Classical and Intuitionistic Subexponential Logics Are Equally Expressive

Abstract

Abstract. It is standard to regard the intuitionistic restriction of a classical logic as increasing the expressivity of the logic because the classical logic can be adequately represented in the intuitionistic logic by double-negation, while the other direction has no truth-preserving propositional encodings. We show here that subexponential logic, which is a family of substructural refinements of classical logic, each parametric over a preorder over the subexponential connectives, does not suffer from this asymmetry if the preorder is systematically modified as part of the encoding. Precisely, we show a bijection between synthetic (i.e., focused) partial sequent derivations modulo a given encoding. Particular instances of our encoding for particular subexponential preorders give rise to both known and novel adequacy theorems for substructural logics.

Computation in focused intuitionistic logic

“…There are many other focused calculi that admit the same analysis, such as intuitionistic linear logic and its extensions with subexponentials [7] or fixpoints [4]. These logics enjoy well-behaved focused calculi, and we expect that an abstract machine can be extracted from cut elimination by following our methodology.…”

Magically Constraining the Inverse Method Using Dynamic Polarity Assignment

“…Most of the material of this section has already appeared in in [4,7,15,8] and in references therefrom. Like other recent accounts of intuitionistic focusing [15,5], we adopt a polarized syntax for formulas. Intuitively, positive formulas (i.e., formulas of the positive polarity) are those formulas whose left sequent rules are invertible and negative formulas are those whose right rules are invertible.…”

“…There will never be any facts generated about lsum − [5,1,2,3,4] x, for instance, because there is never a seed of that form. Thus, as long as there is a well-founded measure on the seeds that is strictly decreasing for every new seed, this implementation of the inverse method will saturate.…”

Undecidability of Multiplicative Subexponential Logic

“…We use the classical dialect of linear logic to show these results. The intuitionistic dialect has the same decision problem because it is possible to faithfully encode (i.e., linearly simulate the sequent proofs of) the classical dialect in the intuitionistic dialect without changing the signature [2].…”