Alternating two-way AC-tree automata

We explore the notion of alternating two-way tree automata modulo the theory of finitely many associative-commutative (AC) symbols. This was prompted by questions arising in cryptographic protocol verification, in particular in modeling group key agreement schemes based on Diffie-Hellman-like functions, where the emptiness question for intersections of such automata is fundamental. This also has independent interest. We show that the use of general push clauses, or of alternation, leads to undecidability, already in the case of one AC symbol, with only functions of arity zero. On the other hand, emptiness is decidable in the general case of several function symbols, including several AC symbols, provided push clauses are unconditional and intersection clauses are final. This class of automata is also shown to be closed under intersection.

[1]  Jean Goubault-Larrecq,et al.  Higher-Order Positive Set Constraints , 2002, CSL.

[2]  S. Ginsburg,et al.  Semigroups, Presburger formulas, and languages. , 1966 .

[3]  Elaine J. Weyuker,et al.  Computability, complexity, and languages , 1983 .

[4]  Florent Jacquemard,et al.  Ground reducibility is EXPTIME-complete , 2003, Inf. Comput..

[5]  Gene Tsudik,et al.  Key Agreement in Dynamic Peer Groups , 2000, IEEE Trans. Parallel Distributed Syst..

[6]  Denis Lugiez,et al.  A Good Class of Tree Automata and Application to Inductive Theorem Proving , 1998, ICALP.

[7]  Jean-Luc Lambert,et al.  A Structure to Decide Reachability in Petri Nets , 1992, Theor. Comput. Sci..

[8]  J. Cheney,et al.  A sequent calculus for nominal logic , 2004, LICS 2004.

[9]  Ernst W. Mayr.  An Algorithm for the General Petri Net Reachability Problem , 1984, SIAM J. Comput..

[10]  Muriel Roger.  Raffinements de la résolution et vérification de protocoles cryptographiques , 2003 .

[11]  Thomas Schwentick,et al.  Numerical document queries , 2003, PODS.

[12]  Grzegorz Rozenberg,et al.  Handbook of Formal Languages , 1997, Springer Berlin Heidelberg.

[13]  M. Minsky.  Recursive Unsolvability of Post's Problem of "Tag" and other Topics in Theory of Turing Machines , 1961 .

[14]  Denis Lugiez.  A Good Class of Tree Automata , 1998 .

[15]  Harald Ganzinger,et al.  Resolution Theorem Proving , 2001, Handbook of Automated Reasoning.

[16]  Tadao Kasami,et al.  Solving a Unification Problem under Constrained Substitutions Using Tree Automata , 1994, FSTTCS.

[17]  Denis Lugiez,et al.  Counting and Equality Constraints for Multitree Automata , 2003, FoSSaCS.

[18]  John E. Hopcroft,et al.  On the Reachability Problem for 5-Dimensional Vector Addition Systems , 1976, Theor. Comput. Sci..

[19]  Robert E. Shostak,et al.  Deciding Linear Inequalities by Computing Loop Residues , 1981, JACM.

[20]  Maurice Nivat,et al.  Resolution of Equations in Algebraic Structures , 1989 .

[21]  Christoph M. Hoffmann,et al.  Pattern Matching in Trees , 1982, JACM.

[22]  Mark E. Stickel,et al.  A Unification Algorithm for Associative-Commutative Functions , 1981, JACM.

[23]  Denis Lugiez,et al.  Tree Automata Help One To Solve Equational Formulae In AC-Theories , 1994, J. Symb. Comput..

[24]  Hubert Comon,et al.  Tree automata techniques and applications , 1997 .

[25]  Elaine J. Weyuker,et al.  Computability, complexity, and languages - fundamentals of theoretical computer science , 2014, Computer science and applied mathematics.

[26]  Kumar Neeraj Verma,et al.  Two-Way Equational Tree Automata for AC-Like Theories: Decidability and Closure Properties , 2003, RTA.

[27]  Kumar Neeraj Verma.  Automates d'arbres bidirectionnels modulo théories équationnelles , 2003 .

[28]  Andrei Voronkov,et al.  Algorithms, Datastructures, and other Issues in Efficient Automated Deduction , 2001, IJCAR.

[29]  Jean Goubault-Larrecq,et al.  Karp-Miller Trees for a Branching Extension of VASS , 2005, Discret. Math. Theor. Comput. Sci..

[30]  Jean Goubault-Larrecq,et al.  A Method for Automatic Cryptographic Protocol Verification , 2000, IPDPS Workshops.

[31]  Martín Abadi,et al.  Security analysis of cryptographically controlled access to XML documents , 2005, PODS '05.

[32]  Thomas Genet,et al.  Rewriting for Cryptographic Protocol Verification , 2000, CADE.

[33]  Andreas Podelski,et al.  Set-Based Analysis of Reactive Infinite-State Systems , 1998, TACAS.

[34]  K. Verma,et al.  Alternating Two − Way AC − Tree Automata Research Report LSV , 2002 .

[35]  M. Rusinowitch,et al.  Reachability is decidable for ground AC Rewrite systems , 1998 .

[36]  B. Courcelle.  On Recognizable Sets and Tree Automata , 1989 .

[37]  François Fages.  Associative-Commutative Unification , 1987, J. Symb. Comput..

[38]  Michael T. Goodrich,et al.  Implementation of an authenticated dictionary with skip lists and commutative hashing , 2001, Proceedings DARPA Information Survivability Conference and Exposition II. DISCEX'01.

[39]  Rohit Parikh,et al.  On Context-Free Languages , 1966, JACM.

[40]  Hitoshi Ohsaki,et al.  Beyond Regularity: Equational Tree Automata for Associative and Commutative Theories , 2001, CSL.

[41]  Wilhelm Ackermann,et al.  Solvable Cases Of The Decision Problem , 1954 .

[42]  Joachim Niehren,et al.  Feature Automata and Recognizable Sets of Feature Trees , 1993, TAPSOFT.

[43]  S. Rao Kosaraju,et al.  Decidability of reachability in vector addition systems (Preliminary Version) , 1982, STOC '82.

[44]  Richard C. T. Lee,et al.  Symbolic logic and mechanical theorem proving , 1973, Computer science classics.

[45]  Nicolas Peltier,et al.  Tree Automata and Automated Model Building , 1997, Fundam. Informaticae.

[46]  Alberto Martelli,et al.  An Efficient Unification Algorithm , 1982, TOPL.

[47]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[48]  Jean H. Gallier,et al.  Tree Pushdown Automata , 1985, J. Comput. Syst. Sci..

[49]  Richard M. Karp,et al.  Parallel Program Schemata , 1969, J. Comput. Syst. Sci..

[50]  Kumar Neeraj Verma.  On Closure under Complementation of Equational Tree Automata for Theories Extending AC , 2003, LPAR.

[51]  G. Denker,et al.  CAPSL and MuCAPSL , 2002, Journal of Telecommunications and Information Technology.

[52]  David Monniaux.  Abstracting cryptographic protocols with tree automata , 2003, Sci. Comput. Program..

[53]  Whitfield Diffie,et al.  New Directions in Cryptography , 1976, IEEE Trans. Inf. Theory.

[54]  Philippe Schnoebelen,et al.  Well-structured transition systems everywhere! , 2001, Theor. Comput. Sci..

[55]  Toshinori Takai,et al.  Decidability and Closure Properties of Equational Tree Languages , 2002, RTA.

[56]  Ferenc Gécseg,et al.  Tree Languages , 1997, Handbook of Formal Languages.

[57]  Alexander Okhotin,et al.  Conjunctive Grammars , 2001, J. Autom. Lang. Comb..

[58]  Andrei Voronkov,et al.  Vampire 1.1 (System Description) , 2001, IJCAR.

[59]  Thomas Genet,et al.  Decidable Approximations of Sets of Descendants and Sets of Normal Forms , 1998, RTA.

[60]  Christophe Reutenauer,et al.  Aspects mathématiques des réseaux de pétri , 1988 .

[61]  Richard L. Tenney,et al.  The decidability of the reachability problem for vector addition systems (Preliminary Version) , 1977, STOC '77.

[62]  Jean-Marc Talbot,et al.  Automata and Logics for Unranked and Unordered Trees , 2005, RTA.

[63]  Helmut Seidl,et al.  Haskell Overloading is DEXPTIME-Complete , 1994, Inf. Process. Lett..

[64]  Sophie Tison,et al.  Equality and Disequality Constraints on Direct Subterms in Tree Automata , 1992, STACS.

[65]  Christoph Weidenbach,et al.  Combining Superposition, Sorts and Splitting , 2001, Handbook of Automated Reasoning.

[66]  Thomas Schwentick,et al.  On the Complexity of Equational Horn Clauses , 2005, CADE.

[67]  Kenneth W. Regan,et al.  Computability , 2022, Algorithms and Theory of Computation Handbook.

[68]  William H. Joyner.  Resolution Strategies as Decision Procedures , 1976, JACM.

[69]  P. Gács,et al.  Algorithms , 1992 .

[70]  Jean H. Gallier,et al.  Linear-Time Algorithms for Testing the Satisfiability of Propositional Horn Formulae , 1984, J. Log. Program..

[71]  Jean-Pierre Jouannaud,et al.  Rewrite Proofs and Computations , 1995 .

[72]  Giora Slutzki,et al.  Alternating Tree Automata , 1983, Theor. Comput. Sci..

[73]  Bruce Kallick.  A decision procedure based on the resolution method , 1968, IFIP Congress.

[74]  Danny Dolev,et al.  On the security of public key protocols , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[75]  John C. Shepherdson,et al.  The Reduction of Two-Way Automata to One-Way Automata , 1959, IBM J. Res. Dev..

[76]  Jean-Pierre Jouannaud,et al.  Automata-Driven Automated Induction , 2001, Inf. Comput..

[77]  Thom W. Frühwirth,et al.  Logic programs as types for logic programs , 1991, [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science.

[78]  Alan Robinson,et al.  The Inverse Method , 2001, Handbook of Automated Reasoning.

[79]  Wolfgang Thomas,et al.  Automata on Infinite Objects , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[80]  Oscar H. Ibarra,et al.  Counter Machines and Verification Problems , 2002, Theor. Comput. Sci..

[81]  Kumar Neeraj Verma,et al.  Alternation in Equational Tree Automata Modulo XOR , 2004, FSTTCS.

[82]  John Mitchell,et al.  Tree Automata with One Memory, Set Constraints, and Ping-Pong Protocols , 2001, ICALP.

[83]  Jean Goubault-Larrecq,et al.  Abstraction and resolution modulo AC: How to verify Diffie-Hellman-like protocols automatically , 2005, J. Log. Algebraic Methods Program..

[84]  Bruno Guillaume,et al.  Vector addition tree automata , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[85]  E. Allen Emerson,et al.  The Complexity of Tree Automata and Logics of Programs , 1999, SIAM J. Comput..

[86]  Harald Ganzinger,et al.  Set constraints are the monadic class , 1993, [1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science.