An Infinitary System for the Least Fixed-Point Logic restricted to Finite Models

Abstract

The notion of the least fixed-point of an operator is widely applied in computer science as, for instance, in the context of query languages for relational databases. Some extensions of first-order classical logic (FOL) with fixed-point operators, as the least fixed-point logic (LFP), were proposed to deal with problems related to the expressivity of FOL. LFP captures the complexity class PTIME over the class of finite ordered structures. The descriptive characterization of computational classes is a central issue within finite model theory (FMT). Trakhtenbrot's theorem states that validity over finite models is not recursively enumerable, that is, completeness fails over finite models. This result is based on an underlying assumption that any deductive system is of finite nature. However, we can relax such assumption as done in the scope of proof theory for arithmetic. Motivated by Gödel incompleteness theorems, proof theory for arithmetic offer an example of a true mathematically meaningful principle non derivable in first-order arithmetic. One way of presenting this proof is based on a definition of a proof system with an infinitary rule, the ω-rule, that establishes the consistency of first-order arithmetic through a proof-theoretical perspective. Inspired by this proof, here we will propose an infinitary natural deduction system for FOL and LFP restricted to finite models, FOLfin and LFPfin, respectively, and we will prove soundness and completeness for them, and also a normalization theorem for fragments of these systems. With this infinitary deductive system for LFPfin, we aim to present a proof theory for a logic traditionally investigated within the scope of FMT. It opens up an alternative way of proving results already obtained within FMT and also new results through a proof theoretical perspective.