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Relativizing Small Complexity Classes and their Theories
 In 16th EACSL Annual Conference on Computer Science and Logic
, 2007
"... Existing definitions of the relativizations of NC 1, L and NL do not preserve the inclusions NC 1 ⊆ L, NL ⊆ AC 1. We start by giving the first definitions that preserve them. Here for L and NL we define their relativizations using Wilson’s stack oracle model, but limit the height of the stack to a c ..."
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Existing definitions of the relativizations of NC 1, L and NL do not preserve the inclusions NC 1 ⊆ L, NL ⊆ AC 1. We start by giving the first definitions that preserve them. Here for L and NL we define their relativizations using Wilson’s stack oracle model, but limit the height of the stack to a constant (instead of log(n)). We show that the collapse of any two classes in {AC 0 (m),TC 0,NC 1,L,NL} implies the collapse of their relativizations. Next we develop theories that characterize the relativizations of subclasses of P by modifying theories previously defined by the second two authors. A function is provably total in a theory iff it is in the corresponding relativized class. Finally we exhibit an oracle α that makes AC k (α) a proper hierarchy. This strengthens and clarifies the separations of the relativized theories in [Takeuti, 1995]. The idea is that a circuit whose nested depth of oracle gates is bounded by k cannot compute correctly the (k + 1) compositions of every oracle function. 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 ..."
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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.
Research Statement
, 2006
"... 2 Background and Past Work The primary motivation for studying propositional proof systems is the theorem of Cook and Reckhow[8, 11] that NP=coNP iff there exists a polynomially bounded proof system for propositional tautologies. Many proof systems are studied and there are some notable successes i ..."
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2 Background and Past Work The primary motivation for studying propositional proof systems is the theorem of Cook and Reckhow[8, 11] that NP=coNP iff there exists a polynomially bounded proof system for propositional tautologies. Many proof systems are studied and there are some notable successes in the search for lower bounds, e.g.[13, 1], but this problem is very hard in general; nevertheless, there are many much more accessible problems than NP vs coNP: at one end of the scale, the detailed study of weak proof systems and their interrelations,and at the other, capturing different forms of reasoning with stronger proof systems.
Proving Infinitude of Prime Numbers Using Binomial Coefficients
, 2008
"... We study the problem of proving in weak theories of Bounded Arithmetic the theorem that there are arbitrarily large prime numbers. We show that the theorem can be proved by some “minimal ” reasoning (i.e., in the theory I∆0) using concepts such as (the logarithm) of a binomial coefficient. In fact w ..."
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We study the problem of proving in weak theories of Bounded Arithmetic the theorem that there are arbitrarily large prime numbers. We show that the theorem can be proved by some “minimal ” reasoning (i.e., in the theory I∆0) using concepts such as (the logarithm) of a binomial coefficient. In fact we prove Bertrand’s Postulate (that there is at least a prime number between n and 2n, for all n> 1) and the fact that the number of prime numbers between n and 2n is of order Θ(n/ln(n)). The proofs that we formalize are much simpler than several existing formalizations, and our theory turns out to be a subtheory of a recent theory proposed by Woods and Cornaros that extends I∆0 by a special counting function. 1