A tutorial on computational classical logic and the sequent calculus

Abstract

We present a model of computation that heavily emphasizes the concept of duality and the interaction between opposites–production interacts with consumption. The symmetry of this framework naturally explains more complicated features of programming languages through relatively familiar concepts. For example, binding a value to a variable is dual to manipulating the flow of control in a program. By looking at the computational interpretation of the sequent calculus, we find a language that lets us speak about duality, control flow, and evaluation order in programs as first-class concepts.We begin by reviewing Gentzen's LK sequent calculus and show how the Curry–Howard isomorphism still applies to give us a different basis for expressing computation. We then illustrate how the fundamental dilemma of computation in the sequent calculus gives rise to a duality between evaluation strategies: strict languages are dual to lazy languages. Finally, we discuss how the concept of focusing, developed in the setting of proof search, is related to the idea of type safety for computation expressed in the sequent calculus. In this regard, we compare and contrast two different methods of focusing that have appeared in the literature, static and dynamic focusing, and illustrate how they are two means to the same end.

Classical (Co)Recursion: Mechanics

“…As such, the built-in negation in every e ÷ A (or α ÷ A) can be removed by swapping between the left-and right-hand sides of the turnstyle (⊢), so that e ÷ A on the right becomes e : A on the left, and α ÷ A on the left becomes α : A on the right. Doing so gives a conventional two-sided sequent calculus as in (Ariola et al, 2009;Downen & Ariola, 2018b), where the rules labeled L with conclusions of the form x i : B i , α j ÷ C j ⊢ e ÷ A correspond to left rules of the form x i : B i | e : A ⊢ α j : C j in the sequent calculus.…”

“…Section 3 presents an abstract machine for both call-by-value and -by-name evaluation, and unifies both into a single presentation (Ariola et al, 2009;Downen & Ariola, 2018b). The lower level nature of the abstract machine explicitly expresses how the recursor of inductive types, like numbers, accumulates a continuation during evaluation, maintaining the progress of recursion.…”

Compiling With Classical Connectives

“…In the realm of logic, Gentzen's sequent calculus [Gen35] provides a clear window to view the symmetry and duality of classical deduction. And through the Curry-Howard correspondence [Wad15], the sequent calculus can be seen as a programming language for bringing out many symmetries that are otherwise hidden in computation [DA18b]. For example, there are dual calling conventions like call-by-value versus call-by-name [CH00,Wad03] and dual programming constructs like functions versus structures [Zei08,MM09].…”

“…Rather than impose a static focusing restriction on the syntax of programs, we instead imply a dynamic focusing behavior-which evaluates the parameters of constructors and observers before (co)pattern matching-during execution. Both static and dynamic notions of focusing are two sides of the same coin, and amount to the same end [DA18b].…”

Dependent Pearl: Normalization by realizability

“…It is known that this presentation is non-confluent; as we explained, the languages of realizers can have very wild computational behaviors. People working with µ μ directly, instead of as a language of realizers, restrict the language to regain confluence by typing typing (enforcing normalization), evaluation strategies [Downen and Ariola 2018] or polarization [Munch-Maccagnoni 2013].…”