Lindstrom Quantifiers Research Articles

Properties of Almost All Graphs and Generalized Quantifiers

We study 0-1 laws for extensions of first-order logic by Lindstrom quantifiers. We state sufficient conditions on a quantifier Q expressing a graph property, for the logic FO[Q] - the extension of first-order logic by means of the quantifier Q - to have a 0-1 law. We use these conditions to show, in particular, that FO[Rig], where Rig is the quantifier expressing rigidity, has a 0-1 law. We also show that extensions of first-order logic with quantifiers for Hamiltonicity, regularity and self-complementarity of graphs do not have a 0-1 law. Blass and Harary pose the question whether there is a logic which is powerful enough to express Hamiltonicity or rigidity and which has a 0-1 law. It is a consequence of our results that there is no such regular logic (in the sense of abstract model theory) in the case of Hamiltonicity, but there is one in the case of rigidity. We also consider sequences of vectorized quantifiers, and show that the extensions of first-order logic obtained by adding such sequences generated by quantifiers that are closed under substructures have 0-1 laws. The positive results also extend to the infinitary logic with finitely many variables.

Quantifying over Quantifiers: (Abstract)

We study existential and universal quantification over quantifiers, i.e. quantification where the objects quantified over are Lindstrom quantifiers. First we consider the fragment where only existential quantification over quantifiers is allowed, denoted ΣQ1. We show that ΣQ1 includes inflationary fixed-point logic extended with the ability to express that two defined structures are non-isomorphic, and that ΣQ1 is included in existential second-order logic with the same extension.The logic ΣQn is defined as the fragment where we alternate existential and universal quantification for n levels. We show that ΣQn+1 is included in the n-th level of the complexity theoretical exponential hierarchy. We also show that there is a hierarchy on the arity of the quantifier variables, by showing that no fragment of ΣQn with restricted arity of the quantifiers can express all ΣQ1 properties.

Regular languages definable by Lindström quantifiers

In our main result, we establish a formal connection between Lindstrom quantifiers with respect to regular languages and the double semidirect product of finite monoids with a distinguished set of generators. We use this correspondence to characterize the expressive power of Lindstrom quantifiers associated with a class of regular languages.

Generalized Quantifiers and Logical Reducibilities

We consider extensions of first order logic (FO) and least fixed point logic (LFP) with generalized quantifiers in the sense of Lindstrom [Lin66]. We show that adding a finite set of such quantifiers to LFP fails to capture all polynomial time properties of structures, even over a fixed signature. We show that this strengthens results in [Hel92] and [KV92a]. We also consider certain regular infinite sets of Lindstrom quantifiers, which correspond to a natural notion of logical reducibility. We show that if there is any recursively enumerable set of quantifiers that can be added to FO (or LFP) to capture P, then there is one with strong uniformity conditions. This is established through a general result, linking the existence of complete problems for complexity classes with respect to the first order translations of [Imm87] or the elementary reductions of [LG77] with the existence of recursive index sets for these classes. Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MSCIS-92-85. This technical report is available at ScholarlyCommons: http://repository.upenn.edu/cis_reports/348 Generalized Quantifiers and Logical Reducibilities MS-CIS-92-85 LINC LAB 240