Leszek Aleksander Kolodziejczyk

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We define a new NP search problem, the “local improvement” principle, about labellings of an acyclic, bounded-degree graph. We show that, provably in PV, it characterizes the ∀Σb1 consequences of V 1 2 and that natural restrictions of it characterize the ∀Σb1 consequences of U 1 2 and of the bounded arithmetic hierarchy. We also show that over V 0 it(More)
We study the long-standing open problem of giving ∀Σ1 separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek’s theories for approximate counting and their subtheories. We show that the ∀Σ1 Herbrandized ordering principle is(More)
Modifying the methods of Z. Adamowicz’s paper Herbrand consistency and bounded arithmetic (Fund. Math. 171 (2002)), we show that there exists a number n such that ⋃ m Sm (the union of the bounded arithmetic theories Sm) does not prove the Herbrand consistency of the finitely axiomatizable theory Sn 3 . From the point of view of bounded arithmetic, the(More)
The paper discusses the notion of finite model truth definitions (or FM–truth definitions), introduced by M. Mostowski as a finite model analogue of Tarski’s classical notion of truth definition. We compare FM–truth definitions with Vardi’s concept of the combined complexity of logics, noting an important difference: the difficulty of defining FM–truth for(More)
We use finite model theory (in particular, the method of FM–truth definitions, introduced in [MM01] and developed in [K04], and a normal form result akin to those of [Ste93] and [G97]) to prove: Let m ≥ 2. Then: (A) If there exists k such that NP ⊆ ΣmTIME(n)∩ΠmTIME(n), then for every r there exists kr such that PNP [n r] ⊆ ΣmTIME(nr)∩ ΠmTIME(nr); (B) If(More)
We show, under the assumption that factoring is hard, that a model of PV exists in which the polynomial hierarchy does not collapse to the linear hierarchy; that a model of S2 exists in which NP is not in the second level of the linear hierarchy; and that a model of S2 exists in which the polynomial hierarchy collapses to the linear hierarchy and in which(More)
We prove that: • If there is a model of I∆0 + ¬exp with cofinal Σ1-definable elements and a Σ1 truth definition for Σ1 sentences, then I∆0 + ¬exp + ¬BΣ1 is consistent, • there is a model of I∆0 + Ω1 + ¬exp with cofinal Σ1-definable elements, both a Σ2 and a Π2 truth definition for Σ1 sentences, and for each n ≥ 2, a Σn truth definition for Σn sentences. The(More)
The Stone tautologies are known to have polynomial size resolution refutations and require exponential size regular refutations. We prove that the Stone tautologies also have polynomial size proofs in both pool resolution and the proof system of regular tree-like resolution with input lemmas (regRTI). Therefore, the Stone tautologies do not separate(More)
Simpson and Yokoyama [Ann. Pure Appl. Logic 164 (2012), 284–293] asked whether there exists a characterization of the natural numbers by a second-order sentence which is provably categorical in the theory RCA0. We answer in the negative, showing that for any characterization of the natural numbers which is provably true in WKL0, the categoricity theorem(More)