Reachability logic has been applied to K rewrite-rule-based language definitions as a language-generic logic of programs. To be able to verify not just code but also distributed system designs, a new rewritetheory-generic reachability logic is presented and proved sound for a wide class of rewrite theories. The logic's automation is increased by means of constructor-based semantic unification, matching, and satisfiability procedures. New methods for proving invariants of possibly never terminating distributed systems are developed, and experiments with a prototype implementation illustrating the new proof methods are presented.

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Abstract. Variant satisfiability is a theory-generic algorithm to decide quantifier-free satisfiability in an initial algebra T Σ{E when the theory pΣ, Eq has the finite variant property and its constructors satisfy a compactness condition. This paper: (i) gives a precise definition of several meta-level sub-algorithms needed for variant satisfiability; (ii) proves them correct; and (iii) presents a reflective implementation in Maude 2.7 of variant satisfiability using these sub-algorithms.

Variant satisfiability is a theory-generic algorithm to decide quantifier-free satisfiability in an initial algebra T Σ{E when the theory pΣ, Eq has the finite variant property and its constructors satisfy a compactness condition. This paper: (i) gives a precise definition of several meta-level sub-algorithms needed for variant satisfiability; (ii) proves them correct; and (iii) presents a reflective implementation in Maude 2.7 of variant satisfiability using these sub-algorithms.The material is adapted from [25,18, 28]. Due to space limitations the following elementary notions, which can be found in [25], are assume known: (i) ordersorted (OS) signature Σ; (ii) set p S of connected components (each denoted rss p S) of a poset of sorts pS, ¤q; (iii) sensible OS signature; (iv) order-sorted Σalgebras and homomorphisms, and its associated category OSAlg Σ ; and (v) the construction of the term algebra T Σ and its initiality in OSAlg Σ when Σ is sensible. Furthermore, for connected components rs 1 s, . . . , rs n s, rss p S, f rs1s...rsns rss tf : s I 1 . . . s I n Ñ s I Σ | s I i rs i s, 1 ¤ i ¤ n, s I rssu denotes the family of "subsort polymorphic" operators f . T Σ will (ambiguously) denote: (i) the term algebra; (ii) its underlying Ssorted set; and (iii) the set T Σ sS T Σ,s . For rss p S, T Σ,rss s I rss T Σ,s I.An OS signature Σ is said to have non-empty sorts iff for each s S, T Σ,s r.We will assume throughout that Σ has non-empty sorts. An OS signature Σ is called preregular [19] iff for each t T Σ the set ts S | t T Σ,s u has a least element, denoted lsptq. We will assume throughout that Σ is preregular.An S-sorted set X tX s u sS of variables, satisfies s s I ñ X s X s I r, and the variables in X are always assumed disjoint from all constants in Σ. The Σ-term algebra on variables X, T Σ pXq, is the initial algebra for the signature ΣpXq obtained by adding to Σ the variables X as extra constants. Since a ΣpXqalgebra is just a pair pA, αq, with A a Σ-algebra, and α an interpretation of the constants in X, i.e., an S-sorted function α rXÑAs, the ΣpXq-initiality of T Σ pXq can be expressed as the following theorem: Theorem 1. (Freeness Theorem). If Σ is sensible, for each A OSAlg Σ and α rXÑAs, there exists a unique Σ-homomorphism, α : T Σ pXq Ñ A extending α, i.e., such that for each s S and x X s we have xα s α s pxq.In particular, when A T Σ pXq, an interpretation of the constants in X, i.e., an S-sorted function σ rXÑT Σ pXqs is called a substitution, and its unique homomorphic extension σ : T Σ pXq Ñ T Σ pXq is also called a substitution. Define dompσq tx X