Classical notions of computation and the Hasegawa-Thielecke theorem (extended version)

TL;DR

This paper explores the semantics of classical logic with involutive negation using a polarised effect calculus, establishing a connection with the Hasegawa-Thielecke theorem through novel categorical structures.

Contribution

It introduces a new denotational semantics framework for classical logic with involutive negation, accommodating different evaluation strategies and proving the Hasegawa-Thielecke theorem in this setting.

Findings

Defined a non-associative categorical framework for semantics.

Interpreted the linear classical L-calculus in dialogue duploids.

Proved the Hasegawa-Thielecke theorem within this semantic setting.

Abstract

In the spirit of the Curry-Howard correspondence between proofs and programs, we define and study a syntax and semantics for classical logic equipped with a computationally involutive negation, using a polarised effect calculus, the linear classical L-calculus. A main challenge in designing a denotational semantics for the calculus is to accommodate both call-by-value and call-by-name evaluation strategies, which leads to a failure of associativity of composition. In order to tackle this issue, we define a notion of adjunction between graph morphisms on non-associative categories, which we use to formulate polarized and non-associative notions of symmetric monoidal closed duploid and of dialogue duploid. We show that they provide a direct style counterpart to adjunction models: linear effect adjunctions for the (linear) call-by-push-value calculus and dialogue chiralities for linear continuations, respectively. In particular, we show that the syntax of the linear classical L-calculus can be interpreted in any dialogue duploid, and that it defines in fact a syntactic dialogue duploid. As an application, we establish, by semantic as well as syntactic means, the Hasegawa-Thielecke theorem, which states that the notions of central map and of thunkable map coincide in any dialogue duploid (in particular, for any double negation monad on a symmetric monoidal category).