Expressive Power Of First-order Logic Research Articles

The accident of logical constants

Work on the nature and scope of formal logic has focused unduly on the distinction between logical and extra-logical vocabulary; which argument forms a logical theory countenances depends not only on its stock of logical terms, but also on its range of grammatical categories and modes of composition. Furthermore, there is a sense in which logical terms are unnecessary. Alexandra Zinke has recently pointed out that propositional logic can be done without logical terms. By defining a logical-term-free language with the full expressive power of first-order logic with identity, I show that this is true of logic more generally. Furthermore, having, in a logical theory, non-trivial valid forms that do not involve logical terms is not merely a technical possibility. As the case of adverbs shows, issues about the range of argument forms logic should countenance can quite naturally arise in such a way that they do not turn on whether we countenance certain terms as logical.

A transformation scheme for infinitary first-order combinatorics presenting computational level of expressiveness in predicate logic

We continue to develop a new first-order combinatorial approach presenting a conceptual framework for investigations concerning expressive power of first-order logic. In this work, we consider the case of infinitary first-order combinatorics. Based on the universal construction of finitely axiomatizable theories, we introduce some common scheme yielding finitely axiomatizable theories with pre-assigned sets of model-theoretic properties. At an initial stage, a maximum common Turing’s computation is performed (one can say, computable Brute Force). Starting from an input block of parameters, the computation yields a computably axiomatizable theory T. Finally, by applying an available version of the universal construction, the theory T is transformed into a finitely axiomatizable theory F that inherits model-theoretic properties of T within the infinitary semantic layer. We also give three demonstrations showing possibilities of this method.

Fusing uncertain knowledge and evidence for maritime situational awareness via Markov Logic Networks

The concepts of event and anomaly are important building blocks for developing a situational picture of the observed environment. We here relate these concepts to the JDL fusion model and demonstrate the power of Markov Logic Networks (MLNs) for encoding uncertain knowledge and compute inferences according to observed evidence. MLNs combine the expressive power of first-order logic and the probabilistic uncertainty management of Markov networks. Within this framework, different types of knowledge (e.g. a priori, contextual) with associated uncertainty can be fused together for situation assessment by expressing unobservable complex events as a logical combination of simpler evidences. We also develop a mechanism to evaluate the level of completion of complex events and show how, along with event probability, it could provide additional useful information to the operator. Examples are demonstrated on two maritime scenarios of rules for event and anomaly detection.

A Framework for Integration of Logical and Probabilistic Knowledge

Integrating the expressive power of first-order logic with the ability of probabilistic reasoning of Bayesian networks has attracted the interest of many researchers for decades. We present an approach to integration that translates logical knowledge into Bayesian networks and uses Bayesian network composition to build a uniform representation that supports both logical and probabilistic reasoning. In particular, we propose a new way of translation of logical knowledge, relation search. Through the use of the proposed framework, without learning new languages or tools, modelers are allowed to 1) specify special knowledge using the most suitable languages, while reasoning in a uniform engine; 2) make use of pre-existing logical knowledge bases for probabilistic reasoning (to complete the model or minimize potential inconsistencies).

Lindstrom theorems for fragments of first-order logic

Lindstr\"om theorems characterize logics in terms of model-theoretic conditions such as Compactness and the L\"owenheim-Skolem property. Most existing characterizations of this kind concern extensions of first-order logic. But on the other hand, many logics relevant to computer science are fragments or extensions of fragments of first-order logic, e.g., k-variable logics and various modal logics. Finding Lindstr\"om theorems for these languages can be challenging, as most known techniques rely on coding arguments that seem to require the full expressive power of first-order logic. In this paper, we provide Lindstr\"om theorems for several fragments of first-order logic, including the k-variable fragments for k>2, Tarski's relation algebra, graded modal logic, and the binary guarded fragment. We use two different proof techniques. One is a modification of the original Lindstr\"om proof. The other involves the modal concepts of bisimulation, tree unraveling, and finite depth. Our results also imply semantic preservation theorems.

Wittgensteinian Tableaux, Identity, and Co-Denotation

Wittgensteinian predicate logic (W-logic) is characterized by the requirement that the objects mentioned within the scope of a quantifier be excluded from the range of the associated bound variable. I present a sound and complete tableaux calculus for this logic and discuss issues of translatability between Wittgensteinian and standard predicate logic in languages with and without individual constants. A metalinguistic co-denotation predicate, akin to Frege’s triple bar of the Begriffsschrift, is introduced and used to bestow the full expressive power of first-order logic with identity on W-logic in the presence of constants.

On second-order generalized quantifiers and finite structures

We consider the expressive power of second-order generalized quantifiers on finite structures, especially with respect to the types of the quantifiers. We show that on finite structures with at most binary relations, there are very powerful second-order generalized quantifiers, even of the simplest possible type. More precisely, if a logic L is countable and satisfies some weak closure conditions, then there is a generalized second-order quantifier Q which is monadic, unary and simple (i.e. of the same type as monadic second-order ∃), and a uniformly obtained sublogic of FO( Q) which is equivalent to L . We show some other results of the above kind, relating other classes of quantifiers to other classes of structures. For example, if the quantifiers are of the simplest non-monadic type, then the result extends to finite structures of any arity. We further show that there are second-order generalized quantifiers which do not increase expressive power of first-order logic but do increase the power significantly when added to stronger logics.

Temporal Logic with Cyclic Counting and the Degree of Aperiodicity of Finite Automata

We define the degree of aperiodicity of finite automata and show that for every set M of positive integers, the class QA_M of finite automata whose degree of aperiodicity belongs to the division ideal generated by M is closed with respect to direct products, disjoint unions, subautomata, homomorphic images and renamings. These closure conditions define q-varieties of finite automata. We show that q-varieties are in a one-to-one correspondence with literal varieties of regular languages. We also characterize QA_M as the cascade product of a variety of counters with the variety of aperiodic (or counter-free) automata. We then use the notion of degree of aperiodicity to characterize the expressive power of first-order logic and temporal logic with cyclic counting with respect to any given set M of moduli. It follows that when M is finite, then it is decidable whether a regular language is definable in first-order or temporal logic with cyclic counting with respect to moduli in M.

Epsilon-logic is more expressive than first-order logic over finite structures

AbstractThere are properties of finite structures that are expressible with the use of Hilbert's ∈-operator in a manner that does not depend on the actual interpretation for ∈-terms. but not expressible in plain first-order. This observation strengthens a corresponding result of Gurevich, concerning the invariant use of an auxiliary ordering in first-order logic over finite structures. The present result also implies that certain non-deterministic choice constructs, which have been considered in database theory, properly enhance the expressive power of first-order logic even as far as deterministic queries are concerned, thereby answering a question raised by Abiteboul and Vianu.

Stability theory, permutations of indiscernibles, and embedded finite models

We show that the expressive power of first-order logic over finite models embedded in a model M M is determined by stability-theoretic properties of M M . In particular, we show that if M M is stable, then every class of finite structures that can be defined by embedding the structures in M M , can be defined in pure first-order logic. We also show that if M M does not have the independence property, then any class of finite structures that can be defined by embedding the structures in M M , can be defined in first-order logic over a dense linear order. This extends known results on the definability of classes of finite structures and ordered finite structures in the setting of embedded finite models. These results depend on several results in infinite model theory. Let I I be a set of indiscernibles in a model M M and suppose ( M , I ) (M,I) is elementarily equivalent to ( M 1 , I 1 ) (M_1,I_1) where M 1 M_1 is | I 1 | + |I_1|^+ -saturated. If M M is stable and ( M , I ) (M,I) is saturated, then every permutation of I I extends to an automorphism of M M and the theory of ( M , I ) (M,I) is stable. Let I I be a sequence of > > -indiscernibles in a model M M , which does not have the independence property, and suppose ( M , I ) (M,I) is elementarily equivalent to ( M 1 , I 1 ) (M_1,I_1) where ( I 1 , > ) (I_1,>) is a complete dense linear order and M 1 M_1 is | I 1 | + |I_1|^+ -saturated. Then ( M , I ) (M,I) -types over I I are order-definable and if ( M , I ) (M,I) is ℵ 1 \aleph _1 -saturated, every order preserving permutation of I I can be extended to a back-and-forth system.

Logics with counting and local properties

The expressive power of first-order logic over finite structures is limited in two ways: it lacks a recursion mechanism, and it cannot count. Overcoming the first limitation has been a subject of extensive study. A number of fixpoint logics have been introduced. and shown to be subsumed by an infinitary logic L ω ∞ω . This logic is easier to analyze than fixpoint logics, and it still lacks counting power, as it has a 0-1 law. On the counting side, there is no analog of L ω ∞ω . There are a number of logics with counting power, usually introduced via generalized quantifiers. Most known expressivityy bounds are based on the fact that counting extensions of first-order logic preserve the locality properties. This article has three main goals. First, we introduce a new logic L * ∞ω ( C ) that plays the same role for counting as L ω ∞ω does for recursion—it subsumes a number of extensions of first-order logic with counting, and has nice properties that make it easy to study. Second, we give simple direct proof that L ω ∞ω ( C ) expresses only local properties: those that depend on the properties of small neighborhoods, but cannot grasp a structure as a whole. This is a general way of saying that a logic lacks a recursion mechanism. Third, we consider a finer analysis of locality of counting logics. In particular, we address the question of how local a logic is, that is, how big are those neighborhoods that local properties depend on. We get a uniform answer for a variety of logics between first-order and L * ∞ω ( C ). This is done by introducing a new form of locality that captures the tightest condition that the duplicator needs to maintain in order to win a game. We also use this technique to give bounds on outputs of L * ∞ω (C)-definable queries.

Local Normal Forms for First-Order Logic with Applications to Games and Automata

Building on work of Gaifman [Gai82] it is shown that every first-order formula is logically equivalent to a formula of the form ∃ x_1,...,x_l, \forall y, φ where φ is r-local around y, i.e. quantification in φ is restricted to elements of the universe of distance at most r from y. \par From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs. \par As another application, automata models are defined that have, on arbitrary classes of relational structures, exactly the expressive power of first-order logic and existential monadic second-order logic, respectively.

Computational model theory: an overview

The computational complexity of a problem is the amount of resources, such as time or space, required by a machine that solves the problem. The descriptive complexity of problems is the complexity of describing problems in some logical formalism over finite structures. One of the exciting developments in complexity theory is the discovery of a very intimate connection between computational and descriptive complexity. It is this connection between complexity theory and finite-model theory that we term computational model theory. In this overview paper we offer one perspective on computational model theory. Two important observations underly our perspective: (1) while computational devices work on encodings of problems, logic is applied directly to the underlying mathematical structures, and this “mismatch” complicates the relationship between logic and complexity significantly, and (2) first-order logic has severely limited expressive power on finite structures, and one way to increase the limited expressive power of first-order logic is by adding fixpoint constructs. These observations motivated the introduction of two absract formalisms: that of finite-variable infinitary logic and that of relational machines. This paper is the story of how these formalisms are used in computational model theory and their interrelationship.

Modulo-counting quantifiers over finite trees

This paper studies logical definability of tree languages (sets of finite trees). The logical systems are consider are located between first-order logic and monadic second-order logic. We obtain results which clarify the expressive power of first-order logic extended by “modulo-counting quantifiers”.