Nominal Equational Problems Modulo Associativity, Commutativity and Associativity-Commutativity

The nominal syntax has been used in many application contexts for almost two decades. It is a powerful tool for dealing with variable binding in a concrete manner that can be applied to any specification in which parameters are used to abstract variables, such as in predicates and functions. In the nominal syntax, syntactically different objects can have the same semantics modulo alpha-conversion, as happens in the lambda calculus. Dealing with equality, and in special with alpha-equivalence, is essential in formal languages and implementations. This work investigates the nominal alpha-equivalence with associative (A), commutative (C) and associative-comutative (AC) function symbols. Equalitychecking, matching and unification modulo A, C and AC are investigated. Regarding equality-checking, nominal alpha-equivalence modulo A, C and AC are specified in Coq and proved sound. An algorithm implemented in OCaml for equality-checking modulo A, C and AC is automatically extracted from the specification and experiments are performed using also an improved algorithm. Upper bounds for solving nominal equality-checking problems are given. A rule-based nominal unification modulo C algorithm is specified in Coq and proved sound and complete. By using protected variables, this unification algorithm solves nominal matching problems modulo C, which is formalised to be sound and complete. The rule-based nominal unification algorithm outputs a finite family of sets of fixed point nominal equations. Each of which might have an infinite set of independent solutions. Therefore, nominal unification modulo C or AC are proved to potentially generate infinite independent solutions. This contrasts with syntactic unification modulo C or AC that are known to be in the finitary class. An OCaml implementation of the nominal unification algorithm is provided and used to build examples.