Universal Proof Theory: Semi-analytic Rules and Craig Interpolation

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 Ł_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 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.


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