Beyond first-order logic

First-order logic augments the expressive power of propositional logic as it links the logical assertions to properties of objects of some non-empty universe: the domain of discourse. This is achieved by allowing the propositional symbols to take arguments that range over elements of the domain of discourse. These are now called predicate symbols and are interpreted as relations on the domain. Elements of the domain of discourse are denoted by terms built up from variables, constants, and functions applied to other terms. First-order logic also expands the lexicon of propositional logic with the quantifiers “for all” and “there exists” that are interpreted consistently with their natural language meaning.This chapter is devoted to classical first-order logic. Our presentation is similar to the one conducted for propositional logic. We first define the syntax of first-order logic, followed by its semantics. Next we define a proof system for it and present the fundamental theoretical results of soundness and completeness. We also discuss the decision problems related to this logic. The remaining sections of the chapter cover variations and extensions of first-order logic, as well as first-order theories.

Approximate pattern matching and transitive closure logics

First-order temporal logic, the extension of first-order logic with operators dealing with time, is a powerful and expressive formalism with many potential applications. This expressive logic can be viewed as a framework in which to investigate problems specified in other logics. The monodic fragment of first-order temporal logic is a useful fragment that possesses good computational properties such as completeness and sometimes even decidability. Temporal logics of knowledge are useful for dealing with situations where the knowledge of agents in a system is involved. In this paper we present a translation from temporal logics of knowledge into the monodic fragment of first-order temporal logic. We can then use a theorem prover for monodic first-order temporal logic to prove properties of the translated formulas. This allows problems specified in temporal logics of knowledge to be verified automatically without needing a specialized theorem prover for temporal logics of knowledge. We present the translation, its correctness, and examples of its use.

We study the expressive power in the finite of the logic Fixed-Point+Counting, the extension of first-order logic which is obtained through adding both the fixed-point constructor and the ability to count.To this end an isomorphism preserving (‘generic’) model of computation is introduced whose PTime restriction exactly corresponds to this level of expressive power, while its PSpace restriction corresponds to While+Counting. From this model we obtain a normal form which shows a rather clear separation of the relational vs. the arithmetical side of the algorithms involved.In parallel, we study the relations of Fixed-Point+Counting with the infinitary logics and the corresponding pebble games.The main result, however, involves the concept of an arithmetical invariant. By this we mean a functor taking every finite relational structure to an expansion of (an initial segment of) the standard arithmetical structure. In particular its values are linearly ordered structures. We establish the existence of a family of arithmetical invariants with the following properties:• The invariants themselves can be evaluated in polynomial time.• A class of finite relational structures is definable in Fixed-Point+Counting if and only if membership can be decided in polynomial time on the basis of the values of one of the invariants.• The invariant r classifies all finite relational structures exactly up to equivalence with respect to the logic We also give a characterization of Fixed-Point+Counting in terms of sequences of formulae in the : It corresponds exactly to the polynomial time computable families (φn)n ∈ ω in these logics.Towards a positive assessment of the expressive power of Fixed-Point+Counting, it is shown that the natural extension of fixed-point logic by Lindström quantifiers, which capture all the PTime computable properties of cardinalities of definable predicates, is strictly weaker than what we get here. This implies in particular that every extension of fixed-point logic by means of monadic Lindström quantifiers, which stays within PTime, must be strictly contained in Fixed-Point+Counting.

Concrete domains have been introduced in description logic (DL) to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. The primary research goal in this context was to find restrictions on the concrete domain such that its integration into certain DLs preserves decidability or tractability. In this article, we complement these investigations by studying the abstract expressive power of both first-order and description logics extended with concrete domains, i.e., we analyze which classes of first-order interpretations can be expressed using these logics, compared to what their counterparts without concrete domains can express. We demonstrate that, under natural conditions on the concrete domain D (which also play a role for decidability), extensions of first-order logic (FOL) or the well-known DL ALC with D share important formal characteristics with FOL, such as the compactness and the Löwenheim-Skolem properties. Nevertheless, these conditions do not ensure that the abstract expressive power of the extensions we consider is contained in that of FOL, though in some cases it is. To be more precise, we show, on the one hand, that unary concrete domains leave the abstract expressive power within FOL if we are allowed to introduce auxiliary predicates. As a by-product, we obtain (semi-)decidability results for some fragments of FOL extended with the concrete domains considered in this article. On the other hand, we show that the ability to express equality between elements of D , another condition employed in the context of showing decidability of ALC ( D ), is sufficient to push the abstract expressive power of most first-order fragments with concrete domains beyond that of FOL. While for such concrete domains D decidability is retained for ALC ( D ), we show that the availability of equality in D causes undecidability of the two-variable fragment of FOL( D ), although the two-variable fragment of FOL is decidable.

Transitive closure logic is a known extension of first-order logic obtained by introducing a transitive closure operator. While other extensions of first-order logic with inductive definitions are a priori parametrized by a set of inductive definitions, the addition of the transitive closure operator uniformly captures all finitary inductive definitions. In this paper we present an infinitary proof system for transitive closure logic which is an infinite descent-style counterpart to the existing (explicit induction) proof system for the logic. We show that, as for similar systems for first-order logic with inductive definitions, our infinitary system is complete for the standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive closure operator allows semantically meaningful complete restrictions to be defined using simple syntactic criteria. Consequently, the restriction to regular infinitary (i.e. cyclic) proofs provides the basis for an effective system for automating inductive reasoning.

Logics for ‘generally’ were introduced as extensions of First-Order Logic (FOL) for handling assertions with vague notions (e.g., ‘generally,’ ‘most,’ and ‘several’) expressed by formulas with generalized quantifiers. Deductive systems have been developed for such logics. Here, we characterize the structure of derivations in natural deduction style for filter logic. This characterization extends the familiar one for FOL.

We study the expressivity and the complexity of various logics in probabilistic team semantics with the Boolean negation. In particular, we study the extension of probabilistic independence logic with the Boolean negation, and a recently introduced logic first-order theory of random variables with probabilistic independence. We give several results that compare the expressivity of these logics with the most studied logics in probabilistic team semantics setting, as well as relating their expressivity to a numerical variant of second-order logic. In addition, we introduce novel entropy atoms and show that the extension of first-order logic by entropy atoms subsumes probabilistic independence logic. Finally, we obtain some results on the complexity of model checking, validity and satisfiability of our logics.

First-order logic (FOL) is evidently insufficient for the many applications of logic in computer science, mainly due to its inability to provide inductive definitions. Therefore, only an extension of FOL which allows finitary inductive definitions can be used as a framework for automated reasoning. The minimal logic that is suitable for this goal is Ancestral Logic (AL), which is an extension of FOL by a transitive closure operator. In order for AL to be able to serve as a reasonable (and better) substitute to the use of FOL in computer science, it is crucial to develop adequate, user-friendly proof systems for it. While the expressiveness of AL renders any effective proof system for it incomplete with respect to the standard semantics, there are useful approximations. In this paper we show that such a Gentzen-style approximation is both sound and complete with respect to a natural, computationally-meaningful Henkin-style semantics for AL.

On the expressive power of counting

The calculus of constructions as a framework for proof search with set variable instantiation

Constraint satisfaction problems (CSPs) are one of the central topics in theoretical computer science, in particular, in the area of artificial intelligence. Their computational complexity is due to relatively recent results from areas of mathematics, including finite-model-theory, algebra and graph homomorphisms. The main conjecture by Feder and Vardi states that any CSP over a finite relational template is either in P or is NP-complete. Further, it amounts to showing that every non NP-complete CSP can be expressed as an extension of first-order logic. A finite template is Mal'tsev, a compatible algebraic operation, which is closely related to an affine space over a finite field. The so-called Bulatov-Dalmau algorithm, a natural generalization of the Gaussian elimination on vector spaces, shows such CSPs are tractable. In this work, we prove that CSPs described over a finite template Mal'tsev are expressible in logic LFP+rnk, providing a logical proof that such CSPs are tractable.

The circuit complexity classes AC0, ACC, and CC (also called pure-ACC) can be characterized as the classes of languages definable in certain extensions of first-order logic. All of the known and conjectured inclusions among these classes have been shown to be equivalent to a single conjecture concerning the form of the formulas required to define the regular languages they contain. (The conjecture states, roughly, that when a formula defines a regular language, predicates representing numerical relations on the positions in a string can be replaced by predicates computed by finite state automata.) Here this conjecture is established in a special case: It is shown that the conjecture holds for the subclasses of AC0, ACC, and CC defined by restricting all the numerical predicates occurring in the logical formulas to be either unary relations, or the order relation <.

A Simple Nominal Type Theory

We have seen that the structure n of natural numbers cannot be characterized in the first-order language corresponding to n. The same situation holds for the field of real numbers and the class of torsion groups. As we showed in Chapter VII, one can, at least in principle, overcome this weakness by a set-theoretical formulation: One introduces a system of axioms for set theory in a first-order language, e.g. ZFC, which is sufficient for mathematics, and then, in this system, carries out the arguments which are required, say, for a definition and characterization of n. However, this approach necessitates an explicit use of set theory to an extent not usual in ordinary mathematical practice.