Decidable Fragments of First-Order Logic and of First-Order Linear Arithmetic with Uninterpreted Predicates

Abstract

First-order logic has a long tradition and is one of the most prominent and most important formalisms in computer science and mathematics. It is well-known that the satisfiability problem for full first-order logic is not solvable algorithmically — we say that first-order logic is undecidable. This fact highlights a fundamental limitation of computing devices in general and of automated reasoning in particular. The classical decision problem, as it is understood today, is the quest for a delineation between the decidable and the undecidable parts of first-order logic based on elegant and computable syntactic criteria. Many researchers have contributed to this endeavor and till today numerous decidable and undecidable fragments of first-order logic have been identified. The present thesis sheds more light on the decidability boundary and aims to open new perspectives on the already known results. In the first part of the present thesis we focus on the syntactic concept of separateness of variables and explore its applicability to the classical decision problem and beyond. Two disjoint sets of first-order variables are separated in a given formula if each atom in that formula contains variables from at most one of the two sets. This simple notion facilitates the definition of decidable extensions of many well-known decidable first-order fragments. We shall demonstrate that for several prefix fragments, several guarded fragments, the two-variable fragment, and for the fluted fragment. Altogether, we will investigate nine such extensions more closely. Interestingly, each of them contains the monadic first-order fragment without equality. Although the extensions exhibit the same expressive power as the respective originals, certain logical properties can be expressed much more succinctly. In at least two cases the succinctness gap cannot be bounded using any elementary function. This observation can be conceived as an indication for computationally hard satisfiability problems associated with the extended fragments. Indeed, we will derive non-elementary lower bounds for an extension of the Bernays–Schonfinkel–Ramsey fragment, called the separated fragment. Furthermore, we shall investigate the effect of separateness of variables at the semantic level, where it may lead to dependences between quantified variables that are weaker than such dependences are in general. Such weak dependences will be studied in the framework of model-checking games. The focus of the second part of the present thesis is on linear arithmetic over the rationals with uninterpreted predicates. Two novel decidable fragments shall be presented, both based on the Bernays–Schonfinkel–Ramsey fragment. On the negative side, we will identify several small fragments of the language for which satisfiability is undecidable.