Results 1 
5 of
5
Theories for Complexity Classes and their Propositional Translations
 Complexity of computations and proofs
, 2004
"... We present in a uniform manner simple twosorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus. ..."
Abstract

Cited by 30 (7 self)
 Add to MetaCart
We present in a uniform manner simple twosorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus.
Theory for TC 0 and Other Small Complexity Classes
 Logical Methods in Computer Science
, 2005
"... Abstract We present a general method for introducing finitely axiomatizable "minimal " secondorder theories for various subclasses of P. We show that our theory VTC 0 for the complexity class TC 0 is RSUV isomorphic to the firstorder theory \Delta b ..."
Abstract

Cited by 9 (4 self)
 Add to MetaCart
Abstract We present a general method for introducing finitely axiomatizable "minimal " secondorder theories for various subclasses of P. We show that our theory VTC 0 for the complexity class TC 0 is RSUV isomorphic to the firstorder theory \Delta b
A thirdorder bounded arithmetic theory for PSPACE
 of Lecture Notes in Computer Science
, 2004
"... Abstract. We present a novel thirdorder theory W 1 1 of bounded arithmetic suitable for reasoning about PSPACE functions. This theory has the advantages of avoiding the smash function symbol and is otherwise much simpler than previous PSPACE theories. As an example we outline a proof in W 1 1 that ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
Abstract. We present a novel thirdorder theory W 1 1 of bounded arithmetic suitable for reasoning about PSPACE functions. This theory has the advantages of avoiding the smash function symbol and is otherwise much simpler than previous PSPACE theories. As an example we outline a proof in W 1 1 that from any configuration in the game of Hex, at least one player has a winning strategy. We then exhibit a translation of theorems of W 1 1 into families of propositional tautologies with polynomialsize proofs in BPLK (a recent propositional proof system for PSPACE and an alternative to G). This translation is clearer and more natural in several respects than the analogous ones for previous PSPACE theories. Keywords: Bounded arithmetic, propositional proof complexity, PSPACE, quantified propositional calculus 1
The Complexity of Proving Discrete Jordan Curve Theorem
 In Proc. 22nd IEEE Symposium on Logic in Computer Science
, 2007
"... The Jordan Curve Theorem (JCT) states that a simple closed curve divides the plane into exactly two connected regions. We formalize and prove the theorem in the context of grid graphs, under different input settings, in theories of bounded arithmetic that correspond to small complexity classes. The ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The Jordan Curve Theorem (JCT) states that a simple closed curve divides the plane into exactly two connected regions. We formalize and prove the theorem in the context of grid graphs, under different input settings, in theories of bounded arithmetic that correspond to small complexity classes. The theory V 0 (2) (corresponding to AC 0 (2)) proves that any set of edges that form disjoint cycles divides the grid into at least two regions. The theory V 0 (corresponding to AC 0) proves that any sequence of edges that form a simple closed curve divides the grid into exactly two regions. As a consequence, the Hex tautologies and the stConnectivity tautologies have polynomial size AC 0 (2)Fregeproofs, which improves results of Buss which only apply to the stronger proof system TC 0Frege.
COMPLEXITY THEORETIC ASPECTS OF PLANAR RESTRICTIONS AND OBLIVIOUSNESS
, 2006
"... In this thesis, we deal largely with complexity theoretic aspects in planar restrictions and obliviousness. Our main motivation was to identify problems for which the planar restriction is much easier, computationally, than the unrestricted version. First, we study constant width polynomialsized ci ..."
Abstract
 Add to MetaCart
In this thesis, we deal largely with complexity theoretic aspects in planar restrictions and obliviousness. Our main motivation was to identify problems for which the planar restriction is much easier, computationally, than the unrestricted version. First, we study constant width polynomialsized circuits of low (polylogarithmic) genus; we show how such circuits characterize exactly the wellknown circuit complexity class ACC0 (given that the unrestricted version captures the whole of NC1). We also give a new circuit characterization of the class NC1. Shifting our focus from circuits to graphs, we look at different notions of connectivity. We investigate the directed planar graph reachability problem, as a possibly more tractable special case of the arbitrary graph reachability problem (which is NLcomplete). We prove that this problem logspacereduces to its complement, and also that reachability questions on genus 1 graphs reduce to that in planar graphs. We also prove that reachability in a particularly simple class of planar graphs (namely, grid graphs) is no easier than the general directed planar reachability question. We then proceed to isolate to several large classes of planar graphs for which the reachability questions are solvable in deterministic logspace. Counting the number of spanning trees in a graph is a useful extension of the task of determining