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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 ..."
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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
V TC 0 : A SecondOrder Theory for TC 0
 In Proc. 19th IEEE Symposium on Logic in Computer Science, 2004. 137
, 2004
"... Abstract We introduce a finitely axiomatizable secondorder theory VTC0, which is associated with the class FOuniform TC0. It consists of the base theory V0 for AC0 reasoningtogether with the axiom NUMONES, which states the existence of a "counting array" Y for any string X: the ith row ..."
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Abstract We introduce a finitely axiomatizable secondorder theory VTC0, which is associated with the class FOuniform TC0. It consists of the base theory V0 for AC0 reasoningtogether with the axiom NUMONES, which states the existence of a "counting array" Y for any string X: the ith rowof Y contains only the number of 1 bits up to (excluding) bit i of X. We introduce the notion of "strong \Delta B1definability"for relations in a theory, and use a recursive characterization of the TC0 relations (rather than functions) to showthat the TC0 relations are strongly \Delta B1definable. It follows that the TC0 functions are \Sigma B1definable in VTC0.We prove a general witnessing theorem for secondorder theories and conclude that the \Sigma B1 theorems of VTC0 arewitnessed by TC0 functions. We prove that VTC0 is RSUVisomorphic to the first order theory \Delta b1CR of Johannsenand Pollett (the "minimal theory for TC0"). \Delta b1CR includes the \Delta b1 comprehension rule, and J and P ask whetherthere is an upper bound to the nesting depth required for this rule. We answer "yes", because VTC0, and therefore \Delta b1CR, are finitely axiomatizable. Finally, we showthat \Sigma B0 theorems of VTC0 translate to families of tautologies which have polynomialsize constantdepth TC0Frege proofs. We also show that PHP is a \Sigma B0 theoremof VTC0. These together imply that the family of propositional tautologies associated with PHP has polynomialsize constantdepth TC0Frege proofs.
Improved Depth Lower Bounds for Small Distance Connectivity
, 1995
"... We consider the problem of determining, given a graph G and specified nodes s and t, whether or not there is a path of at most k edges in G from s to t. We show that solving this problem on polynomialsize unbounded fanin circuits, requires depth \Omega\Gammapth log k), improving on a depth lower b ..."
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We consider the problem of determining, given a graph G and specified nodes s and t, whether or not there is a path of at most k edges in G from s to t. We show that solving this problem on polynomialsize unbounded fanin circuits, requires depth \Omega\Gammapth log k), improving on a depth lower bound of \Omega\Gamma/16 k) when k = log O(1) n given in [2, 8]. In addition we show that there is a constant c such that for k log n, any depth d unbounded fanin circuit for this problem requires size at least n ck ffl d where ffl d = OE \Gamma2d =3 and OE is the golden mean. This latter result improves on an n \Omega\Gamma711 (d+3) k) bound from [2, 8] where log (i) is the ifold composition of log with itself. The key to our technique is a new form of `switching lemma' which combines some of the features of iteratively shortening terms due to Furst, Saxe, and Sipser [13] and Ajtai [1] with the kinds of switching lemma arguments introduced by Yao [18], Hastad [14], and Cai [9]...
Combinatorics in Bounded Arithmetics
, 2004
"... A basic aim of logic is to consider what axioms are used in proving various theorems of mathematics. This thesis will be concerned with such issues applied to a particular area of mathematics: combinatorics. We will consider two widely known groups of proof methods in combinatorics, namely, probabil ..."
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A basic aim of logic is to consider what axioms are used in proving various theorems of mathematics. This thesis will be concerned with such issues applied to a particular area of mathematics: combinatorics. We will consider two widely known groups of proof methods in combinatorics, namely, probabilistic methods and methods using linear algebra. We will consider certain applications of such methods, both of which are significant to Ramsey theory. The systems we choose to work in are various theories of bounded arithmetic. For the probabilistic method, the key point is that we use the weak pigeonhole principle to simulate the probabilistic reasoning. We formalize various applications of the ordinary probabilistic method and linearity of expectations, making partial progress on the Local Lemma. In the case of linearity of expectations, we show how to eliminate the weak pigeonhole principle by simulating the derandomization technique of “conditional probabilities.” We consider linear algebra methods applied to various set system theorems. We formalize some theorems using a linear algebra principle as an extra axiom. We also show how weaker results can be attained by giving alternative proofs that avoid linear algebra, and thus also avoid the extra axiom. We formalize upper and lower Ramsey bounds. For the lower bounds, both the probabilistic methods and the linear algebra methods are used. We provide a stratification of the various Ramsey lower bounds, showing that stronger bounds can be proved in stronger theories. A natural question is whether or not the axioms used are necessary. We provide “reversals” in a few cases, showing that the principle used to prove the theorem is in fact a consequence of the theorem (over some base theory). Thus this work can be seen as a (humble) beginning in the direction of developing the Reverse Mathematics of finite combinatorics.
Graph Isomorphism is not AC 0 reducible to Group Isomorphism
"... We give a new upper bound for the Group and Quasigroup Isomorphism problems when the input structures are given explicitly by multiplication tables. We show that these problems can be computed by polynomial size nondeterministic circuits of unbounded fanin with O(log log n) depth and O(log 2 n) non ..."
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We give a new upper bound for the Group and Quasigroup Isomorphism problems when the input structures are given explicitly by multiplication tables. We show that these problems can be computed by polynomial size nondeterministic circuits of unbounded fanin with O(log log n) depth and O(log 2 n) nondeterministic bits, where n is the number of group elements. This improves the existing upper bound from [Wol94] for the problems. In the previous upper bound the circuits have bounded fanin but depth O(log 2 n) and also O(log 2 n) nondeterministic bits. We then prove that the kind of circuits from our upper bound cannot compute the Parity function. Since Parity is AC 0 reducible to Graph Isomorphism, this implies that Graph Isomorphism is strictly harder than Group or Quasigroup Isomorphism under the ordering defined by AC 0 reductions.