A Semantics for Evaluation Logic (extended version)

This paper proposes an internal semantics for the modalities and evaluation predicate of Pitts’ Evaluation Logic, and introduces several predicate calculi (ranging from Horn sequents to Higher Order Logic), which are sound and complete w.r.t. natural classes of models. It is shown (by examples) that many computational monads satisfy the additional properties required by the proposed semantics. Introduction Evaluation logic ELT is a typed predicate logic (see [CP92, Pit91]) based on the metalanguage for computational monads MLT (a typed calculus introduced in [Mog91]), which permits statements about the evaluation of programs to values by the use of evaluation modalities. In particular, ELT might be used for axiomatising computation-related properties of a monad or devising computationally adequate theories (see [Pit91]), and it appears useful when addressing the question of logical principles for reasoning about the behaviour of programs. Ideally, ELT should provide a uniform framework for presenting programming languages and program logics (as in Scott’s and Milner’s LCF approach [GMW79], one should view programs as terms and assertions as formulas), which hopefully will support a modular approach to their description. This paper addresses the issue of finding general logical principles for evaluation modalities by following the same methodology used to find the equational axioms for MLT , i.e. first the categorical semantics, then sound and complete formal systems (see the introduction of [Mog91]). This issue is addressed also in [Pit91], but our approach differs from that by Pitts mainly in the categorical semantics of the evaluation modalities. In fact, our interpretation is uniquely determined by a strong monad T (but it is defined only if T satisfies some additional properties), while Pitts’ interpretation depends on some additional structure (which has to be found). However, there are important differences also at the level of logical principles, due to the fact that Pitts allows nonstandard semantics for formulas (e.g. when formulas over A are interpreted by subsets of A×S, rather than subsets of A), while we want to stick to standard semantics. By means of examples, we will show that there is no need to allow non-standard semantics of formulas, and that in our semantics the interpretation of evaluation modalities is “almost always the expected one”. The paper is organised as follows. Section 1 explains the intuition about ELT , presents a settheoretic semantics with a few simple examples, and discusses alternative semantics. Section 2 introduces several formal systems, that will be proved sound and complete w.r.t. suitable classes of categorical models, and establishes some definability results (see Theorem 2.12 and 2.13). Section 3 recalls the external and internal approaches to interpreting typed predicate logics. Section 4 defines our internal semantics for (the necessity modality of) ELT , establishes soundness results for those This work is supported by ESPRIT BRA 6811 (Categorical Logic In Computer Science II) and EC SCIENCE twinning ERBSC1*CT920795 (Progr. Lang. Semantics and Program Logics).