Universal Proof Theory: Semi-analytic Rules and Craig Interpolation

Abstract

In [6], Iemhoff introduced the notion of a focused axiom and a focused rule as the building blocks for a certain form of sequent calculus which she calls a focused proof system. She then showed how the existence of a terminating focused system implies the uniform interpolation property for the logic that the calculus captures. In this paper we first generalize her focused rules to semi-analytic rules, a dramatically powerful generalization, and then we will show how the semi-analytic calculi consisting of these rules together with our generalization of her focused axioms, lead to the feasible Craig interpolation property. Using this relationship, we first present a uniform method to prove interpolation for different logics from sub-structural logics FL e , FL ec , FL ew and IPC to their appropriate classical and modal extensions, including the intuitionistic and classical linear logics. Then we will use our theorem negatively, first to show that so many sub-structural logics including L n , G n , BL, R and RM e and almost all super-intutionistic logics (except at most seven of them) do not have a semi-analytic calculus. To investigate the case that the logic actually has the Craig interpolation property, we will first define a certain specific type of semi-analytic calculus which we call PPF systems and we will then present a sound and complete PPF calculus for classical logic. However, we will show that all such PPF calculi are exponentially slower than the classical Hilbert-style proof system (or * The authors are supported by the ERC Advanced Grant 339691 (FEALORA).equivalently LK `Cut). We will then present a similar exponential lower bound for a certain form of complete PPF calculi, this time for any super-intuitionistic logic.

Universal Proof Theory: Feasible Admissibility in Intuitionistic Modal Logics

“…Universal proof theory [1,2] is a recent research project designed to investigate the generic behavior of proof systems considered as the independent mathematical objects standing on their own feet. It opposes the usual attitude in proof theory that sees the proof systems as the auxiliary objects by which we investigate the logical systems they capture.…”

“…The focused rules are very limited in their form and it seems that even the intuitionistic propositional logic cannot have a system only consisting of these focused rules. To solve this problem, later, in [1,2], the focussed rules were strengthened to a more general form (called semi-analytic), where both Craig and uniform interpolation properties were studied. Moreover, the scope of the logics were also extended to cover the substructural (modal) logics.…”

“…They form a basis for the admissible rules of intuitionistic logic and any intermediate logic in which they are admissible. In this paper, we follow the universal proof theory program introduced and developed in [1,2,24,25] to establish a connection between the form of the rules in a sequent calculus for an intuitionistic modal logic and the admissibility of Visser's rules in that logic. More precisely, by investigating the form of the constructively acceptable rules, we first introduce a very general family of rules called the constructive rules.…”

Uniform Lyndon Interpolation for Basic Non-normal Modal Logics

“…Over the last years a method to prove uniform (Lyndon) interpolation has been developed by the authors that applies to various (intuitionistic) modal and intermediate logics [9,10,12,1,2,3]. Uniform interpolation is a strengthening of interpolation in which the interpolant depends only on the premise or the conclusion of an implication.…”