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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>A tree stack automaton<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> (plural: tree stack automata) is a <a href="/wiki/Formalism_(philosophy_of_mathematics)" title="Formalism (philosophy of mathematics)">formalism</a> considered in <a href="/wiki/Automata_theory" title="Automata theory">automata theory</a>. It is a <a href="/wiki/Finite_state_automaton" class="mw-redirect" title="Finite state automaton">finite state automaton</a> with the additional ability to manipulate a <a href="/wiki/Tree_(set_theory)#Tree_(automata_theory)" title="Tree (set theory)">tree</a>-shaped <a href="/wiki/Stack_(abstract_data_type)" title="Stack (abstract data type)">stack</a>. It is an automaton with storage<sup id="cite_ref-Sco67_3-0" class="reference"><a href="#cite_note-Sco67-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> whose storage roughly resembles the configurations of a <a href="/wiki/Thread_automaton" title="Thread automaton">thread automaton</a>. A restricted class of tree stack automata recognises exactly the <a href="/wiki/Formal_language" title="Formal language">languages</a> generated by multiple <a href="/wiki/Context-free_grammar" title="Context-free grammar">context-free grammars</a><sup id="cite_ref-Den16_4-0" class="reference"><a href="#cite_note-Den16-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> (or <a href="/wiki/Generalized_context-free_grammar#Linear_Context-free_Rewriting_Systems_(LCFRSs)" title="Generalized context-free grammar">linear context-free rewriting systems</a>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tree_stack_automaton&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Tree_stack">Tree stack</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tree_stack_automaton&action=edit&section=2" title="Edit section: Tree stack"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Tree_stack.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Tree_stack.svg/220px-Tree_stack.svg.png" decoding="async" width="220" height="352" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Tree_stack.svg/330px-Tree_stack.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Tree_stack.svg/440px-Tree_stack.svg.png 2x" data-file-width="303" data-file-height="485" /></a><figcaption>A tree stack with stack pointer 1.2 and domain {<i>ε</i>, 1, 42, 1.2, 1.5, 1.5.3}</figcaption></figure> <p>For a finite and non-empty set <span class="texhtml"><i>Γ</i></span>, a <i>tree stack over <span class="texhtml"><var style="padding-right: 1px;">Γ</var></span></i> is a tuple <span class="texhtml">(<i>t</i>, <i>p</i>)</span> where </p> <ul><li><span class="texhtml"><i>t</i></span> is a <a href="/wiki/Partial_function" title="Partial function">partial function</a> from strings of positive integers to the set <span class="texhtml"><i>Γ</i> ∪ {@</span>} with <a href="/wiki/Substring#Prefix" title="Substring">prefix</a>-closed<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> (called <i>tree</i>),</li> <li><span class="texhtml">@</span> (called <i>bottom symbol</i>) is not in <span class="texhtml"><i>Γ</i></span> and appears exactly at the root of <span class="texhtml"><i>t</i></span>, and</li> <li><span class="texhtml"><i>p</i></span> is an element of the domain of <span class="texhtml"><i>t</i></span> (called <i>stack pointer</i>).</li></ul> <p>The set of all tree stacks over <span class="texhtml"><i>Γ</i></span> is denoted by <span class="texhtml">TS(<i>Γ</i>)</span>. </p><p>The set of <i>predicates</i> on <span class="texhtml">TS(<i>Γ</i>)</span>, denoted by <span class="texhtml">Pred(<i>Γ</i>)</span>, contains the following <a href="/wiki/Unary_operation" title="Unary operation">unary</a> <a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">predicates</a>: </p> <ul><li><span class="texhtml">true</span> which is true for any tree stack over <span class="texhtml"><i>Γ</i></span>,</li> <li><span class="texhtml">bottom</span> which is true for tree stacks whose stack pointer points to the bottom symbol, and</li> <li><span class="texhtml">equals(<i>γ</i>)</span> which is true for some tree stack <span class="texhtml">(<i>t</i>, <i>p</i>)</span> if <span class="texhtml"><i>t</i>(<i>p</i>) = <i>γ</i></span>,</li></ul> <p>for every <span class="texhtml"><i>γ</i> ∈ <i>Γ</i></span>. </p><p>The set of <i>instructions</i> on <span class="texhtml">TS(<i>Γ</i>)</span>, denoted by <span class="texhtml">Instr(<i>Γ</i>)</span>, contains the following partial functions: </p> <ul><li><span class="texhtml">id: TS(<i>Γ</i>) → TS(<i>Γ</i>)</span> which is the identity function on <span class="texhtml">TS(<i>Γ</i>)</span>,</li> <li><span class="texhtml">push<sub><i>n</i>,<i>γ</i></sub>: TS(<i>Γ</i>) → TS(<i>Γ</i>)</span> which adds for a given tree stack <span class="texhtml">(<i>t</i>,<i>p</i>)</span> a pair <span class="texhtml">(<i>pn</i> ↦ <i>γ</i>)</span> to the tree <span class="texhtml"><i>t</i></span> and sets the stack pointer to <span class="texhtml"><i>pn</i></span> (i.e. it pushes <span class="texhtml"><i>γ</i></span> to the <span class="texhtml"><i>n</i></span>-th child position) if <span class="texhtml"><i>pn</i></span> is not yet in the domain of <span class="texhtml"><i>t</i></span>,</li> <li><span class="texhtml">up<sub><i>n</i></sub>: TS(<i>Γ</i>) → TS(<i>Γ</i>)</span> which replaces the current stack pointer <span class="texhtml"><i>p</i></span> by <span class="texhtml"><i>pn</i></span> (i.e. it moves the stack pointer to the <span class="texhtml"><i>n</i></span>-th child position) if <span class="texhtml"><i>pn</i></span> is in the domain of <span class="texhtml"><i>t</i></span>,</li> <li><span class="texhtml">down: TS(<i>Γ</i>) → TS(<i>Γ</i>)</span> which removes the last symbol from the stack pointer (i.e. it moves the stack pointer to the parent position), and</li> <li><span class="texhtml">set<sub><i>γ</i></sub>: TS(<i>Γ</i>) → TS(<i>Γ</i>)</span> which replaces the symbol currently under the stack pointer by <span class="texhtml"><i>γ</i></span>,</li></ul> <p>for every positive integer <span class="texhtml"><i>n</i></span> and every <span class="texhtml"><i>γ</i> ∈ <i>Γ</i></span>. </p> <table style="margin: 0 auto;"> <tbody><tr> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Id_on_a_tree_stack.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Id_on_a_tree_stack.svg/220px-Id_on_a_tree_stack.svg.png" decoding="async" width="220" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Id_on_a_tree_stack.svg/330px-Id_on_a_tree_stack.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Id_on_a_tree_stack.svg/440px-Id_on_a_tree_stack.svg.png 2x" data-file-width="611" data-file-height="485" /></a><figcaption>Illustration of the instruction id on a tree stack</figcaption></figure> </td> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Push_on_a_tree_stack.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Push_on_a_tree_stack.svg/220px-Push_on_a_tree_stack.svg.png" decoding="async" width="220" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Push_on_a_tree_stack.svg/330px-Push_on_a_tree_stack.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Push_on_a_tree_stack.svg/440px-Push_on_a_tree_stack.svg.png 2x" data-file-width="709" data-file-height="485" /></a><figcaption>Illustration of the instruction push on a tree stack</figcaption></figure> </td> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Up_and_down_on_a_tree_stack.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Up_and_down_on_a_tree_stack.svg/220px-Up_and_down_on_a_tree_stack.svg.png" decoding="async" width="220" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Up_and_down_on_a_tree_stack.svg/330px-Up_and_down_on_a_tree_stack.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Up_and_down_on_a_tree_stack.svg/440px-Up_and_down_on_a_tree_stack.svg.png 2x" data-file-width="642" data-file-height="485" /></a><figcaption>Illustration of the instructions up and down on a tree stack</figcaption></figure> </td> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Set_on_a_tree_stack.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Set_on_a_tree_stack.svg/220px-Set_on_a_tree_stack.svg.png" decoding="async" width="220" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Set_on_a_tree_stack.svg/330px-Set_on_a_tree_stack.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Set_on_a_tree_stack.svg/440px-Set_on_a_tree_stack.svg.png 2x" data-file-width="642" data-file-height="485" /></a><figcaption>Illustration of the instruction set on a tree stack</figcaption></figure> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Tree_stack_automata">Tree stack automata</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tree_stack_automaton&action=edit&section=3" title="Edit section: Tree stack automata"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i>tree stack automaton</i> is a 6-tuple <span class="texhtml"><i>A</i> = (<i>Q</i>, <i>Γ</i>, <i>Σ</i>, <i>q</i><sub>i</sub>, <i>δ</i>, <i>Q</i><sub>f</sub>)</span> where </p> <ul><li><span class="texhtml"><i>Q</i></span>, <span class="texhtml"><i>Γ</i></span>, and <span class="texhtml"><i>Σ</i></span> are finite sets (whose elements are called <i>states</i>, <i>stack symbols</i>, and <i>input symbols</i>, respectively),</li> <li><span class="texhtml"><i>q</i><sub>i</sub> ∈ <i>Q</i></span> (the <i>initial state</i>),</li> <li><span class="texhtml"><i>δ</i> ⊆<sub>fin.</sub> <i>Q</i> × (<i>Σ</i> ∪ {<i>ε</i>}) × Pred(<i>Γ</i>) × Instr(<i>Γ</i>) × <i>Q</i></span> (whose elements are called <i>transitions</i>), and</li> <li><span class="texhtml"><i>Q</i><sub>f</sub> ⊆ TS(<i>Γ</i>)</span> (whose elements are called <i>final states</i>).</li></ul> <p>A <i>configuration of <span class="texhtml"><var style="padding-right: 1px;">A</var></span></i> is a tuple <span class="texhtml">(<i>q</i>, <i>c</i>, <i>w</i>)</span> where </p> <ul><li><span class="texhtml"><i>q</i></span> is a state (the <i>current state</i>),</li> <li><span class="texhtml"><i>c</i></span> is a tree stack (the <i>current tree stack</i>), and</li> <li><span class="texhtml"><i>w</i></span> is a word over <span class="texhtml"><i>Σ</i></span> (the <i>remaining word</i> to be read).</li></ul> <p>A transition <span class="texhtml"><i>τ</i> = (<i>q</i><sub>1</sub>, <i>u</i>, <i>p</i>, <i>f</i>, <i>q</i><sub>2</sub>)</span> is <i>applicable</i> to a configuration <span class="texhtml">(<i>q</i>, <i>c</i>, <i>w</i>)</span> if </p> <ul><li><span class="texhtml"><i>q</i><sub>1</sub> = <i>q</i></span>,</li> <li><span class="texhtml"><i>p</i></span> is true on <span class="texhtml"><i>c</i></span>,</li> <li><span class="texhtml"><i>f</i></span> is defined for <span class="texhtml"><i>c</i></span>, and</li> <li><span class="texhtml"><i>u</i></span> is a prefix of <span class="texhtml"><i>w</i></span>.</li></ul> <p>The <i>transition relation of <span class="texhtml"><var style="padding-right: 1px;">A</var></span></i> is the <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> <span class="texhtml">⊢</span> on configurations of <span class="texhtml"><var style="padding-right: 1px;">A</var></span> that is the union of all the relations <span class="texhtml">⊢<sub><i>τ</i></sub></span> for a transition <span class="texhtml"><i>τ</i> = (<i>q</i><sub>1</sub>, <i>u</i>, <i>p</i>, <i>f</i>, <i>q</i><sub>2</sub>)</span> where, whenever <span class="texhtml"><i>τ</i></span> is applicable to <span class="texhtml">(<i>q</i>, <i>c</i>, <i>w</i>)</span>, we have <span class="texhtml">(<i>q</i>, <i>c</i>, <i>w</i>) ⊢<sub><i>τ</i></sub> (<i>q</i><sub>2</sub>, <i>f</i>(<i>c</i>), <i>v</i>)</span> and <span class="texhtml"><i>v</i></span> is obtained from <span class="texhtml"><i>w</i></span> by removing the prefix <span class="texhtml"><i>u</i></span>. </p><p>The <i>language of <span class="texhtml"><var style="padding-right: 1px;">A</var></span></i> is the set of all words <span class="texhtml"><i>w</i></span> for which there is some state <span class="texhtml"><i>q</i> ∈ <i>Q</i><sub>f</sub></span> and some tree stack <span class="texhtml"><i>c</i></span> such that <span class="texhtml">(<i>q</i><sub>i</sub>, <i>c</i><sub>i</sub>, w) ⊢<sup>*</sup> (<i>q</i>, <i>c</i>, <i>ε</i>)</span> where </p> <ul><li><span class="texhtml">⊢<sup>*</sup></span> is the <a href="/wiki/Binary_relation#Operations" title="Binary relation">reflexive transitive closure</a> of <span class="texhtml">⊢</span> and</li> <li><span class="texhtml"><i>c</i><sub>i</sub> = (<i>t</i><sub>i</sub>, <i>ε</i>)</span> such that <span class="texhtml"><i>t</i><sub>i</sub></span> assigns for <span class="texhtml"><i>ε</i></span> the symbol <span class="texhtml">@</span> and is undefined otherwise.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Related_formalisms">Related formalisms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tree_stack_automaton&action=edit&section=4" title="Edit section: Related formalisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tree stack automata are equivalent to <a href="/wiki/Turing_machine" title="Turing machine">Turing machines</a>. </p><p>A tree stack automaton is called <i><span class="texhtml"><var style="padding-right: 1px;">k</var></span>-restricted</i> for some positive natural number <span class="texhtml"><var style="padding-right: 1px;">k</var></span> if, during any run of the automaton, any position of the tree stack is accessed at most <span class="texhtml"><i>k</i></span> times from below. </p><p>1-restricted tree stack automata are equivalent to <a href="/wiki/Pushdown_automaton" title="Pushdown automaton">pushdown automata</a> and therefore also to <a href="/wiki/Context-free_grammar" title="Context-free grammar">context-free grammars</a>. <span class="texhtml"><i>k</i></span>-restricted tree stack automata are equivalent to <a href="/wiki/Generalized_context-free_grammar#Linear_Context-free_Rewriting_Systems_(LCFRSs)" title="Generalized context-free grammar">linear context-free rewriting systems</a> and multiple context-free grammars of fan-out at most <span class="texhtml"><i>k</i></span> (for every positive integer <span class="texhtml"><i>k</i></span>).<sup id="cite_ref-Den16_4-1" class="reference"><a href="#cite_note-Den16-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tree_stack_automaton&action=edit&section=5" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">not to be confused with a device with the same name introduced in 1990 by Wolfgang Golubski and Wolfram-M. Lippe <sup id="cite_ref-GolLip90_1-0" class="reference"><a href="#cite_note-GolLip90-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">A set of strings is <i>prefix-closed</i> if for every element <span class="texhtml"><i>w</i></span> in the set, all prefixes of <span class="texhtml"><i>w</i></span> are also in the set.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tree_stack_automaton&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-GolLip90-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-GolLip90_1-0">^</a></b></span> <span class="reference-text">Golubski, Wolfgang and Lippe, Wolfram-M. (1990). <i>Tree-stack automata</i>. Proceedings of the 15th Symposium on Mathematical Foundations of Computer Science (MFCS 1990). Lecture Notes in Computer Science, Vol. 452, pages 313–321, <a rel="nofollow" class="external text" href="https://www.dx.doi.org/10.1007/BFb0029624">doi:10.1007/BFb0029624</a>.</span> </li> <li id="cite_note-Sco67-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Sco67_3-0">^</a></b></span> <span class="reference-text">Scott, Dana (1967). <i>Some Definitional Suggestions for Automata Theory</i>. Journal of Computer and System Sciences, Vol. 1(2), pages 187–212, <a rel="nofollow" class="external text" href="https://www.dx.doi.org/10.1016/s0022-0000(67)80014-x">doi:10.1016/s0022-0000(67)80014-x</a>.</span> </li> <li id="cite_note-Den16-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Den16_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Den16_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Denkinger, Tobias (2016). <i>An automata characterisation for multiple context-free languages</i>. Proceedings of the 20th International Conference on Developments in Language Theory (DLT 2016). Lecture Notes in Computer Science, Vol. 9840, pages 138–150, <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-3-662-53132-7_12">doi:10.1007/978-3-662-53132-7_12</a>.</span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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class="navbox-abovebelow" style="font-weight:bold;"><a href="/wiki/Chomsky_hierarchy" title="Chomsky hierarchy">Chomsky hierarchy</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/Formal_grammar" title="Formal grammar">Grammars</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/Formal_language" title="Formal language">Languages</a></td><td class="navbox-abovebelow" style="border-left:2px solid #fdfdfd;font-weight:bold;"><a href="/wiki/Abstract_machine" title="Abstract machine">Abstract machines</a></td></tr><tr style="vertical-align:top"><td class="navbox-list" style="padding:0px;text-align: center;width:10em;"><div> <ul><li>Type-0</li> <li>—</li> <li>Type-1</li> <li>—</li> <li>—</li> <li>—</li> <li>—</li> <li>—</li> <li>Type-2</li> <li>—</li> <li>—</li> <li>Type-3</li> <li>—</li> <li>—</li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;width:10em;"><div> <ul><li><a href="/wiki/Unrestricted_grammar" title="Unrestricted grammar">Unrestricted</a></li> <li>(no common name)</li> <li><a href="/wiki/Context-sensitive_grammar" title="Context-sensitive grammar">Context-sensitive</a></li> <li><span style="white-space:nowrap;">Positive <a href="/wiki/Range_concatenation_grammars" class="mw-redirect" title="Range concatenation grammars">range concatenation</a></span></li> <li><a href="/wiki/Indexed_grammar" title="Indexed grammar">Indexed</a></li> <li>—</li> <li><a href="/wiki/Linear_context-free_rewriting_system" class="mw-redirect" title="Linear context-free rewriting system">Linear context-free rewriting systems</a></li> <li><a href="/wiki/Tree-adjoining_grammar" title="Tree-adjoining grammar">Tree-adjoining</a></li> <li><a href="/wiki/Context-free_grammar" title="Context-free grammar">Context-free</a></li> <li><a href="/wiki/Deterministic_context-free_grammar" title="Deterministic context-free grammar">Deterministic context-free</a></li> <li><a href="/wiki/Nested_word" title="Nested word">Visibly pushdown</a></li> <li><a href="/wiki/Regular_grammar" title="Regular grammar">Regular</a></li> <li>—</li> <li><a href="/wiki/Non-recursive_grammar" class="mw-redirect" title="Non-recursive grammar">Non-recursive</a></li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;width:10em;"><div> <ul><li><a href="/wiki/Recursively_enumerable_language" title="Recursively enumerable language">Recursively enumerable</a></li> <li><a href="/wiki/Recursive_language" title="Recursive language">Decidable</a></li> <li><a href="/wiki/Context-sensitive_language" title="Context-sensitive language">Context-sensitive</a></li> <li><span style="white-space:nowrap;">Positive <a href="/wiki/Range_concatenation_language" class="mw-redirect" title="Range concatenation language">range concatenation</a><sup>*</sup></span></li> <li><a href="/wiki/Indexed_language" title="Indexed language">Indexed</a><sup>*</sup></li> <li>—</li> <li><a href="/wiki/Linear_context-free_rewriting_language" class="mw-redirect" title="Linear context-free rewriting language">Linear context-free rewriting language</a></li> <li><a href="/wiki/Tree-adjoining_grammar" title="Tree-adjoining grammar">Tree-adjoining</a></li> <li><a href="/wiki/Context-free_language" title="Context-free language">Context-free</a></li> <li><a href="/wiki/Deterministic_context-free_language" title="Deterministic context-free language">Deterministic context-free</a></li> <li><a href="/wiki/Nested_word" title="Nested word">Visibly pushdown</a></li> <li><a href="/wiki/Regular_language" title="Regular language">Regular</a></li> <li><a href="/wiki/Star-free_language" title="Star-free language">Star-free</a></li> <li><a href="/wiki/Finite_language" class="mw-redirect" title="Finite language">Finite</a></li></ul> </div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;width:10em;"><div> <ul><li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a href="/wiki/Decider_(Turing_machine)" title="Decider (Turing machine)">Decider</a></li> <li><a href="/wiki/Linear_bounded_automaton" title="Linear bounded automaton">Linear-bounded</a></li> <li><a href="/wiki/PTIME" class="mw-redirect" title="PTIME">PTIME</a> Turing Machine</li> <li><a href="/wiki/Nested_stack_automaton" title="Nested stack automaton">Nested stack</a></li> <li><a href="/wiki/Thread_automaton" title="Thread automaton">Thread automaton</a></li> <li>restricted <a class="mw-selflink selflink">Tree stack automaton</a></li> <li><a href="/wiki/Embedded_pushdown_automaton" title="Embedded pushdown automaton">Embedded pushdown</a></li> <li><a href="/wiki/Pushdown_automaton" title="Pushdown automaton">Nondeterministic pushdown</a></li> <li><a href="/wiki/Deterministic_pushdown_automaton" title="Deterministic pushdown automaton">Deterministic pushdown</a></li> <li><a href="/wiki/Nested_word" title="Nested word">Visibly pushdown</a></li> <li><a href="/wiki/Finite-state_machine" title="Finite-state machine">Finite</a></li> <li><a href="/wiki/Aperiodic_finite_state_automaton" title="Aperiodic finite state automaton">Counter-free (with aperiodic finite monoid)</a></li> <li><a href="/wiki/Deterministic_acyclic_finite_state_automaton" title="Deterministic acyclic finite state automaton">Acyclic finite</a></li></ul> </div></td></tr></tbody></table></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><span style="white-space:nowrap;">Each category of languages, except those marked by a <sup>*</sup>, is a <a href="/wiki/Proper_subset" class="mw-redirect" title="Proper subset">proper subset</a> of the category directly above it.</span> <span style="white-space:nowrap;">Any language in each category is generated by a grammar and by an automaton in the category in the same line.</span></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Tree_stack_automaton&oldid=1148268438">https://en.wikipedia.org/w/index.php?title=Tree_stack_automaton&oldid=1148268438</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Models_of_computation" title="Category:Models of computation">Models of computation</a></li><li><a href="/wiki/Category:Automata_(computation)" title="Category:Automata (computation)">Automata (computation)</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> 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