A Generic Cyclic Theorem Prover

Abstract

Abstract. We describe the design and implementation of an automated theorem prover realising a fully general notion of cyclic proof. Our tool, called Cyclist, is able to construct proofs obeying a very general cycle scheme in which leaves may be linked to any other matching node in the proof, and to verify the general, global infinitary condition on such proof objects ensuring their soundness. Cyclist is based on a new, generic theory of cyclic proofs that can be instantiated to a wide variety of logics. We have developed three such concrete instantiations, based on: (a) first-order logic with inductive definitions; (b) entailments of pure separation logic; and (c) Hoare-style termination proofs for pointer programs. Experiments run on these instantiations indicate that Cyclist offers significant potential as a future platform for inductive theorem proving.

Disproving Inductive Entailments in Separation Logic via Base Pair Approximation

“…Our method for checking invalidity, using Game 3 and the optimisations given by Proposition 4.12 and Theorem 4.13, has been implemented in OCaml (openly available at [1]). We used the theorem prover Cyclist as the basis for our implementation, as it provides facilities for separation logic entailments with inductive predicates [13], including a procedure for computing the base pairs of formulas [10].…”

“…We have implemented our disproof algorithm in the Cyclist theorem proving framework [1,13]. Our experimental evaluation indicates that our disproof method can identify a significant proportion of invalid entailments arising in three different benchmark suites, and that it is inexpensive on average.…”

Cyclic Abduction of Inductively Defined Safety and Termination Preconditions

“…An alternative would be to use (Cut) with an entailment theorem prover to establish the required logical entailment F F (such steps are needed for some proofs). We did implement such a tactic, calling on the separation logic entailment prover in Cyclist [11], but found the costs to be prohibitive in the absence of a sophisticated lemma speculation mechanism.…”

“…Caber is built on top of the open-source theorem prover Cyclist, a generic framework for constructing cyclic theorem provers [11]. It essentially provides an instantiation of the proof system in Section 3 (based on an earlier version in [11]), and an abductive proof search algorithm implementing the tactics in Section 4. Safety versus termination is handled via a prover switch.…”

“…We have implemented our abduction procedure as an automatic tool, Caber, that builds on the generic cyclic theorem prover Cyclist [11]. Caber is able to automatically abduce safety and/or termination preconditions for a fairly wide variety of common small programs, including the majority of those tested in the Mutant tool, where the (list-based) preconditions previously had to be supplied by hand [2].…”