Handling Transitive Relations in First-Order Automated Reasoning

We present a number of alternative ways of handling transitive binary relations that commonly occur in first-order problems, in particular equivalence relations, total orders, and transitive relations in general. We show how such relations can be discovered syntactically in an input theory, and how they can be expressed in alternative ways. We experimentally evaluate different such ways on problems from the TPTP, using resolution-based reasoning tools as well as instance-based tools. Our conclusions are that (1) it is beneficial to consider different treatments of binary relations as a user, and that (2) reasoning tools could benefit from using a preprocessor or even built-in support for certain types of binary relations

Proof Automation in the Theory of Finite Sets and Finite Set Relation Algebra

{log} ('setlog') is a satisfiability solver for formulas of the theory of finite sets and finite set relation algebra (FSTRA). As such, it can be used as an automated theorem prover (ATP) for this theory. {log} is able to automatically prove a number of FSTRA theorems, but not all of them. Nevertheless, we have observed that many theorems that {log} cannot automatically prove can be divided into a few subgoals automatically dischargeable by {log}. The purpose of this work is to present a prototype interactive theorem prover (ITP), called {log}-ITP, providing evidence that a proper integration of {log} into world-class ITP's can deliver a great deal of proof automation concerning FSTRA. An empirical evaluation based on 210 theorems from the TPTP and Coq's SSReflect libraries shows a noticeable reduction in the size and complexity of the proofs with respect to Coq

TLA+ Model Checking Made Symbolic

International audienceTLA + is a language for formal specification of all kinds of computer systems. System designers use this language to specify concurrent, distributed, and fault-tolerant protocols, which are traditionally presented in pseudo-code. TLA + is extremely concise yet expressive: The language primitives include Booleans, integers, functions, tuples, records, sequences, and sets thereof, which can be also nested. This is probably why the only model checker for TLA + (called TLC) relies on explicit enumeration of values and states. In this paper, we present APALACHE-a first symbolic model checker for TLA +. Like TLC, it assumes that all specification parameters are fixed and all states are finite structures. Unlike TLC, APALACHE translates the underlying transition relation into quantifier-free SMT constraints, which allows us to exploit the power of SMT solvers. Designing this translation is the central challenge that we address in this paper. Our experiments show that APALACHE outperforms TLC on examples with large state spaces

A decision procedure for sets, binary relations and partial functions

In this paper we present a decision procedure for sets, binary relations and partial functions. The language accepted by the decision procedure includes untyped, hereditarily finite sets, where some of their elements can be variables, and basically all the classic set and relational operators used in formal languages such as B and Z. Partial functions are encoded as binary relations which in turn are just sets of ordered pairs. Sets are first-class entities in the language, thus they are not encoded in lower level theories. The decision procedure exploits set unification and set constraint solving as primitive features. The procedure is proved to be sound, complete and terminating. A Prolog implementation is presented