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On Proofs About Threshold Circuits and Counting Hierarcies (Extended Abstract)
, 1998
"... ) Jan Johannsen Chris Pollett Department of Mathematics Department of Computer Science University of California, San Diego Boston University La Jolla, CA 910930112 Boston, MA 02215 Abstract We dene theories of Bounded Arithmetic characterizing classes of functions computable by constantdepth t ..."
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) Jan Johannsen Chris Pollett Department of Mathematics Department of Computer Science University of California, San Diego Boston University La Jolla, CA 910930112 Boston, MA 02215 Abstract We dene theories of Bounded Arithmetic characterizing classes of functions computable by constantdepth threshold circuits of polynomial and quasipolynomial size. Then we dene certain secondorder theories and show that they characterize the functions in the Counting Hierarchy. Finally we show that the former theories are isomorphic to the latter via the socalled RSUV isomorphism. 1 Introduction A phenomenon that is commonly observed in Complexity Theory is that proofs of results about counting complexity classes (#P , Mod p P etc.) can often be scaled down to yield results about small depth circuit classes with the corresponding counting gates. For example, Toda's result [17] that every problem in the Polynomial Hierarchy can be solved in polynomial time with an oracle for #P correspond...
Quantified Propositional Calculus and a SecondOrder Theory for NC¹
, 2004
"... Let H be a proof system for the quantified propositional calculus (QPC). We j witnessing problem for H to be: given a prenex S j formula A, an Hproof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S ..."
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Let H be a proof system for the quantified propositional calculus (QPC). We j witnessing problem for H to be: given a prenex S j formula A, an Hproof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S witnessing problems for the systems G 1 and G 1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce
Dynamic ordinals – universal measures for implicit computational complexity
, 2002
"... We extend the definition of dynamic ordinals to generalised dynamic ordinals. We compute generalised dynamic ordinals of all fragments of relativised bounded arithmetic by utilising methods from Boolean complexity theory, similar to Krajíček in [14]. We indicate the role of generalised dynamic ordin ..."
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We extend the definition of dynamic ordinals to generalised dynamic ordinals. We compute generalised dynamic ordinals of all fragments of relativised bounded arithmetic by utilising methods from Boolean complexity theory, similar to Krajíček in [14]. We indicate the role of generalised dynamic ordinals as universal measures for implicit computational complexity. I.e., we describe the connections between generalised dynamic ordinals and witness oracle Turing machines for bounded arithmetic theories. In particular, through the determination of generalised dynamic ordinals we reobtain wellknown independence results for relativised bounded arithmetic theories.
Thirdorder computation and bounded arithmetic
 University of Wales Swansea
, 2006
"... Abstract. We describe a natural generalization of ordinary computation to a thirdorder setting and give a function calculus with nice properties and recursiontheoretic characterizations of several large complexity classes. We then present a number of thirdorder theories of bounded arithmetic whos ..."
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Abstract. We describe a natural generalization of ordinary computation to a thirdorder setting and give a function calculus with nice properties and recursiontheoretic characterizations of several large complexity classes. We then present a number of thirdorder theories of bounded arithmetic whose definable functions are the classes of the EXPtime hierarchy in the thirdorder setting.
Consistency and Gamesin Search of New Combinatorial Principles
, 2004
"... We show that a semantical interpretation of Herbrand's disjunctions can be used to obtain 2 independent sentences whose nature is more combinatorial than the nature of the usual consistency statements. Then we apply this method to Bounded Arithmetic and present 8 1 combinatorial sentences tha ..."
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We show that a semantical interpretation of Herbrand's disjunctions can be used to obtain 2 independent sentences whose nature is more combinatorial than the nature of the usual consistency statements. Then we apply this method to Bounded Arithmetic and present 8 1 combinatorial sentences that characterize all 8 1 sentences provable in S 2 . We use the concept of a two player game to describe these sentences.
Fragments of Approximate Counting
, 2012
"... We study the longstanding open problem of giving ∀Σ b 1 separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeˇrábek’s theories for approximate counting and their subtheories. ..."
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We study the longstanding open problem of giving ∀Σ b 1 separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeˇrábek’s theories for approximate counting and their subtheories. We show that the ∀Σ b 1 Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective weak pigeonhole principle for FP NP functions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of T 1 2 augmented with the surjective weak pigeonhole principle for polynomial time functions.
Higher complexity search problems for bounded arithmetic and
, 2010
"... a formalized nogap theorem ..."
b Mathematical /nstitute at Prague, Czechoslovakia
"... In this article we study relations between various fragments of bounded arithmetic. The fragments of interest bere are the theories S ~ and T ~ introduced by Buss in [1]. The reader may recall that the principal axioms of S ~ resp. of T ~ are!tPIND resp.!tIND axioms, where the former are instances ..."
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In this article we study relations between various fragments of bounded arithmetic. The fragments of interest bere are the theories S ~ and T ~ introduced by Buss in [1]. The reader may recall that the principal axioms of S ~ resp. of T ~ are!tPIND resp.!tIND axioms, where the former are instances of
A Improved Witnessing and Local Improvement Principles for SecondOrder Bounded Arithmetic
"... This paper concerns the second order systems U1 2 and V1 2 of bounded arithmetic, which have proof theoretic strengths corresponding to polynomial space and exponential time computation. We formulate improved witnessing theorems for these two theories by using S1 2 as a base theory for proving the c ..."
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This paper concerns the second order systems U1 2 and V1 2 of bounded arithmetic, which have proof theoretic strengths corresponding to polynomial space and exponential time computation. We formulate improved witnessing theorems for these two theories by using S1 2 as a base theory for proving the correctness of the polynomial space or exponential time witnessing functions. We develop the theory of nondeterministic polynomialspace computation, includingSavitch’s theorem,inU 1 2.Kolodziejczyk, Nguyen, and Thapen have introduced local improvement properties to characterize the provably total NP functions of these second order theories. We show that the strengths of their local improvement principles over U1 2 and V1 2 depend primarily on the topology of the underlying graph, not the number of rounds in the local improvement games. The theory U1 2 proves the local improvement principle for linear graphs even without restricting to logarithmically many rounds. The local improvement principle for grid graphs with only logarithmically many rounds is complete for the provably total NP search problems of V1 2. Related results are obtained for local improvement principles with one improvement round, and for local improvement over rectangular grids.