> <span class="action-btn__button-text inline-block align-baseline"> Like </span> </div> </a> <span class="before:middot font-sans text-s text-color-text-low-emphasis"></span> <a href="https://www.linkedin.com/signup/cold-join?session_redirect=%2Fpulse%2Frecursion-theory-recursive-functions-hasanain-alsabonchi-eh9bc&trk=article-ssr-frontend-pulse_x-social-details_comments-action_comment_reply" class="comment__action inline-block font-sans text-xs text-color-text-low-emphasis cursor-pointer comment__reply" data-feed-action="replyToComment" data-feed-control="reply" data-feed-action-category="EXPAND" data-feed-action-type="expandReplyBox" data-tracking-control-name="article-ssr-frontend-pulse_x-social-details_comments-action_comment_reply" data-tracking-will-navigate> <div class="flex flex-row items-center "> <icon class="action-btn__icon w-2 mr-0.5" data-svg-class-name="action-btn__icon--svg" data-delayed-url="https://static.licdn.com/aero-v1/sc/h/4ol9mo4lxvobj5ww3va90wz1o"> </icon> <span class="action-btn__button-text inline-block align-baseline"> Reply </span> </div> </a>  <span class="comment__reactions-count font-sans text-xs text-color-text-low-emphasis border-l-1 border-solid border-color-border-low-emphasis pl-1 mx-1 hidden"> 1 Reaction </span> </div> </div> </section> <div class="comments__see-more-wrapper"> <a class="comments__see-more" href="https://www.linkedin.com/signup/cold-join?session_redirect=%2Fpulse%2Frecursion-theory-recursive-functions-hasanain-alsabonchi-eh9bc&trk=article-ssr-frontend-pulse_x-social-details_comments_comment-see-more" data-tracking-control-name="article-ssr-frontend-pulse_x-social-details_comments_comment-see-more" data-tracking-will-navigate> See more comments </a> </div> </div> </div> <p class="feed-cta-banner__text border-t-1 border-color-surface-border border-solid p-3 text-center text-md text-color-text-low-emphasis"> To view or add a comment, <a href="https://www.linkedin.com/signup/cold-join?session_redirect=%2Fpulse%2Frecursion-theory-recursive-functions-hasanain-alsabonchi-eh9bc&trk=article-ssr-frontend-pulse_x-social-details_feed-cta-banner-cta" target="" aria-label="To view or add a comment, sign in" data-tracking-control-name="article-ssr-frontend-pulse_x-social-details_feed-cta-banner-cta" data-tracking-will-navigate="true" class="feed-cta-banner__link link">sign in</a> </p>  </div> </section> <section class="right-rail papabear:w-right-rail-width papabear:ml-column-gutter mamabear:max-w-[790px] mamabear:px-mobile-container-padding babybear:max-w-[790px] babybear:px-mobile-container-padding">  <section class="aside-section-container mb-4 pulse-to-x"> <h2 class="aside-section-container__title section-title"> Insights from the community </h2>  <div class="aside-section-container__content break-words"> <div class="show-more-less">  <ul data-max-num-to-show="6" class="show-more-less__list show-more-less__list--hide-after-6" data-impression-id="article-ssr-frontend-pulse_show-more-less"> <li class="articles-side-panel__list-item -ml-3"> <a class="x-article-card__link hover:no-underline focus:no-underline active:no-underline" href="https://www.linkedin.com/advice/1/how-can-you-determine-optimal-substructure-problem-0jnec" data-tracking-will-navigate="true" data-tracking-control-name="recommended_article" data-feed-action-category="view" data-feed-action-type="viewArticle" target="_self"> <div class="x-article-card container-lined ml-0 my-2 min-h-max py-1 px-1.5 hover:shadow-md active:shadow-md md:m-2 w-100%"> <div class="x-article-card__tag tag-sm tag-enabled inline break-words font-normal"> Algorithms </div> <div class="x-article-card__title text-base break-words pt-1 text-md font-normal text-color-text"> How can you determine the optimal substructure of a problem? 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Recursion Theory and Recursive Functions