Closure Properties of Weak Systems of Bounded Arithmetic

@inproceedings{Kolokolova2005ClosurePO,
  title={Closure Properties of Weak Systems of Bounded Arithmetic},
  author={Antonina Kolokolova},
  booktitle={CSL},
  year={2005}
}
  • A. Kolokolova
  • Published in CSL 22 August 2005
  • Computer Science, Mathematics
In this paper we study the properties of systems of bounded arithmetic capturing small complexity classes and state conditions sufficient for such systems to capture the corresponding complexity class tightly. Our class of systems of bounded arithmetic is the class of second-order systems with comprehension axiom for a syntactically restricted class of formulas Φ ⊂ Σ$_{\rm 1}^{B}$ based on a logic in the descriptive complexity setting. This work generalizes the results of [8] and [9]. We… 

Expressing vs . proving : relating forms of complexity in logic

This work shows how the complexity of logics in the setting of finite model theory is used to obtain results in bounded arithmetic, stating which functions are provably total in certain weak systems of arithmetic.

Expressing versus Proving: Relating Forms of Complexity in Logic

It is shown how the complexity of logics in the setting of finite model theory is used to obtain results in bounded arithmetic, stating which functions are provably total in certain weak systems of arithmetic.

Many Facets of Complexity in Logic

This work shows how the complexity of logics in the setting of finite model theory is used to obtain results in bounded arithmetic, stating which functions are provably total in certain weak systems of arithmetic.

Arithmetic and Modularity in Declarative Languages for Knowledge Representation

This thesis develops an ideal KR language that captures the complexity class NP for arithmetical search problems and guarantees universality and efficiency for solving such problems, and introduces a framework to language-independently combine modules from different KR languages.

References

SHOWING 1-10 OF 34 REFERENCES

Systems of bounded arithmetic from descriptive complexity

A general method of constructing systems of bounded arithmetic from descriptive complexity logics of known complexity, which works for small complexity classes (P and below) which have simple proofs of closure under complementation.

Theories for TC0 and Other Small Complexity Classes

An elegant theory VTC^0 is presented in which the provably-total functions are those associated with TC^0, and the isomorphism proof is defining binary number multiplication in terms a bit-counting function, and showing how to formalize the proofs of its algebraic properties.

Capturing Complexity Classes by Fragments of Second-Order Logic

Arithmetic, proof theory, and computational complexity

This chapter discusses Frege Proofs, Open Induction, Tennenbaum Phenomena, and Complexity Theory, and how to use Herbrand-type Theorems to Separate Strong Fragments of Arithmetic.

Generalized first-order spectra, and polynomial. time recognizable sets

The spectrum of a first-order sentence σ is the set of cardinalities of its finite models. Jones and Selman showed that a set C of numbers (written in binary) is a spectrum if and only if C is in the

A second-order system for polytime reasoning based on Grädel's theorem

A second-order system for polytime reasoning using Gradel's theorem

  • S. CookA. Kolokolova
  • Computer Science
    Proceedings 16th Annual IEEE Symposium on Logic in Computer Science
  • 2001
It is shown that V/sub 1/-Horn is finitely, axiomatizable, and, as a corollary, that the class of /spl forall//spl Sigma//sub 1//sup b/ consequences of S/sub 2//sup 1/ is finally axiomatsizable as well, thus answering an open question.

Finite model theory

The text presents the main results of descriptive complexity theory, the connection between axiomatizability of classes of finite structures and their complexity with respect to time and space bounds.

The complexity of satisfiability problems

An infinite class of satisfiability problems is considered which contains these two particular problems as special cases, and it is shown that every member of this class is either polynomial-time decidable or NP-complete.

Relational queries computable in polynomial time (Extended Abstract)

This paper shows that the Fixpoint Hierarchy collapses at the first fixpoint level, and shows weaker languages for expressing less complex queries, and gives a simple syntactic categorization of those queries which can be answered in polynomial time.