The expressive power of fixed-point logic with counting

We consider various extensions of first-order logic. Informally, a logic 𝓛 is an extension of first-order logic if every sentence of first-order logic is also a sentence of 𝓛. We also require that 𝓛 is closed under conjunction and negation and has other basic properties of a logic. In Section 9.4, we list the properties that formally define the notion of an extension of first-order logic. Prior to Section 9.4, we provide various natural examples of such extensions. In Sections 9.1–9.3, we consider, respectively, second-order logic, infinitary logics, and logics with fixed-point operators. We do not provide a thorough treatment of any one of these logics. Indeed, we could easily devote an entire chapter to each. Rather, we define each logic and provide examples that demonstrate the expressive power of the logics. In particular, we show that none of these logics has compactness. In the final Section 9.4, we prove that if a proper extension of first-order logic has compactness, then the Downward Löwenhiem–Skolem theorem must fail for that logic. This is Lindstrom’s theorem. The Compactness theorem and Downward Löwenheim–Skolem theorem are two crucial results for model theory. Every property of first-order logic from Chapter 4 is a consequence of these two theorems. Lindström’s theorem implies that the only extension of first-order logic possessing these properties is first-order logic itself. Second-order logic is the extension of first-order logic that allows quantification of relations. The symbols of second-order logic are the same symbols used in first-order logic. The syntax of second-order logic is defined by adding one rule to the syntax of first-order logic. The additional rule makes second-order logic far more expressive than first-order logic. Specifically, the syntax of second-order logic is defined as follows. Any atomic first-order formula is a formula of second-order logic. Moreover, we have the following four rules: (R1) If φ is a formula then so is ￢φ. (R2) If φ and ψ are formulas then so is φ ∧ ψ. (R3) If φ is a formula, then so is ∃x φ for any variable x.

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.

A number of efficient methods for evaluating first-order and monadic-second order queries on finite relational structures are based on tree-decompositions of structures or queries. We systematically study these methods.In the first part of the article, we consider arbitrary formulas on tree-like structures. We generalize a theorem of Courcelle [1990] by showing that on structures of bounded tree-width a monadic second-order formula (with free first- and second-order variables) can be evaluated in time linear in the structure size plus the size of the output.In the second part, we study tree-like formulas on arbitrary structures. We generalize the notions of acyclicity and bounded tree-width from conjunctive queries to arbitrary first-order formulas in a straightforward way and analyze the complexity of evaluating formulas of these fragments. Moreover, we show that the acyclic and bounded tree-width fragments have the same expressive power as the well-known guarded fragment and the finite-variable fragments of first-order logic, respectively.

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.

On the expressive power of counting

Invertible map equivalences are approximations of graph isomorphism that refine the well-known Weisfeiler-Leman method. They are parameterized by a number k and a set Q of primes. The intuition is that two equivalent graphs G equiv^IM_{k, Q} H cannot be distinguished by means of partitioning the set of k-tuples in both graphs with respect to any linear-algebraic operator acting on vector spaces over fields of characteristic p, for any p in Q. These equivalences have first appeared in the study of rank logic, but in fact they can be used to delimit the expressive power of any extension of fixed-point logic with linear-algebraic operators. We define {LA^{k}}(Q), an infinitary logic with k variables and all linear-algebraic operators over finite vector spaces of characteristic p in Q and show that equiv^IM_{k, Q} is the natural notion of elementary equivalence for this logic. The logic LA^{omega}(Q) = Cup_{k in omega} LA^{k}(Q) is then a natural upper bound on the expressive power of any extension of fixed-point logics by means of Q-linear-algebraic operators. By means of a new and much deeper algebraic analysis of a generalized variant, for any prime p, of the CFI-structures due to Cai, Furer, and Immerman, we prove that, as long as Q is not the set of all primes, there is no k such that equiv^IM_{k, Q} is the same as isomorphism. It follows that there are polynomial-time properties of graphs which are not definable in LA^{omega}(Q), which implies that no extension of fixed-point logic with linear-algebraic operators can capture PTIME, unless it includes such operators for all prime characteristics. Our analysis requires substantial algebraic machinery, including a homogeneity property of CFI-structures and Maschke's Theorem, an important result from the representation theory of finite groups.

Towards finite homomorphism-homogeneous relational structures

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

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.

In the constraint satisfaction problem CSP(H) corresponding to a finite relational structure H, the aim is to decide, given a relational structure G, whether there exists a homomorphism from G to H. In (Bulatov, 2003), we proved that if H is a conservative structure, then it can be associated with a complete edge-3-colored graph whose vertex set is the universe of H. The complexity and a solution algorithm for CSP(H) strongly depend on certain properties of the associated graph. In this paper we show how a similar edge-3-colored graph can be defined for an arbitrary finite relational structure H. Then we study properties of the defined graph and find a solution algorithm for CSP(H), where G(H) satisfies some restrictions. The latter result substantially generalizes the results (2000,2002,1998,1997) concerning max-closed constraints and constraints with a 2-semilattice, semigroup or conservative groupoid polymorphism. Finally, we complete the study of the complexity of maximal constraint languages started in (Bulatov et al., 2001).

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 conservative extension of first order logic, suitable for circumscriptive inference, is introduced.

Approximate pattern matching and transitive closure logics

We present two, quite different, logical characterizations of the computational complexity class PSPACE on unordered, finite relational structures. The first of these, the closure of second-order logic under the formation of partial fixed points is well-known in the folklore but does not seem to be in the literature. The second, the closure of first-order logic under taking partial fixed points and under an operator for nondeterministic choice, is novel. We also present syntactic normal forms for the two logics and compare the second with other choice-based fixed-point logics found in the literature.

We show that if a finite relational structure has a near unanimity polymorphism, then the constraint satisfaction problem with that structure as its fixed template has bounded pathwidth duality, putting the problem in nondeterministic logspace. This generalizes the analogous result of Dalmau and Krokhin for majority polymorphisms and lends further support to a conjecture suggested by Larose and Tesson.