The standard classes of a first-order theory T are certain classes of prenex T -sentences defined by restrictions on prefix, number of monadic, dyadic, etc. predicate variables, and number of monadic, dyadic, etc. operation variables. In [3] it is shown that, for any theory T , (1) the decision problem for any class of prenex T -sentences specified by such restrictions reduces to that for the standard classes, and (2) there are finitely many standard classes K 1 , …, K n such that any undecidable standard class contains one of K 1 , …, K n . These results give direction to the study of the decision problem. Below T is predicate logic with identity and operation variables. The Main Theorem solves the decision problem for the standard classes admitting at least one operation variable.

The decision problem for standard classes

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.

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

The first-order theory of addition over the natural numbers, known as Presburger arithmetic , is decidable in double exponential time. Adding an uninterpreted unary predicate to the language leads to an undecidable theory. We sharpen the known boundary between decidable and undecidable in that we show that the purely universal fragment of the extended theory is already undecidable. Our proof is based on a reduction of the halting problem for two-counter machines to unsatisfiability of sentences in the extended language of Presburger arithmetic that does not use existential quantification. On the other hand, we argue that a single ∀∃ quantifier alternation turns the set of satisfiable sentences of the extended language into a Σ(1,1)-complete set. Some of the mentioned results can be transfered to the realm of linear arithmetic over the ordered real numbers. This concerns the undecidability of the purely universal fragment and the Σ(1,1)-hardness for sentences with at least one quantifier alternation. Finally, we discuss the relevance of our results to verification. In particular, we derive undecidability results for quantified fragments of separation logic, the theory of arrays, and combinations of the theory of equality over uninterpreted functions with restricted forms of integer arithmetic. In certain cases our results even imply the absence of sound and complete deductive calculi.

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The Universal Fragment of Presburger Arithmetic with Unary Uninterpreted Predicates is Undecidable

Mechanical Proof-Search and the Theory of Logical Deduction in the Ussr

The reachability problem for Petri nets and decision problems for Skolem arithmetic

We call a formula of E Д type an arbitrary formula of predicate calculus with equality and functional symbols which has the form 

$$\exists \,x\,D,$$

where x is the subject variable, D is the disjunction of elementary formulas.†

V. A. Lifshits

Papers

Some Reduction Classes and Undecidable Theories