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Many inductive theorem provers generate induction schemes from the recursive calls appearing in terminating functions defined recursively in the specification. This strategy is called coverset induction in the context of algebraic specifications, and has been shown to be quite useful in a wide variety of applications. One challenge is that coverset induction typically requires using a total recursive function, while many operations are only meaningful on a subset of their inputs. A generalization of coverset induction method that supports partial constructors and operations specified in membership equational logic is proposed. The generalization has been implemented in the Maude ITP, and used to perform an extensive case study involving powerlists — a data structure introduced by J. Misra to elegantly formalize parallel algorithms based on divide and conquer strategy. Powerlists are constructed by partial operations, and it has been a challenge to naturally reason about powerlists using a formal logic that only supports total operations. We show how theorems about powerlists can be elegantly proven using the generalized coverset induction scheme implemented in the Maude ITP.

We introduce a logic of functional fixed-points. It is suitable for analyzing heap-manipulating programs and can encode several logics used for program verification with different ways of expressing reachability. While full fixed-point logic remains undecidable, several subsets admit decision procedures. In particular, for the logic of linear functional fixed-points, we develop an abstraction refinement integration of the SMT solver Z3 and a satisfiability checker for propositional linear-time temporal logic. The integration refines the temporal abstraction by generating safety formulas until the temporal abstraction is unsatisfiable or a model for it is also a model for the functional fixed-point formula.

In the paper, we introduce a new tree automata framework, called propositional tree automata, capturing the class of tree languages that are closed under an equational theory and Boolean operations. This framework originates in work on developing a sufficient completeness checker for specifications with rewriting modulo an equational theory. Propositional tree automata recognize regular equational tree languages. However, unlike regular equational tree automata, the class of propositional tree automata is closed under Boolean operations. This extra expressiveness does not affect the decidability of the membership problem. This paper also analyzes in detail the emptiness problem for propositional tree automata with associative theories. Though undecidable in general, we present a semi-algorithm for checking emptiness based on machine learning that we have found useful in practice.

We present a tool for checking the sufficient completeness of left-linear, order-sorted equational specifications modulo associativity, commutativity, and identity. Our tool treats this problem as an equational tree automata decision problem using the tree automata library CETA, which we also introduce in this paper. CETA implements a semi-algorithm for checking the emptiness of a class of tree automata that are closed under Boolean operations and an equational theory containing associativity, commutativity and identity axioms. Though sufficient completeness for this class of specifications is undecidable in general, our tool is a decision procedure for subcases known to be decidable, and has specialized techniques that are effective in practice for the general case.

We present the Maude sufficient completeness tool, which explicitly supports sufficient completeness reasoning for partial conditional specifications having sorts and subsorts and with domains of functions defined by conditional memberships. Our tool consists of two main components: (i) a sufficient completeness analyzer that generates a set of proof obligations which if discharged, ensures sufficient completeness; and (ii) Maude’s inductive theorem prover (ITP) that is used as a backend to try to automatically discharge those proof obligations.

This paper presents a rule-based algorithm for performing order-sorted E-unification using an unsorted E-unification decision procedure under assumptions about E that are commonly satisfied in practice. We have implemented this algorithm in Maude for use with the Maude-NRL protocol analyzer and have used CiME for unsorted E-unification for E any set of AC and ACU axioms. In many examples of interest, using order-sorted unification over unsorted unification is able to reduce the total number of unifiers considered, and dramatically improve the performance of the Maude-NRL tool.