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Notes on Polynomially Bounded Arithmetic
"... We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general modeltheoretical investigations on fragments of bounded arithmetic. Contents 0 Introduction and motivation. 1 1 Preliminaries. 3 1.1 The p ..."
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We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general modeltheoretical investigations on fragments of bounded arithmetic. Contents 0 Introduction and motivation. 1 1 Preliminaries. 3 1.1 The polynomially bounded hierarchy. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.2 The axioms of secondorder bounded arithmetic. : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.3 Rudimentary functions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.4 Other fragments. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.5 Polynomial time computable functions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.6 Relations among fragments. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 1.7 Relations with Buss' bounded arithmetic. : : : :...
Structure and Definability in General Bounded Arithmetic Theories
, 1999
"... This paper is motivated by the questions: what are the \Sigma ..."
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This paper is motivated by the questions: what are the \Sigma
Relating the PSPACE reasoning power of Boolean Programs and Quantified Boolean Formulas
, 2000
"... We present a new propositional proof system based on a recent new characterization of
polynomial space (PSPACE) called Boolean Programs, due to Cook and Soltys. We show
that this new system, BPLK, is polynomially equivalent to the system G, which is based
on the familiar and very different quantifie ..."
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Cited by 13 (9 self)
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We present a new propositional proof system based on a recent new characterization of
polynomial space (PSPACE) called Boolean Programs, due to Cook and Soltys. We show
that this new system, BPLK, is polynomially equivalent to the system G, which is based
on the familiar and very different quantified Boolean formula (QBF) characterization of
PSPACE due to Stockmeyer and Meyer. We conclude with a discussion of some closely
related open problems and their implications.
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
End Extensions of Models of Linearly Bounded Arithmetic
, 1994
"... We show that every model of I \Delta 0 has an end extension to a model of a theory (extending Buss' S 0 2 ) where logspace computable function are formalizable. We also show the existence of an isomorphism between models of I \Delta 0 and models of linear arithmetic LA (i.e., secondorder Presb ..."
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We show that every model of I \Delta 0 has an end extension to a model of a theory (extending Buss' S 0 2 ) where logspace computable function are formalizable. We also show the existence of an isomorphism between models of I \Delta 0 and models of linear arithmetic LA (i.e., secondorder Presburger arithmetic with finite comprehension for bounded formulas). 0 Introduction. In the last two decades the research on bounded arithmetic has been focussed mainly on the theories of I \Delta 0 and I \Delta 0+\Omega 1 and on their fragments. The interest in these theories is in part motivated by complexity theoretical considerations. It is well known that the provably recursive function of I \Delta 0+\Omega 1 are those computable by algorithms in the polynomial time hierarchy while the provably recursive functions of I \Delta 0 correspond to the linear time hierarchy. Some open problems in bounded arithmetic are known to be equivalent to problems of complexity theory. More precisely, to the s...
The provable total search problems of bounded arithmetic
, 2007
"... We give combinatorial principles GIk, based on kturn games, which are complete for the class of NP search problems provably total at the kth level T k 2 of the bounded arithmetic hierarchy and hence characterize the ∀ ˆ Σ b 1 consequences of T k 2, generalizing the results of [20]. Our argument use ..."
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We give combinatorial principles GIk, based on kturn games, which are complete for the class of NP search problems provably total at the kth level T k 2 of the bounded arithmetic hierarchy and hence characterize the ∀ ˆ Σ b 1 consequences of T k 2, generalizing the results of [20]. Our argument uses a translation of first order proofs into large, uniform propositional proofs in a system in which the soundness of the rules can be witnessed by polynomial time reductions between games. We show that ∀ ˆ Σ b 1(α) conservativity of of T i+1 2 (α) over T i 2(α) already implies ∀ ˆ Σ b 1(α) conservativity of T2(α) over T i 2(α). We translate this into propositional form and give a polylogarithmic width CNF GI3 such that if GI3 has small R(log) refutations then so does any polylogarithmic width CNF which has small constant depth refutations. We prove a resolution lower bound for GI3. We use our characterization to give a sufficient condition for the totality of a relativized NP search problem to be unprovable in T i 2(α) in terms of a nonlogical question about multiparty communication protocols.
Characterising Definable Search Problems in Bounded Arithmetic via Proof Notations
, 2009
"... The complexity class of Π p kPolynomial Local Search (PLS) problems with Π p ℓgoal is introduced, and is used to give new characterisations of definable search problems in fragments of Bounded Arithmetic. The characterisations are established via notations for propositional proofs obtained by tran ..."
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The complexity class of Π p kPolynomial Local Search (PLS) problems with Π p ℓgoal is introduced, and is used to give new characterisations of definable search problems in fragments of Bounded Arithmetic. The characterisations are established via notations for propositional proofs obtained by translating Bounded Arithmetic proofs using the ParisWilkietranslation. For ℓ ≤ k, the Σb ℓ+1definable search problems of T k+1 2 are exactly characterised by Π p kPLS problems with Πp ℓgoals. These Π p kPLS problems can be defined in a weak base theory such as S1 2, and proved to be total in T k+1 2. Furthermore, the Π p kPLS definitions can be Skolemised with simple polynomial time functions. The Skolemised Π p kPLS definitions give rise to a new ∀Σb1(α) principle conjectured to separate Tk 2(α) from T k+1 2 (α). 1
Comparing Constructive Arithmetical Theories Based On NPPIND and coNPPIND
"... In this note we show that the intuitionistic theory of polynomial induction on Π b+ 1formulas does not imply the intuitionistic theory IS1 2 of polynomial induction on Σ b+ 1formulas. We also show the converse assuming the Polynomial Hierarchy does not collapse. Similar results hold also for lengt ..."
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In this note we show that the intuitionistic theory of polynomial induction on Π b+ 1formulas does not imply the intuitionistic theory IS1 2 of polynomial induction on Σ b+ 1formulas. We also show the converse assuming the Polynomial Hierarchy does not collapse. Similar results hold also for length induction in place of polynomial induction. We also investigate the relation between various other intuitionistic firstorder theories of bounded arithmetic. Our method is mostly semantical, we use Kripke models of the theories.