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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.
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
The Equivalence of Theories that Characterize
, 2007
"... A number of theories have been developed to characterize ALogTime (or uniform NC 1, or just NC 1), the class of languages accepted by alternating logtime Turing machines, in the same way that Buss’s theory S 1 2 characterizes polytime functions. Among these, ALV ′ (by Clote) is particularly interest ..."
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A number of theories have been developed to characterize ALogTime (or uniform NC 1, or just NC 1), the class of languages accepted by alternating logtime Turing machines, in the same way that Buss’s theory S 1 2 characterizes polytime functions. Among these, ALV ′ (by Clote) is particularly interesting because it is developed based on Barrington’s theorem that the word problem for the permutation group S5 is complete for ALogTime. On the other hand, ALV (by Clote), T 0 NC 0 (by Clote and Takeuti) as well as Arai’s theory AID + Σ B 0CA and its twosorted version VNC 1 (by Cook and Morioka) are based on the circuit characterization of ALogTime. While the last three theories have been known to be equivalent, their relationship to ALV ′ has been an open problem. Here we show that ALV ′ is indeed equivalent to the other theories. 1
Abstract Theories and Proof Systems for PSPACE and the EXPTime Hierarchy
, 2006
"... This dissertation concerns theories of bounded arithmetic and propositional proof systems associated with PSPACE and classes from the exponentialtime hierarchy. The secondorder viewpoint of Zambella and Cook associates secondorder theories of bounded arithmetic with various complexity classes by ..."
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This dissertation concerns theories of bounded arithmetic and propositional proof systems associated with PSPACE and classes from the exponentialtime hierarchy. The secondorder viewpoint of Zambella and Cook associates secondorder theories of bounded arithmetic with various complexity classes by studying the definable functions of strings, rather than numbers. This approach simplifies presentation of the theories and their propositional translations, and furthermore is applicable to complexity classes that previously had no corresponding theories. We adapt this viewpoint to large complexity classes from the exponentialtime hierarchy by adding a third sort, intended to represent exponentially long strings (“superstrings”), and capable of coding, for example, the computation of an exponentialtime Turing machine. Specifically, our main theories W i 1 and T W i 1 are associated with PSPACEΣp i−1 and EXPΣp i−1, respectively. We also develop a model for computation in this thirdorder setting including a function calculus, and define thirdorder analogues of ordinary complexity classes. We then obtain recursiontheoretic characterizations of our function classes for FP, FPSPACE and FEXP. We use our characterization of FPSPACE as the basis for an open theory for PSPACE that is a
Abstract
, 2010
"... 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
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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.