We analyze the expressive resources of \ logic that do not stem from Henkin quantification. When one restricts attention to regular \ sentences, this amounts to the study of the fragment of \ logic which is individuated by the game-theoretical property of action recall. We prove that the fragment of prenex AR sentences can express all existential second-order properties. We then show that the same can be achieved in the non-prenex fragment of AR, by using “signalling by disjunction” instead of (...) Henkin or signalling patterns. We also study irregular IF logic and analyze its correspondence to regular IF logic. By using new methods, we prove that the game-theoretical property of knowledge memory is a first-order syntactical constraint also for irregular sentences, and we identify another new first-order fragment. Finally we discover that irregular prefixes behave quite differently in finite and infinite models. In particular, we show that, over infinite structures, every irregular prefix is equivalent to a regular one; and we present an irregular prefix which is second order on finite models but collapses to a first-order prefix on infinite models. (shrink)

We introduce a new framework for classifying logics on finite structures and studying their expressive power. This framework is based on the concept of almost everywhere equivalence of logics, that is to say, two logics having the same expressive power on a class of asymptotic measure 1. More precisely, if L, L ′ are two logics and μ is an asymptotic measure on finite structures, then $\scr{L}\equiv _{\text{a.e.}}\scr{L}^{\prime}(\mu)$ means that there is a class C of finite structures with μ (C)=1 (...) and such that L and L ′ define the same queries on C. We carry out a systematic investigation of $\equiv _{\text{a.e.}}$ with respect to the uniform measure and analyze the $\equiv _{\text{a.e.}}$ -equivalence classes of several logics that have been studied extensively in finite model theory. Moreover, we explore connections with descriptive complexity theory and examine the status of certain classical results of model theory in the context of this new framework. (shrink)

The concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity type t there is a generalized quantifier of type t which is not definable in the extension of first order logic by all generalized quantifiers of type smaller than t. This was proved for unary similarity types by Per Lindström [17] with (...) a counting argument. We extend his method to arbitrary similarity types. (shrink)

We study generalized quantifiers on finite structures.With every function : we associate a quantifier Q by letting Q x say there are at least (n) elementsx satisfying , where n is the sizeof the universe. This is the general form ofwhat is known as a monotone quantifier of type .We study so called polyadic liftsof such quantifiers. The particular lifts we considerare Ramseyfication, branching and resumption.In each case we get exact criteria fordefinability of the lift in terms of simpler quantifiers.

The concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity typetthere is a generalized quantifier of typetwhich is not definable in the extension of first order logic by all generalized quantifiers of type smaller thant. This was proved for unary similarity types by Per Lindström [17] with a counting argument. We extend (...) his method to arbitrary similarity types. (shrink)

We carry out a systematic investigation of the definability of linear order on classes of finite rigid structures. We obtain upper and lower bounds for the expressibility of linear order in various logics that have been studied extensively in finite model theory, such as least fixpoint logic LFP, partial fixpoint logic PFP, infinitary logic Lω∞ω with a finite number of variables, as well as the closures of these logics under implicit definitions. Moreover, we show that the upper and lower bounds (...) established here cannot be made substantially tighter, unless outstanding conjectures in complexity theory are resolved at the same time. (shrink)

Many known tools for proving expressibility bounds for first-order logic are based on one of several locality properties. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notions: one deals with sentences and one with open formulae. We prove that the former implies Hanf's notion of locality, which in turn implies Gaifman's locality for open formulae. Each of these implies the bounded degree property, which is one of (...) the easiest tools for proving expressibility bounds. These results apply beyond the first-order case. We use them to derive expressibility bounds for first-order logic with unary quantifiers and counting. We also characterize the notions of locality on structures of small degree. (shrink)

We consider the problem of obtaining logical characterisations of oracle complexity classes. In particular, we consider the complexity classes LOGSPACENP and PTIMENP. For these classes, characterisations are known in terms of NP computable Lindström quantifiers which hold on ordered structures. We show that these characterisations are unlikely to extend to arbitrary structures, since this would imply the collapse of certain exponential complexity hierarchies. We also observe, however, that PTIMENP can be characterised in terms of Lindström quantifers , though it remains (...) open whether this can be done for LOGSPACENP. (shrink)

In this paper we prove that thek-ary fragment of transitive closure logic is not contained in the extension of the (k−1)-ary fragment of partial fixed point logic by all (2k−1)-ary generalized quantifiers. As a consequence, the arity hierarchies of all the familiar forms of fixed point logic are strict simultaneously with respect to the arity of the induction predicates and the arity of generalized quantifiers.Although it is known that our theorem cannot be extended to the sublogic deterministic transitive closure logic, (...) we show that an extension is possible when we close this logic under congruence. (shrink)

Motivated by constraint satisfaction problems, Feder and Vardi (SIAM Journal of Computing, 28, 57–104, 1998) set out to search for fragments of satisfying the dichotomy property: every problem definable in is either in P or else NP-complete. Feder and Vardi considered in this connection two logics, strict NP (or SNP) and monadic, monotone, strict NP without inequalities (or MMSNP). The former consists of formulas of the form , where is a quantifier-free formula in a relational vocabulary; and the latter is (...) the fragment of SNP whose formulas involve only negative occurrences of relation symbols, only monadic second-order quantifiers, and no occurrences of the equality symbol. It remains an open problem whether MMSNP enjoys the dichotomy property. In the present paper, SNP and MMSNP are characterized in terms of partially ordered connectives. More specifically, SNP is characterized using the logic D of partially ordered connectives introduced in Blass and Gurevich (Annals of Pure and Applied Logic, 32, 1–16, 1986), Sandu and Väänänen (Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 38, 361–372 1992), and MMSNP employing a generalization C of D introduced in the present paper. (shrink)

We prove that if ℵα is uncountable and regular, then the Beth-closure of Lωω(Qα) is not a sublogic of L∞ω(Qn), where Qn is the class of all n-ary generalized quantifiers. In particular, B(Lωω(Qα)) is not a sublogic of any finitely generated logic; i.e., there does not exist a finite set Q of Lindstrom quantifiers such that B(Lωω(Qα)) ≤ Lωω(Q).

We prove that if $\aleph_\alpha$ is uncountable and regular, then the Beth-closure of $\mathscr{L}_{\omega\omega}(Q_\alpha)$ is not a sublogic of $\mathscr{L}_{\infty\omega}(\mathbf{Q}_n)$, where $\mathbf{Q}_n$ is the class of all $n$-ary generalized quantifiers. In particular, $B(\mathscr{L}_{\omega\omega}(Q_\alpha))$ is not a sublogic of any finitely generated logic; i.e., there does not exist a finite set $\mathbf{Q}$ of Lindstrom quantifiers such that $B(\mathscr{L}_{\omega\omega}(Q_\alpha)) \leq \mathscr{L}_{\omega\omega}(\mathbf{Q})$.

We give a sufficient condition for the inexpressibility of the k-th extended vectorization of a generalized quantifier $\sf Q$ in ${\rm FO}({\vec Q}_k)$ , the extension of first-order logic by all k-ary quantifiers. The condition is based on a model construction which, given two ${\rm FO}({\vec Q}_1)$ -equivalent models with certain additional structure, yields a pair of ${\rm FO}({\vec Q}_k)$ -equivalent models. We also consider some applications of this condition to quantifiers that correspond to graph properties, such as connectivity and (...) planarity. (shrink)

We provide a sound and complete proof system for an extension of Kleene’s ternary logic to predicates. The concept of theory is extended with, for each function symbol, a formula that specifies when the function is defined. The notion of “is defined” is extended to terms and formulas via a straightforward recursive algorithm. The “is defined” formulas are constructed so that they themselves are always defined. The completeness proof relies on the Henkin construction. For each formula, precisely one of the (...) formula, its negation, and the negation of its “is defined” formula is true on the constructed model. Many other ternary logics in the literature can be reduced to ours. Partial functions are ubiquitous in computer science and even in (in)equation solving at schools. Our work was motivated by an attempt to precisely explain, in terms of logic, typical informal methods of reasoning in such applications. (shrink)