Focusing and polarization in linear, intuitionistic, and classical logics

Abstract

A focused proof system provides a normal form to cut-free proofs in which the application of invertible and non-invertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cut-free proofs. Within intuitionistic and classical logics, there are various different proof systems in the literature that exhibit focusing behavior. These focused proof systems have been applied to both the proof search and the proof normalization approaches to computation. We present a new, focused proof system for intuitionistic logic, called LJF, and show how other intuitionistic proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing. We also use LJF to design a focused proof system LKF for classical logic. Our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in Girard's LC and LU proof systems.

A Proposal for Broad Spectrum Proof Certificates

“…Notice that if the formula has n occurrences of these four logical connectives then there are 2 n different polarizations of that formula. The following theorem is proved in [12].…”

“…To illustrate these general comments about focused proof systems more concretely, we now present the LKF proof system for first-order classical logic [12]. We shall adopt a presentation of first-order classical logic in which negations are applied only to atomic formulas (i.e., negation normal form) and where the propositional connectives t, f , ∧, and ∨ are replaced by two "polarized" versions: t − , t + , f − , f + , ∧ − , ∧ + , ∨ − , ∨ + .…”

Foundational Proof Certificates in First-Order Logic

“…Formulas in LKU contain a mix of classical and linear logic connectives and (first-order) quantifiers. Unrestricted, LKU is essentially a verbose presentation of the focused classical logic LKF [14]. The LKU proof system allows for various restrictions to be placed on its structural rules.…”

“…The LKU proof system allows for various restrictions to be placed on its structural rules. One set of restrictions (reminiscent of Gentzen's restricting of classical inference rules to only single-conclusion sequents) gives rise to the LJF [14] focused proof systems for intuitionistic logic. Another set of restrictions (reminiscent of Girard's restricting of classical inference rules so that weakening and contraction are not available) gives rise to a focused proof system for (multiplicative-additive) linear logic [1].…”

“…On the other hand, the clerks are responsible for taking information released by the experts and performing some computations on them, including their indexing and storing. The justification of this division of effort between clerks and experts comes from the structure of focused sequent proof systems [1,14,15]: experts operate during the synchronous phase of proof construction while the clerks operation during the asynchronous phase. Furthermore, the vocabulary of decide, release, and store is reused as structural rules within the focused proof system.…”