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Regulated Pushdown Automata

This presentation discusses regulated pushdown automata (PDA), a computational model that enhances traditional pushdown automata with control languages. Presented by Alexander Meduna at Kyoto Sangyo University in 2006, the talk explores their defining principles, their correlation with regulated grammars, and their classifications within language families such as context-free and recursively enumerable languages. Key results highlight the implications of regulated PDAs for computational linguistics and automata theory, supported by foundational references and examples illustrating their operation.

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Regulated Pushdown Automata

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  1. Regulated Pushdown Automata 1/22 Alexander Meduna Faculty of Information Technology Brno University of Technology Brno, Czech Republic, Europe Presented at Kyoto Sangyo University, Kyoto, Japan March 9, 2006

  2. Fundamental References 2/22 • Meduna Alexander, Kolář Dušan: • Regulated Pushdown Automata,Acta Cybernetica, • Vol. 2000, No. 4, p. 653-664 • Meduna Alexander, Kolář Dušan: • One-Turn Regulated Pushdown Automata and Their Reduction, Fundamenta Informatica, • Vol. 2002, No. 16, p. 399-405

  3. Inspiration: Regulated Grammars 3/22 • Grammar G: 1.S AC 2.A aAb 3.A ab 4.C Cc 5.C c •  = {1}{24}*{35}

  4. Regulated Grammars 1/2 4/22 • Grammar G: • Without , G • generates aabbccc: 1.S AC 2.A aAb 3.A ab 4.C Cc 5.C c S AC[1]  aAbC[2]  aAbCc[4]  aabbCc[3]  aabbCcc[4]  aabbccc[5]  = {1}{24}*{35} L(G) = {anbncm: n, m 1}

  5. Regulated Grammars 2/2 5/22 • with,Gdoesnotgenerateaabbccc,because • 124345   = {1}{24}*{35} • with, G generates aabbcc: S AC[1]  aAbC[2]  aAbCc[4]  aabbCc[3]  aabbcc[5] and 12435   L(G,) = {anbncn: n 1}

  6. PDA: Notation 6/22 • A PDA is based on a finite set of rules of the form: pushdown symbol states Aqaxp input symbol or  pushdown string

  7. New Concept: Regulated PDAs 7/22 • PDA M: 1.Ssa Sas 2.asa aas 3.asb q 4.aqb q 5.Sqc Sq 6.Sqc f •  = {12m34n5n6: m, n 0}

  8. Regulated PDAs 1/2 8/22 • PDA M: • Without , M • accepts aabbccc: 1.Ssa Sas 2.asa aas 3.asb q 4.aqb q 5.Sqc Sq 6.Sqc f Ssaabbccc  Sasabbccc[1]  Saasbbccc [2]  Saqbccc [3]  Sqccc[4]  Sqcc[5]  Sqc[5]  f[6]  = {12m34n5n6: m, n 0} L(M) = {anbncm: n, m 1}

  9. Regulated PDAs 2/2 9/22 • with, Mdoes notacceptaabbcccbecause • 1234556 = {12m34n5n6: m, n 0} • with, Maccepts aabbcc: Ssaabbcc  Sasabbcc[1]  Saasbbcc [2]  Saqbcc [3]  Sqcc [4]  Sqc[5]  f[6] and123456   L(M,) = {anbncn: n 1}

  10. Gist: Regulated PDAs 10/22 • Consider a pushdown automaton, M, and • control language, . • M accepts a string, x, if and only if  • contains a control string according to • which M makes a sequence of moves soit • reaches a final configuration after readingx.

  11. Definition: Regulated PDA 1/4 11/22 A pushdown automaton is a 7-tuple M = (Q, , , R, s, S, F), where • Q is a finite set of states, •  is an input alphabet, •  is a pushdown alphabet, • Ris a finiteset of rules of the form: • Apa  wq,where • A ,p,q Q, a    {},w  * • s Q is the start state • S is the start symbol • F  Q is a set of final states

  12. Definition: Regulated PDA 2/4 12/22 • Let  be an alphabet of rule labels. Let every • ruleApawqbelabeledwithauniqueas • . Apa  wq. • AconfigurationofM,,isanystringfrom*Q* • For every x *, y *, and . Apa  wq  R, • M makes a move from configuration xApay to • configuration xwqy according to , written as • xApay  xwqy []

  13. Definition: Regulated PDA 3/4 13/22 • Let  be any configuration of M. M makes zero • moves from  to  according to , writtenas •  0  [] • Let there exist a sequence of configurations • 0, 1, ..., nforsomen  1suchthati-1i[i], • wherei,fori= 1,...,n,thenMmakes n moves • from 0 to n according to [1 …n], written as • 0 n n [1... n]

  14. Definition: Regulated PDA 3/4 14/22 • If forsomen  0,0 n n [1... n], we write • 0 * n [1... n] • Letbeacontrollanguageover,thatis,*. • With , M accepts its language, L(M, ), as • L(M, ) = {w: w *,Ssw*f [],  }

  15. Language Families 15/22 • LIN- the family of linear languages • CF- the family of context-free languages • RE- the family of recursively enumerable • languages • RPD(REG)- the family of languages accepted • by PDAs regulated by regular • languages • RPD(LIN)- the family of languages accepted • by PDAs regulated by linear languages

  16. Theorem 1 and its Proof 1/2 16/22 RPD(REG) = CF Proof: I.CF RPD(REG) is clear. II.RPD(REG) CF: • Let L = L(M, ), PDA Regular language • Let  = L(G), G - regular grammar based • on rules: A  aB,A  a

  17. Theorem 1 and its Proof 2/2 17/22 Transform M regulated by  to a PDAN as follows: 1) for every a.Cqbxp from M and every AaB from G, add C<qA>b  x<pB> to N 2) for every a.Cqbxp from M and every Aa from G, add C<qA>b  x<pf> to N New symbol 3)The set of final states in N: {<pf>: p is a final state in M}

  18. Theorem 2 18/22 RPD(LIN) = RE Proof: • See [Meduna Alexander, Kolář Dušan: • Regulated Pushdown Automata,Acta Cybernetica,Vol. 2000, No. 4, p. 653-664]

  19. Simplification of RPDAs 1/2 19/22 • I. consider two consecutive moves made by a pushdown automaton, M. • If during the first moveM does not shorten its pushdown and during the second move it does, then M makes a turn during the second move. • A pushdown automatonis one-turn if it makes no more than one turn during any computation starting from an initial configuration.

  20. One-Turn PDA: Illustration 20/22 Length of pushdown One-turn Moves

  21. Simplification of RPDAs 2/2 21/22 II. During a move, an atomic regulated PDA changes a state and, in addition, performs exactly one of the following actions: 1. pushes a symbol onto the pushdown 2. pops a symbol from the pushdown 3. reads an input symbol

  22. Theorem 3 22/22 • Every L  RE is accepted by an atomic one-turn PDA regulated by , where   LIN. Proof: • See [Meduna Alexander, Kolář Dušan: • One-Turn Regulated Pushdown Automata and Their Reduction, Fundamenta Informatica,Vol. 2002, No. 16, p. 399-405] End

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