Level 0/1 Prover: A Tutorial

This paper gives an overview of a prototype implementation of a fragment of the logic Linc [5, 8, 13] (also referred to as FOλ∆∇ in [5]), which we tentatively call ‘Level 0/1 Prover’ here. This implementation is part of a larger project, Parsifal, at INRIA Futurs (see http://www.lix.polytechnique.fr/~dale/parsifal). The logic Linc is an extension of first-order intuitionistic logic with a proof-theoretic notion of definitions [3, 11, 2, 4], proof rules for induction and co-induction and a new quantifier ∇. Our aim is to use Linc as a logic for specifying and reasoning about various operational semantics of computation systems. It is not necessary to go through the detailed description of the logic in order to use the prover; this is the intent of this tutorial, of course. For those who are interested, details of the logic are available in the related papers on Linc mentioned previously. We assume that users are familiar with Prolog, or better, λProlog, as we follow closely the syntax of both languages, especially λProlog. The current implementation is very much experimental so we choose to restrict the features of the language to purely logical ones, that is, most non-logical features commonly found in Prolog-like languages, e.g., arithmetic, string processing, I/O, etc., are not implemented. This tutorial is organized as follows. Section 2 gives an overview of the language of the prover and its operational semantics (informally), along with some examples illustrating its differences with λProlog. Section 3 shows how to configure and run the prover. Section 4 discusses the treatment of eigenvariables in Level 0/1 prover, and shows how it is different from the use of eigenvariables in λProlog. This section also shows how to give a logically sound interpretation to certain non-logical features in Prolog or λProlog. Section 5 discusses a non-logical feature of Level 0/1 prover which allows users to perform analysis on a logic program. What it does is basically transforming a logic program into a certain abstract syntax, called λ-tree syntax, and vice versa. Section 6 shows how to use this feature to do type checking on logic programs. The type checking program itself would be just another logic program that is fed the λ-tree syntax of another program. Section 7 and Section 8 illustrate the use of Level 0/1 prover to reason about transition systems. The examples cover the specification and verification of Peterson’s algorithm (for guaranteeing mutual exclusion), and bisimulation of π-calculus. Section 9 mentions some possible directions for future work.