In the days of its foundation, the field of science covered by UNIF – a series of annual international workshops on unification – was still in its infancy. With the advent of automated reasoning, term rewriting, logic programming, natural language processing, and program analysis, the areas of computer science concerned by unification were seething with excitement. With the coming out of researches in constraint solving and admissibility of inference rules and with the breaking out of applications, such as type checking, query answering, and cryptographic protocol analysis, the development of unification was not long in going at full speed.

Ruitenburg’s Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ultimately periodic if f fixes all the generators but one. More precisely, there is N ≥ 0 such that fN+2 = fN, thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms of free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators.

The theory ACUI of an associative, commutative, and idempotent binary function symbol + with unit 0 was one of the first equational theories for which the complexity of testing solvability of unification problems was investigated in detail. In this paper, we investigate two extensions of ACUI. On one hand, we consider approximate ACUI-unification, where we use appropriate measures to express how close a substitution is to being a unifier. On the other hand, we extend ACUI-unification to ACUIG-unification, that is, unification in equational theories that are obtained from ACUI by adding a finite set G of ground identities. Finally, we combine the two extensions, that is, consider approximate ACUI-unification. For all cases we are able to determine the exact worst-case complexity of the unification problem.

We consider anti-unification for simply typed lambda terms in theories defined by associativity, commutativity, identity (unit element) axioms and their combinations and develop a sound and complete algorithm which takes two lambda terms and computes their equational generalizations in the form of higher-order patterns. The problem is finitary: the minimal complete set of such generalizations contains finitely many elements. We define the notion of optimal solution and investigate special restrictions of the problem for which the optimal solution can be computed in linear or polynomial time.

We consider the problem of the unification modulo an equational theory associativity and commutativity (ACh), which consists of a function symbol h that is homomorphic over an associative–commutative operator +. Since the unification modulo ACh theory is undecidable, we define a variant of the problem called bounded ACh unification. In this bounded version of ACh unification, we essentially bound the number of times h can be applied to a term recursively and only allow solutions that satisfy this bound. There is no bound on the number of occurrences of h in a term, and the + symbol can be applied an unlimited number of times. We give inference rules for solving the bounded version of the problem and prove that the rules are sound, complete, and terminating. We have implemented the algorithm in Maude and give experimental results. We argue that this algorithm is useful in cryptographic protocol analysis.

We study decision procedures for two knowledge problems critical to the verification of security protocols, namely the intruder deduction and the static equivalence problems. These problems can be related to particular forms of context matching and context unification. Both problems are defined with respect to an equational theory and are known to be decidable when the equational theory is given by a subterm convergent term rewrite system (TRS). In this work, we extend this to consider a subterm convergent TRS defined modulo an equational theory, like Commutativity. We present two pairs of solutions for these important problems. The first solves the deduction and static equivalence problems in rewrite systems modulo shallow theories such as Commutativity. The second provides a general procedure that solves the deduction and static equivalence problems in subterm convergent systems modulo syntactic permutative theories, provided a finite measure is ensured. Several examples of such theories are also given.

We consider matching, rewriting, critical pairs and the Knuth–Bendix confluence test on rewrite rules in a nominal setting extended by atom-variables. We utilize atom-variables instead of atoms to formulate and rewrite rules on constrained expressions, which is an improvement of expressiveness over previous approaches. Nominal unification and nominal matching are correspondingly extended. Rewriting is performed using nominal matching, and computing critical pairs is done using nominal unification. We determine the complexity of several problems in a quantified freshness logic. In particular we show that nominal matching is $$\prod _2^p$$ -complete. We prove that the adapted Knuth–Bendix confluence test is applicable to a nominal rewrite system with atom-variables, and thus that there is a decidable test whether confluence of the ground instance of the abstract rewrite system holds. We apply the nominal Knuth–Bendix confluence criterion to the theory of monads and compute a convergent nominal rewrite system modulo alpha-equivalence.