Equational methods in first order predicate calculus

We show that the application of the resolution principle to a set of clauses can be regarded as the construction of a term rewriting system confluent on valid formulas. This result allows the extension of usual properties and methods of equational theories (such as Birkhoff's theorem and the Knuth and Bendix completion algorithm) to quantifier-free first order theories. These results are extended to first order predicate calculus in an equational theory, as studied by Plotkin (1972), Slagle (1974) and Lankford (1975). This paper is a continuation of the work of Hsiang & Dershowitz (1983), who have already shown that rewrite methods can be used in first order predicate calculus. The main difference is the following: Hsiang uses rewrite methods only as a refutational proof technique, the initial set of formulas being unsatisfiable iff the equation TRUE = FALSE is generated by the completion algorithm. We generalise these methods to satisfiable theories; in particular, we show that the concept of confluent rewriting system, which is the main tool for studying equational theories, can be extended to any quantifier-free first order theory. Furthermore, we show that rewrite methods can be used even if formulas are kept in clausal form

Deduction modulo theory

This paper is a survey on Deduction modulo theor

Automated verification of refinement laws

Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back's atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs

UTP2: Higher-Order Equational Reasoning by Pointing

We describe a prototype theorem prover, UTP2, developed to match the style of hand-written proof work in the Unifying Theories of Programming semantical framework. This is based on alphabetised predicates in a 2nd-order logic, with a strong emphasis on equational reasoning. We present here an overview of the user-interface of this prover, which was developed from the outset using a point-and-click approach. We contrast this with the command-line paradigm that continues to dominate the mainstream theorem provers, and raises the question: can we have the best of both worlds?Comment: In Proceedings UITP 2014, arXiv:1410.785

Implicit complexity for coinductive data: a characterization of corecurrence

We propose a framework for reasoning about programs that manipulate coinductive data as well as inductive data. Our approach is based on using equational programs, which support a seamless combination of computation and reasoning, and using productivity (fairness) as the fundamental assertion, rather than bi-simulation. The latter is expressible in terms of the former. As an application to this framework, we give an implicit characterization of corecurrence: a function is definable using corecurrence iff its productivity is provable using coinduction for formulas in which data-predicates do not occur negatively. This is an analog, albeit in weaker form, of a characterization of recurrence (i.e. primitive recursion) in [Leivant, Unipolar induction, TCS 318, 2004].Comment: In Proceedings DICE 2011, arXiv:1201.034

Towards mechanized correctness proofs for cryptographic algorithms: Axiomatization of a probabilistic Hoare style logic

In [Corin, den Hartog in ICALP 2006] we build a formal verification technique for game based correctness proofs of cryptograhic algorithms based on a probabilistic Hoare style logic [den Hartog, de Vink in IJFCS 13(3), 2002]. An important step towards enabling mechanized verification within this technique is an axiomatization of implication between predicates which is purely semantically defined in [den Hartog, de Vink in IJFCS 13(3), 2002]. In this paper we provide an axiomatization and illustrate its place in the formal verification technique of [Corin, den Hartog in ICALP 2006]