Term Algebras with Length Function and Bounded Quantifier Alternation

Term algebras have wide applicability in computer science. Unfortunately, the decision problem for term algebras has a nonelementary lower bound, which makes the theory and any extension of it intractable in practice. However, it is often more appropriate to consider the bounded class, in which formulae can have arbitrarily long sequences of quantifiers but the quantifier alternation depth is bounded. In this paper we present new quantifier elimination procedures for the first-order theory of term algebras and for its extension with integer arithmetic. The elimination procedures deal with a block of quantifiers of the same type in one step. We show that for the bounded class of at most k quantifier alternations, regardless of the total number of quantifiers, the complexity of our procedures is k-fold exponential (resp. 2k fold exponential) for the theory of term algebras (resp. for the extended theory with integers).

[1]  C. Ward Henson,et al.  A Uniform Method for Proving Lower Bounds on the Computational Complexity of Logical Theories , 1990, Ann. Pure Appl. Log..

[2]  Henny B. Sipma,et al.  Decision Procedures for Recursive Data Structures with Integer Constraints , 2004, IJCAR.

[3]  Kenneth Kunen,et al.  Negation in Logic Programming , 1987, J. Log. Program..

[4]  Rolf Backofen.  A Complete Axiomatization of a Theory with Feature and Arity Constraints , 1995, J. Log. Program..

[5]  Hubert Comon-Lundh,et al.  Equational Formulae with Membership Constraints , 1994, Inf. Comput..

[6]  Martin Fürer,et al.  The Complexity of Presburger Arithmetic with Bounded Quantifier Alternation Depth , 1982, Theor. Comput. Sci..

[7]  Herbert B. Enderton,et al.  A mathematical introduction to logic , 1972 .

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

[9]  Andrei Voronkov,et al.  Knuth--bendix constraint solving is NP-complete , 2002, TOCL.

[10]  L. Lovász.  Combinatorial problems and exercises , 1979 .

[11]  Derek C. Oppen,et al.  Reasoning about recursively defined data structures , 1978, POPL.

[12]  Andrei Voronkov,et al.  A decision procedure for term algebras with queues , 2001, TOCL.

[13]  Wilfrid Hodges,et al.  Model Theory: The existential case , 1993 .

[14]  J. Ferrante,et al.  The computational complexity of logical theories , 1979 .

[15]  Sergei G. Vorobyov.  An Improved Lower Bound for the Elementary Theories of Trees , 1996, CADE.

[16]  Andrei Voronkov,et al.  A decision procedure for the existential theory of term algebras with the Knuth-Bendix ordering , 2000, Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332).

[17]  Viktor Kuncak,et al.  On the Theory of Structural Subtyping , 2004, ArXiv.

[18]  Michael J. Maher.  Complete axiomatizations of the algebras of finite, rational and infinite trees , 1988, [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science.

[19]  Donald W. Loveland,et al.  Presburger arithmetic with bounded quantifier alternation , 1978, STOC.

[20]  Viktor Kuncak,et al.  Structural subtyping of non-recursive types is decidable , 2003, 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings..

[21]  Hubert Comon-Lundh,et al.  Equational Problems and Disunification , 1989, J. Symb. Comput..

[22]  A. I. Malcev.  Axiomatizable classes of locally free algebras of various types , 1971 .

[23]  Henny B. Sipma,et al.  The Decidability of the First-Order Theory of Knuth-Bendix Order , 2005, CADE.

[24]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[25]  Andrei Voronkov,et al.  Knuth-Bendix Constraint Solving Is NP-Complete , 2001, ICALP.