Semantics of a Relational {\lambda}-Calculus (Extended Version)

TL;DR

This paper introduces an extended { extlambda}-calculus with relational and logic programming features, ensuring confluence through a novel approach that avoids higher-order unification and supports a sound denotational semantics.

Contribution

It presents a new { extlambda}-calculus extension with non-determinism, unification, and freshness, using location annotations to maintain confluence without higher-order unification.

Findings

Confluent small-step operational semantics achieved.

Soundness of reduction with respect to denotational semantics established.

System supports relational and functional-logic programming constructs.

Abstract

We extend the {\lambda}-calculus with constructs suitable for relational and functional-logic programming: non-deterministic choice, fresh variable introduction, and unification of expressions. In order to be able to unify {\lambda}-expressions and still obtain a confluent theory, we depart from related approaches, such as {\lambda}Prolog, in that we do not attempt to solve higher-order unification. Instead, abstractions are decorated with a location, which intuitively may be understood as its memory address, and we impose a simple coherence invariant: abstractions in the same location must be equal. This allows us to formulate a confluent small-step operational semantics which only performs first-order unification and does not require strong evaluation (below lambdas). We study a simply typed version of the system. Moreover, a denotational semantics for the calculus is proposed and reduction is shown to be sound with respect to the denotational semantics.