Disjoint Equational Theories Research Articles

Decidability and Combination Results for Two Notions of Knowledge in Security Protocols

In formal approaches, messages sent over a network are usually modeled by terms together with an equational theory, axiomatizing the properties of the cryptographic functions (encryption, exclusive or, ...). The analysis of cryptographic protocols requires a precise understanding of the attacker knowledge. Two standard notions are usually considered: deducibility and indistinguishability. Those notions are well-studied and several decidability results already exist to deal with a variety of equational theories. Most of the existing results are dedicated to specific equational theories and only few results, especially in the case of indistinguishability, have been obtained for equational theories with associative and commutative properties $(\textsf{AC})$ . In this paper, we show that existing decidability results can be easily combined for any disjoint equational theories: if the deducibility and indistinguishability relations are decidable for two disjoint theories, they are also decidable for their union. We also propose a general setting for solving deducibility and indistinguishability for an important class (called monoidal) of equational theories involving $\textsf{AC}$ operators. As a consequence of these two results, new decidability and complexity results can be obtained for many relevant equational theories.

Tractable and intractable instances of combination problems for unification and disunification

The algorithms for combining decision procedures for (dis)unification in the union of arbitrary disjoint equational theories known from the literature generate an exponential number of 'pure' output problems over the component theories. Despite several optimization techniques, no general results are known that describe in which cases a polynomial-time combination procedure is (im)possible. In the first part of this paper we characterize a large class of equational theories E where E-unification with free function symbols is NP-hard and where a polynomial-time optimization of the combination algorithm is impossible (unless P = NP). This supports the conjecture that for the theories in this class there is no polynomial-time algorithm for combining E-unification with constants with free unification. In the second part of the paper we introduce another class of equational theories where (dis)unification algorithms can be combined deterministically and in polynomial time. Under reasonable assumptions on the form of the given unification procedure, each unitary, regular and collapse-free equational theory with a polynomial-time unification algorithm belongs to this class.

Combining Decision Algorithms for Matching in the Union of Disjoint Equational Theories

This paper addresses the problem of systematically building a matching algorithm for the union of two disjoint theoriesE1∪E2provided that matching algorithms are known in both theoriesE1andE2. In general, the blind use of combination techniques introduces unification. Two different restrictions are considered in order to reduce this unification to matching. First, we show that combining matching algorithms (with linear constant restriction) is always sufficient for solving a pure fragment of combined matching problems. Second, the investigated method is complete for the largest class of theories where unification is not needed, including regular collapse-free theories and linear theories. Syntactic conditions are given to define this class of theories in which solving the combined matching problem is performed in a modular way.

Combination techniques and decision problems for disunification

Previous work on combination techniques considered the question of how to combine unification algorithms for disjoint equational theories E 1,…, E n in order to obtain a unification algorithm for the union E 1 ∪ ⋯ ∪ E n of the theories. Here we want to show that variants of this method may be used to decide solvability and ground solvability of disunification problems in E 1 ∪ ⋯ ∪ E n . Our first result says that solvability of disunification problems in the free algebra of the combined theory E 1 ∪ ⋯ ∪ E n is decidable if solvability of disunification problems with linear constant restrictions in the free algebras of the theories E i ( i = 1,…, n) is decidable. In order to decide ground solvability (i.e., solvability in the initial algebra) of disunification problems in E 1 ∪ ⋯ ∪ E n we have to consider a new kind of subproblem for the particular theories E i , namely solvability (in the free algebra) of disunification problems with linear constant restriction under the additional constraint that values of variables are not E i -equivalent to variables. The correspondence between ground solvability and this new kind of solvability holds, (1) if one theory E i is the free theory with at least one function symbol and one constant, or (2) if the initial algebras of all theories E i are infinite. Our results can be used to show that the existential fragment of the theory of the (ground) term algebra modulo associativity of a finite number of function symbols is decidable; the same result follows for function symbols which are associative and commutative, or associative, commutative and idempotent.

Combining Symbolic Constraint Solvers on Algebraic Domains

In the context of constraint logic programming and theorem proving, the development of constraint solvers on algebraic domains and their combination is of prime interest. As an example, a constraint solver in finite algebras is presented for a constraint language including for instance equations, disequations and inequations. By extending techniques used for the combination of unification in disjoint equational theories, we show how to combine constraint solvers on different algebraic domains that may share some constant symbols. We illustrate this technique by combining the constraint solver in finite algebras with other unification algorithms, and with another constraint solver on a different finite algebra.

Unification in a combination of arbitrary disjoint equational theories

The unification problem in a disjoint combination of equational theories, E 1 +...+E n , is reduced to a combination of two kinds of problems in the E j 's : the pure unification, problem with free (Uninterpreted) constants and the constant-elimination problem. The constant-elimination problem is to find, given terms t i , 1≤i≤m and free constants c i , 1≤i≤m , all substitutions σ such that for all i with 1≤i≤m , σt i is equal to some t′ i that does not contain c i . The soundness and completeness of the method shows, that a disjoint combination of theories is finitary, provided every theory is finitary and constant-elimination problems in every theory are finitary solvable. In particular, any combination of finitary unifying regular theories, of Boolean rings, of Abelian groups or of theories with terminating basic narrowing, is of unification-type finitary. As a special case, unification in a combination of a free Boolean ring with free function symbols is decidable and finitary; the same holds for Abelian groups. This paper extends the known results on unification in a combination of regular and collapse-free theories in two aspects: i) Arbitrary theories are admissable. ii) The described general procedure can be used for enumerating a complete set of unifiers, even if some unification procedure for a paticular theory produces an infinite complete set of unifiers. It is demonstrated that unifiability of unification problems is decidable in the general case E 1 +...+E n if for every i∈{1,…,n} , there is a method to decide unification problems in a combination of E i with free function symbols.