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14
Improved Bounds on the Weak Pigeonhole Principle and Infinitely Many Primes from Weaker Axioms
, 2001
"... We show that the known boundeddepth proofs of the Weak Pigeonhole Principle PHP 2n n in size n O(log(n)) are not optimal in terms of size. More precisely, we give a sizedepth tradeoff upper bound: there are proofs of size n O(d(log(n)) 2=d ) and depth O(d). This solves an open problem ..."
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We show that the known boundeddepth proofs of the Weak Pigeonhole Principle PHP 2n n in size n O(log(n)) are not optimal in terms of size. More precisely, we give a sizedepth tradeoff upper bound: there are proofs of size n O(d(log(n)) 2=d ) and depth O(d). This solves an open problem of Maciel, Pitassi and Woods (2000). Our technique requires formalizing the ideas underlying Nepomnjascij's Theorem which might be of independent interest. Moreover, our result implies a proof of the unboundedness of primes in I \Delta 0 with a provably weaker `large number assumption' than previously needed.
Factorization in generalized power series
 Trans. Amer. Math. Soc
"... Abstract. The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group G is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring R((G≤0)) consisting of the generalized power series with non ..."
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Abstract. The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group G is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring R((G≤0)) consisting of the generalized power series with nonpositive exponents. The following candidate for such an irreducible series was given by Conway (1976): � n t−1/n + 1. Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway’s series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If G =(R,+,0,≤) we can give the following test for irreducibility based only on the order type of the support of the series: if the order type is either ω or of the form ωωα and the series is not divisible by any monomial, then it is irreducible. To handle the general case we use a suggestion of M.H. Mourgues, based on an idea of Gonshor, which allows us to reduce to the special case G = R. In the final part of the paper we study the irreducibility of series with finite support. 1.
The Complexity of ResourceBounded Propositional Proofs
, 2001
"... Propositional Proof Complexity is an active area of research whose main focus is the study of the length of proofs in propositional logic. There are several motivations for such a study, the main of which is probably its connection to the P vs NP problem in Computational Complexity. The experience i ..."
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Propositional Proof Complexity is an active area of research whose main focus is the study of the length of proofs in propositional logic. There are several motivations for such a study, the main of which is probably its connection to the P vs NP problem in Computational Complexity. The experience in the field has revealed that the most interesting parameter of a proof system is the set of allowed formulas. The exact set of rules and axioms of the system is often irrelevant as long as they remain sound and reasonable. This parameterization by the set of allowed formulas establishes a link with the field of Boolean complexity where bounds are imposed on different computational resources as a classification tool. This idea is adopted in the proof complexity setting too. In this thesis we study the complexity of several ’resourcebounded ’ proof systems. The first set of results of the thesis is about a resourcebounded proof system that we call the Monotone Sequent Calculus (MLK). This is the standard propositional Gentzen Calculus (LK) when negation is not allowed in the formulas. The main result is that the use of negation does not yield exponential savings in the length of proofs. More precisely, we show that MLK quasipolynomially simulates LK on monotone sequents. We also show that, as refutation systems, MLK
A BottomUp Approach to Foundations of Mathematics
"... this paper is to survey some results which should give an idea to an outsider of what is going on in this eld and explain motivations for the studied problems. We recommend [3, 5, 15, 11, 34] to those who want to learn more about this subject ..."
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this paper is to survey some results which should give an idea to an outsider of what is going on in this eld and explain motivations for the studied problems. We recommend [3, 5, 15, 11, 34] to those who want to learn more about this subject
Real closures of models of weak arithmetic
, 2011
"... D’Aquino et al. (J. Symb. Log. 75(1)(2010)) have recently shown that every realclosed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by I∆0. We prove that the theorem holds if IΣ4 is replaced by weak subtheori ..."
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D’Aquino et al. (J. Symb. Log. 75(1)(2010)) have recently shown that every realclosed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by I∆0. We prove that the theorem holds if IΣ4 is replaced by weak subtheories of Buss ’ bounded arithmetic: PV or Σ b 1IND xk. It also holds for I∆0 (and even its subtheory IE 2) under a rather mild assumption on cofinality. On the other hand, it fails for the extension of IOpen by an axiom expressing the Bézout property, even under the same assumption on cofinality. A discretely ordered subring A of a realclosed field (henceforth often: rcf) R is an integer part of R if for every r ∈ R there exists a ∈ A such that a ≤ r < a + 1. It is wellknown that every rcf has an integer part [MR93], which is then a model of the weak arithmetic theory IOpen (induction for quantifierfree formulas in the language of ordered rings). On the other hand, every model of IOpen is an integer part of its real closure (or, more precisely, the real closure of its fraction field). Recently, d’Aquino et al. [DKS10] studied the question which rcfs have integer parts satisfying more arithmetic, e.g. Peano Arithmetic. It turns out
A NOTE ON THE JOINT EMBEDDING PROPERTY IN FRAGMENTS OF ARITHMETIC
"... It is known that full Peano Arithmetic does not have the joint embedding property (JEP). At the other extreme of the hierarchy, Open Induction also fails to have this property. We prove, using some conservation results about fragments of arithmetic, that if T is a theory consistent with PA and T\ I ..."
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It is known that full Peano Arithmetic does not have the joint embedding property (JEP). At the other extreme of the hierarchy, Open Induction also fails to have this property. We prove, using some conservation results about fragments of arithmetic, that if T is a theory consistent with PA and T\ IE ^ (bounded existential parameterfree induction), then any two models of PA which jointly embed in a model of T also jointly embed in an elementary extension of one of them. In particular, any fragment of PA extending IE [ fails to have JEP. 1.
Real closures of models of weak arithmetic
, 2011
"... D’Aquino et al. (J. Symb. Log. 75(1)(2010)) have recently shown that every realclosed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by I∆0. We prove that the theorem holds if IΣ4 is replaced by weak subtheori ..."
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D’Aquino et al. (J. Symb. Log. 75(1)(2010)) have recently shown that every realclosed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by I∆0. We prove that the theorem holds if IΣ4 is replaced by weak subtheories of Buss ’ bounded arithmetic: PV or Σ b 1IND xk. It also holds for I∆0 (and even its subtheory IE 2) under a rather mild assumption on cofinality. On the other hand, it fails for the extension of IOpen by an axiom expressing the Bézout property, even under the same assumption on cofinality. A discretely ordered subring A of a realclosed field (henceforth often: rcf) R is an integer part of R if for every r ∈ R there exists a ∈ A such that a ≤ r < a + 1. It is wellknown that every rcf has an integer part [MR93], which is then a model of the weak arithmetic theory IOpen (induction for quantifierfree formulas in the language of ordered rings). On the other hand, every model of IOpen is an integer part of its real closure (or, more precisely, the real closure of its fraction field). Recently, d’Aquino et al. [DKS10] studied the question which rcfs have integer parts satisfying more arithmetic, e.g. Peano Arithmetic. It turns out
On Ordered Fields with Infinitely Many Integer Parts
"... We investigate integer parts of ordered fields. We prove the existence of normal integer parts for a class of ordered fields. Along with the normal one we construct infinitely many elementary nonequivalent integer parts for each field from this class. ..."
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We investigate integer parts of ordered fields. We prove the existence of normal integer parts for a class of ordered fields. Along with the normal one we construct infinitely many elementary nonequivalent integer parts for each field from this class.
On Normal Integer Parts of Real Closed Fields
"... SHEPHERDSON [6]: The models of Open Induction (OI) are the Integer Parts (IP) of real closed fields (RCF). WILKIE [7]: Each discretely ordered Zring can be embedded in a model of OI. ..."
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SHEPHERDSON [6]: The models of Open Induction (OI) are the Integer Parts (IP) of real closed fields (RCF). WILKIE [7]: Each discretely ordered Zring can be embedded in a model of OI.