Some connections between mathematical logic and complexity theory

Lecture Notes in Mathematics, 1979

1979

Annals of Pure and Applied Logic, 1989

The complexity of subclasses of Magical theories (mainly Presburger and Skolem arithmetic) is studied. The subclasses are defined by the structure of the quantifier prefix.

2011

Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorov-random strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems C defined in terms of polynomial-time truth-table reducibility to R K (the set of Kolmogorov-random strings) that lies between BPP and PSPACE . In this paper, we investigate improving this upper bound from PSPACE to PSPACE ∩ P/poly. More precisely, we present a collection of true statements in the language of arithmetic, (each provable in ZF) and show that if these statements can be proved in certain extensions of Peano arithmetic, then BPP ⊆ C ⊆ PSPACE ∩ P/poly.

Texts in Theoretical Computer Science an EATCS Series, 2007

Lecture Notes in Computer Science, 2004

We present a probability logic (essentially a first order language extended with quantifiers that count the fraction of elements in a model that satisfy a first order formula) which, on the one hand, captures uniform circuit classes such as AC 0 and TC 0 over arithmetic models, namely, finite structures with linear order and arithmetic relations, and, on the other hand, their semantics, with respect to our arithmetic models, can be closely approximated by giving interpretations of their formulas on finite structures where all relations (including the order) are restricted to be "modular" (i.e. to act subject to an integer modulo). In order to give a precise measure of the proximity between satisfaction of a formula in an arithmetic model and satisfaction of the same formula in the "approximate" model, we define the approximate formulas and work on a notion of approximate truth. We also indicate how to enhance the expressive power of our probability logic in order to capture polynomial time decidable queries, There are various motivations for this work. As of today, there is not known logical description of any computational complexity class below NP which does not requires a built-in linear order. Also, it is widely recognized that many model theoretic techniques for showing definability in logics on finite structures become almost useless when order is present. Hence, if we want to obtain significant lower bound results in computational complexity via the logical description we ought to find ways of by-passing the ordering restriction. With this work we take steps towards understanding how well can we approximate, without a true order, the expressive power of logics that capture complexity classes on ordered structures.

2012

Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorov-random strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems C defined in terms of polynomial-time truth-table reducibility to RK (the set of Kolmogorov-random strings) that lies between BPP and PSPACE [4, 3]. In this paper, we investigate improving this upper bound from PSPACE to PSPACE ∩ P/poly. More precisely, we present a collection of true statements in the language of arithmetic, (each provable in ZF) and show that if these statements can be proved in certain extensions of Peano arithmetic, then BPP ⊆C⊆PSPACE ∩ P/poly. We conjecture that C is equal to P, and discuss the possibility this might be an avenue for trying to prove the equality of BPP and P.

2002

Abstract In [3] the tautology problem for Hájek's Basic Logic BL is proved to be co-NP-complete by showing that if a formula ϕ is not a tautology of BL then there exists an integer m> 0, polynomially bounded by the length of ϕ, such that ϕ fails to be a tautology in the infinite-valued logic mL corresponding to the ordinal sum of m copies of the Lukasiewicz t-norm.

Proceedings of the twelfth annual ACM symposium on Theory of computing - STOC '80, 1980