Unification in Boolean Rings and Abelian Groups

Abstract A complete unification algorithm is presented for the combination of two theories E in T(F,X) and E ’ in T(F’,X) where F and F ’ denote two disjoint sets of function symbols, E and E ’ are arbitrary equational theories for which are given, for E : a complete unification algorithm for terms in T(F ∪ C,X ), where C is a set of free constants and a complete constant elimination algorithm for eliminating a constant c from a term s; for E ’: a complete unification algorithm. E ’ is supposed to be cycle free, i.e., equations x=t where x is a variable occurring in t have no E ’-solution. The method adapts to unification of infinite trees. It is applied to two well-known open problems, when E is the theory of Boolean Rings or the theory of Abelian Groups, and E ’ is the free theory. Our interest to Boolean Rings originates in VLSI verification.

[1]  Michaël Rusinowitch,et al.  On Word Problems in Equational Theories , 1987, ICALP.

[2]  Manfred Schmidt-Schauß.  Unification in a Combination of Arbitrary Disjoint Equational Theories , 1988, CADE.

[3]  Tobias Nipkow,et al.  Unification in Boolean Rings , 1986, CADE.

[4]  Claude Kirchner.  Computing Unification Algorithms , 1986, LICS.

[5]  Erik Tidén.  Unification in Combinations of Collapse-Free Theories with Disjoint Sets of Function Symbols , 1986, CADE.

[6]  J. A. Robinson,et al.  A Machine-Oriented Logic Based on the Resolution Principle , 1965, JACM.

[7]  Mark E. Stickel.  A Complete Unification Algorithm for Associative-Commutative Functions , 1975, IJCAI.

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

[9]  Nachum Dershowitz,et al.  Rewrite Methods for Clausal and Non-Clausal Theorem Proving , 1983, ICALP.

[10]  Robert E. Shostak,et al.  Deciding Combinations of Theories , 1982, JACM.

[11]  Jean-Marie Hullot,et al.  Canonical Forms and Unification , 1980, CADE.

[12]  Jacques Herbrand.  Recherches sur la théorie de la démonstration , 1930 .

[13]  Franz Baader,et al.  Characterization of Unification Type Zero , 1989, RTA.

[14]  Katherine A. Yelick,et al.  Combining Unification Algorithms for Confined Regular Equational Theories , 1985, RTA.

[15]  M. Schmidt-Schauβ.  Unification in a combination of arbitrary disjoint equational theories , 1989 .

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

[17]  Nachum Dershowitz,et al.  Orderings for Equational Proofs , 1986, LICS.

[18]  Yoshihito Toyama,et al.  On the Church-Rosser property for the direct sum of term rewriting systems , 1984, JACM.

[19]  Jörg H. Siekmann,et al.  A short survey on the state of the art in matching and unification problems , 1979, SIGS.

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

[21]  Wayne Snyder,et al.  A General Complete E-Unification Procedure , 1987, RTA.

[22]  Hans-Jürgen Bürckert.  Matching - A Special Case of Unification? , 1989, J. Symb. Comput..

[23]  Wolfram Büttner,et al.  Embedding Boolean Expressions into Logic Programming , 1987, J. Symb. Comput..