Class Of Finite Structures Research Articles

0-1 laws for maps

A class of finite structures has a 0–1 law with respect to a logic if every property expressible in the logic has a probability approaching a limit of 0 or 1 as the structure size grows. To formulate 0–1 laws for maps (i.e., embeddings of graphs in a surface), it is necessary to represent maps as logical structures. Three such representations are given, the most general being the full cross representation based on Tutte's theory of combinatorial maps. The main result says that if a class of maps has two properties, richness and large representativity, then the corresponding class of full cross representations has a 0–1 law with respect to first-order logic. As a corollary the following classes of maps on a surface of fixed type have a first-order 0–1 law: all maps, smooth maps, 2-connected maps, 3-connected maps, triangular maps, 2-connected triangular maps, and 3-connected triangular maps. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14, 215–237, 1999

Connectedness, Classes and Cycle Index

This paper begins with the observation that half of all graphs containing no induced path of length 3 are disconnected. We generalize this in several directions. First, we give necessary and sufficient conditions (in terms of generating functions) for the probability of connectedness in a suitable class of graphs to tend to a limit strictly between zero and one. Next we give a general framework in which this and related questions can be posed, involving operations on classes of finite structures. Finally, we discuss briefly an algebra associated with such a class of structures, and give a conjecture about its structure.

Finite Variable Logics in Descriptive Complexity Theory

Throughout the development of finite model theory, the fragments of first-order logic with only finitely many variables have played a central role. This survey gives an introduction to the theory of finite variable logics and reports on recent progress in the area.For each k ≥ 1 we let Lk be the fragment of first-order logic consisting of all formulas with at most k (free or bound) variables. The logics Lk are the simplest finite-variable logics. Later, we are going to consider infinitary variants and extensions by so-called counting quantifiers.Finite variable logics have mostly been studied on finite structures. Like the whole area of finite model theory, they have interesting model theoretic, complexity theoretic, and combinatorial aspects. For finite structures, first-order logic is often too expressive, since each finite structure can be characterized up to isomorphism by a single first-order sentence, and each class of finite structures that is closed under isomorphism can be characterized by a first-order theory. The finite variable fragments seem to be promising candidates with the right balance between expressive power and weakness for a model theory of finite structures. This may have motivated Poizat [67] to collect some basic model theoretic properties of the Lk. Around the same time Immerman [45] showed that important complexity classes such as polynomial time (PTIME) or polynomial space (PSPACE) can be characterized as collections of all classes of (ordered) finite structures definable by uniform sequences of first-order formulas with a fixed number of variables and varying quantifier-depth.

Modal Logic over Finite Structures

We investigate properties of propositional modal logic over the class of finite structures. In particular, we show that certain known preservation theorems remain true over this class. We prove that a class of finite models is defined by a first-order sentence and closed under bisimulations if and only if it is definable by a modal formula. We also prove that a class of finite models defined by a modal formula is closed under extensions if and only if it is defined by a ♦-modal formula.

Parametrization over inductive relations of a bounded number of variables

We present a Parametrization Theorem for (positive elementary) inductions that use a bounded number of variables. We investigate associated halting problem(s) on classes of finite structures and on solitary ‘unreasonable’ structures. These results involve the complexity of the inductive relations—and the complexity of the structure or class of structures on which these relations live. We also apply this Parametrization Theorem to Moschovakis closure ordinals, to determine when the closure ordinal is greater than ω, and to investigate the closure ordinals of unreasonable structures.