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Logical Approaches to the Complexity of Search Problems: Proof Complexity, Quantified Propositional Calculus, and Bounded Arithmetic
, 2005
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The strength of replacement in weak arithmetic
 Proceedings of the Nineteenth Annual IEEE Symposium on Logic in Computer Science
, 2004
"... The replacement (or collection or choice) axiom scheme BB(Γ) asserts bounded quantifier exchange as follows: ∀i<a  ∃x<aφ(i,x) → ∃w ∀i<aφ(i,[w]i) proves the scheme where φ is in the class Γ of formulas. The theory S1 2 BB(Σb 1), and thus in S1 2 every Σb1 formula is equivalent to a stri ..."
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The replacement (or collection or choice) axiom scheme BB(Γ) asserts bounded quantifier exchange as follows: ∀i<a  ∃x<aφ(i,x) → ∃w ∀i<aφ(i,[w]i) proves the scheme where φ is in the class Γ of formulas. The theory S1 2 BB(Σb 1), and thus in S1 2 every Σb1 formula is equivalent to a strict Σb1 formula (in which all nonsharplybounded quantifiers are in front). Here we prove (sometimes subject to an assumption) that certain theories weaker than S1 2 do not prove either BB(Σb1) or BB(Σb0). We show (unconditionally) that V 0 does not prove BB(ΣB 0), where V 0 (essentially IΣ 1,b 0) is the twosorted theory associated with the complexity class AC0. We show that PV does not prove BB(Σb 0), assuming
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
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]...
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.
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.
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
The Equivalence of Theories that Characterize ALogTime
, 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.