Logical Definability of Counting Functions

Logical Definability of Counting Functions

The relationship between counting functions and logical expressibility is explored. The most well studied class of counting functions is #P, which consists of the functions counting the accepting computation paths of a nondeterministic polynomial-time Turing machine. For a logic L, #L is the class of functions on nite structures counting the tuples (T ; c) satisfying a given formula (T ; c) in ...

Logical Definability of NP Optimization Problems

We investigate here NP optimization problems from a logical deenability standpoint. We show that the class of optimization problems whose optimum is deenable using rst-order formulae coincides with the class of polynomially bounded NP optimization problems on nite structures. After this, we analyze the relative expressive power of various classes of optimization problems that arise in this fram...

Generalized Rational Relations and their Logical Definability

The family of rational subsets of a direct product of free monoids Σ∗ 1×. . .×Σ ∗ n (the rational relations) is not closed under Boolean operations, except when n = 1 or when all Σi’s are empty or singletons. In this paper we introduce the family of generalized rational subsets of an arbitrary monoid as the closure of the singletons under the Boolean operations, concatenation and Kleene star (j...

Logical Definability Versus Computational Complexity: Another Equivalence

Improving Model Counting by Leveraging Definability

We present a new preprocessing technique for propositional model counting. This technique leverages definability, i.e., the ability to determine that some gates are implied by the input formula ⌃. Such gates can be exploited to simplify ⌃ without modifying its number of models. Unlike previous techniques based on gate detection and replacement, gates do not need to be made explicit in our appro...