2014/04/15 by Lauri Hella, Hella, Lauri, Antti Kuusisto +1
Computer Science · #FOS: Computer and information sciences #FOS: Mathematics #Formal Methods in Verification #Logic (math.LO) #Logic in Computer Science (cs.LO) #Logic, Reasoning, and Knowledge #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.1404.4004
openalex publication_date 2014/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a novel decidable fragment of first-order logic. The fragment is one-dimensional in the sense that quantification is limited to applications of blocks of existential (universal) quantifiers such that at most one variable remains free in the quantified formula. The fragment is closed under Boolean operations, but additional restrictions (called uniformity conditions) apply to combinations of atomic formulae with two or more variables. We argue that the notions of one-dimensionality and uniformity together offer a novel perspective on the robust decidability of modal logics. We also establish that minor modifications to the restrictions of the syntax of the one-dimensional fragment lead to undecidable formalisms. Namely, the two-dimensional and non-uniform one-dimensional fragments are shown undecidable. Finally, we prove that with regard to expressivity, the one-dimensional fragment is incomparable with both the guarded negation fragment and two-variable logic with counting. Our proof of the decidability of the one-dimensional fragment is based on a technique involving a direct reduction to the monadic class of first-order logic. The novel technique is itself of an independent mathematical interest.