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13
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.
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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Cited by 5 (3 self)
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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
A simple proof of Parsons' theorem
"... Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is ..."
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Cited by 3 (2 self)
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Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is a consequence of Herbrand's theorem concerning the of universal theories. We give a selfcontained proof requiring only basic knowledge of mathematical logic.
A QuantifierFree String Theory Alogtime Reasoning
, 2000
"... The main contribution of this work is the definition of a quantifierfree string theory T1 suitable for formalizing ALOGTIME reasoning. After describing L1—a new, simple, algebraic characterization of the complexity class ALOGTIME based on strings instead of numbers—the theory T1 is defined (based ..."
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Cited by 2 (0 self)
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The main contribution of this work is the definition of a quantifierfree string theory T1 suitable for formalizing ALOGTIME reasoning. After describing L1—a new, simple, algebraic characterization of the complexity class ALOGTIME based on strings instead of numbers—the theory T1 is defined (based on L1), and a detailed formal development of T1 is given. Then, theorems of T1 are shown to translate into families of propositional tautologies that have uniform polysize Frege proofs, T1 is shown to prove the soundness of a particular Frege system F, and F is shown to provably psimulate any proof system whose soundness can be proved in T1. Finally, T1 is compared with other theories for ALOGTIME reasoning in the literature. To our knowledge, this is the first formal theory for ALOGTIME reasoning whose basic objects are strings instead of numbers, and the first quantifierfree theory formalizing ALOGTIME reasoning in which a direct proof of the soundness of some Frege system has been given (in the case of firstorder theories, such a proof was first given by Arai for his theory AID). Also, the polysize Frege proofs we give for the propositional translations of theorems of T1 are considerably simpler than those for other theories, and so is our proof of the soundness of a particular
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.
EXAMINING FRAGMENTS OF THE QUANTIFIED PROPOSITIONAL CALCULUS
"... Abstract. When restricted to proving Σ q i formulas, the quantified propositional proof system G ∗ i is closely related to the Σbi theorems of Buss’s theory Si 2. Namely, G∗i has polynomialsize proofs of the translations of theorems of S i 2, and Si 2 proves that G∗ i is sound. However, little is k ..."
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Abstract. When restricted to proving Σ q i formulas, the quantified propositional proof system G ∗ i is closely related to the Σbi theorems of Buss’s theory Si 2. Namely, G∗i has polynomialsize proofs of the translations of theorems of S i 2, and Si 2 proves that G∗ i is sound. However, little is known about G ∗ i when proving more complex formulas. In this paper, we prove a witnessing theorem for G ∗ i similar in style to the KPT witnessing theorem for T i 2. This witnessing theorem is then used to show that Si 2 proves G∗ i is sound with respect to Σ q i+1 formulas. Note that unless the polynomialtime hierarchy collapses S i 2 is the weakest theory in the S2 hierarchy for which this is true. The witnessing theorem is also used to show that G ∗ 1 is pequivalent to a quantified version of extendedFrege for prenex formulas. This is followed by a proof that Gi psimulates G ∗ i+1. We finish by proving that S2 can be axiomatized by S 1 2 plus axioms stating that the cutfree version of G ∗ 0 is sound. All together this shows that the connection between G∗
On the Lengths of Proofs of Consistency  a Survey of Results
"... This article is essentially a part of my thesis for the degree DrSc (Doctor of Sciences). Therefore it mainly surveys my articles [41, 42, 43, 28, 29, 44, 22], and it is structured according to the requirements for such theses. I made only minor changes in the original text and added a few further r ..."
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This article is essentially a part of my thesis for the degree DrSc (Doctor of Sciences). Therefore it mainly surveys my articles [41, 42, 43, 28, 29, 44, 22], and it is structured according to the requirements for such theses. I made only minor changes in the original text and added a few further references. Since Godel's main achievement concerns the problem of consistency and some of the problems that I am going to describe had been considered by him, I think that it is appropriate to publish this article in Godel Society. 1 Historical remarks The question that we are going to consider in is interesting per se and is related to some more practical questions, especially in complexity theory, but the original motivation for it comes from foundational studies. Among the variety of streams in foundations of mathematics, the one which had the biggest influence and which very much determined later development of mathematical logic was Hilbert's<F1
Approximate counting by hashing in bounded arithmetic
, 2008
"... We show how to formalize approximate counting via hash functions in subsystems of bounded arithmetic, using variants of the weak pigeonhole principle. We discuss several applications, including a proof of the tournament principle, and an improvement on the known relationship of the collapse of the b ..."
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We show how to formalize approximate counting via hash functions in subsystems of bounded arithmetic, using variants of the weak pigeonhole principle. We discuss several applications, including a proof of the tournament principle, and an improvement on the known relationship of the collapse of the bounded arithmetic hierarchy to the collapse of the polynomialtime hierarchy.
Approximate counting in bounded arithmetic
, 2007
"... We develop approximate counting of sets definable by Boolean circuits in bounded arithmetic using the dual weak pigeonhole principle (dWPHP(P V)), as a generalization of results from [15]. We discuss applications to formalization of randomized complexity classes (such as BPP, APP, MA, AM) in P V1 + ..."
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We develop approximate counting of sets definable by Boolean circuits in bounded arithmetic using the dual weak pigeonhole principle (dWPHP(P V)), as a generalization of results from [15]. We discuss applications to formalization of randomized complexity classes (such as BPP, APP, MA, AM) in P V1 + dWPHP(P V).
Bounded Arithmetic
, 2008
"... Definable functions Language of Bounded Arithmetic (BA) Language of first order arithmetic similar to Peano Arithmetic Nonlogical symbols: {0,1,+, ·, ≤} + {.,#,...} x  = length of binary representation of x x#y = 2 x·y  produces polynomial growth rate Arnold Beckmann (joint work with Klaus ..."
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Definable functions Language of Bounded Arithmetic (BA) Language of first order arithmetic similar to Peano Arithmetic Nonlogical symbols: {0,1,+, ·, ≤} + {.,#,...} x  = length of binary representation of x x#y = 2 x·y  produces polynomial growth rate Arnold Beckmann (joint work with Klaus Aehlig)