Leszek Aleksander Kołodziejczyk

The provably total NP search problems of weak second order bounded arithmetic

Abstract 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 ∀ Σ 1 b consequences of V 2 1 and that natural restrictions of it characterize the ∀ Σ 1 b consequences of U 2 1 and of the bounded arithmetic hierarchy. We also show that over V 0 it characterizes the ∀ Σ 0 B consequences of V 1 and hence that, in some sense, a miniaturized version of the principle gives a new characterization of the ∀ Π 1 b consequences of S 2 1 . Throughout our search problems are “type-2” NP search problems, which take second-order objects as parameters.

Fragments of approximate counting

We study the long-standing open problem of giving ∀Σb1 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 Jeřabek’s theories for approximate counting and their subtheories. We show that the ∀Σb1 Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective weak pigeonhole principle for FP functions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of T 1 2 augmented with the surjective weak pigeonhole principle for polynomial time functions. §

Small stone in pool

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 resolution from DPLL with clause learning.

Solutions in XML data exchange

The task of XML data exchange is to restructure a document conforming to a source schema under a target schema according to certain mapping rules. The rules are typically expressed as source-to-target dependencies using various kinds of patterns, involving horizontal and vertical navigation, as well as data comparisons. The target schema imposes complex conditions on the structure of solutions, possibly inconsistent with the mapping rules. In consequence, for some source documents there may be no solutions. We investigate three problems: deciding if all documents of the source schema can be mapped to a document of the target schema (absolute consistency), deciding if a given document of the source schema can be mapped (solution existence), and constructing a solution for a given source document (solution building). We show that the complexity of absolute consistency is rather high in general, but within the polynomial hierarchy for bounded depth schemas. The combined complexity of solution existence and solution building behaves similarly, but the data complexity turns out to be very low. In addition to this we show that even for much more expressive mapping rules, based on MSO definable queries, absolute consistency is decidable and data complexity of solution existence is polynomial.

Collapsing modular counting in bounded arithmetic and constant depth propositional proofs

Jeřabek introduced fragments of bounded arithmetic which are axiomatized with weak surjective pigeonhole principles and support a robust notion of approximate counting. We extend these fragments of bounded arithmetic to accommodate modular counting quantifiers. These theories can formalize and prove the relativized versions of Toda’s theorem on the collapse of the polynomial hierarchy with modular counting. We introduce a version of the Paris-Wilkie translation for converting formulas and proofs of bounded arithmetic with modular counting quantifiers into constant depth propositional logic with modular counting gates. We also define Paris-Wilkie translations to Nullstellensatz and polynomial calculus refutations. As an application, we The first author was supported in part by NSF grant DMS-1101228. In the preliminary stages of this work, the second and third authors were supported by grant no. N N201 382234 of the Polish Ministry of Science and Higher Education. Most of this work was carried out while the second author was visiting the University of California, San Diego, supported by Polish Ministry of Science and Higher Education programme “Mobilnośc Plus” with additional support from a grant from the Simons Foundation (#208717 to Sam Buss).

Independence results for variants of sharply bounded induction

Abstract The theory T 2 0 , axiomatized by the induction scheme for sharply bounded formulae in Buss’ original language of bounded arithmetic (with ⌊ x / 2 ⌋ but not ⌊ x / 2 y ⌋ ), has recently been unconditionally separated from full bounded arithmetic S 2 . The method used to prove the separation is reminiscent of those known from the study of open induction. We make the connection to open induction explicit, showing that models of T 2 0 can be built using a “nonstandard variant” of Wilkie’s well-known technique for building models of I O p e n . This makes it possible to transfer many results and methods from open to sharply bounded induction with relative ease. We provide two applications: (i) the Shepherdson model of I O p e n can be embedded into a model of T 2 0 , which immediately implies some independence results for T 2 0 ; (ii) T 2 0 extended by an axiom which roughly states that every number has a least 1 bit in its binary notation, while significantly stronger than plain T 2 0 , does not prove the infinity of primes.

Well-behaved principles alternative to bounded induction

Abstract We introduce some Π1-expressible combinatorial principles which may be treated as axioms for some bounded arithmetic theories. The principles, denoted Sk(Σ n b ,length log k ) and Sk(Σ n b ,depth log k ) (where ‘Sk’ stands for ‘Skolem’), are related to the consistency of Σnb induction: for instance, they provide models for Σnb induction. However, the consistency is expressed indirectly, via the existence of evaluations for sequences of terms. The evaluations do not have to satisfy Σnb induction, but must determine the truth value of Σnb statements. Our principles have the property that Sk(Σ n b ,depth log k ) proves Sk(Σ n+1 b ,length log k ) . Additionally, Sk(Σ n b ,length log k−2 ) proves Sk(Σ n+1 b ,length log k ) . Thus, some provability is involved where conservativity is known in the case of Σnb induction on an initial segment and induction for higher Σmb classes on smaller segments.

New Bounds on the Strength of Some Restrictions of Hindman’s Theorem

We prove upper and lower bounds on the effective content and logical strength for a variety of natural restrictions of Hindmans Finite Sums Theorem. For example, we show that Hindmans Theorem for sums of length at most 2 and 4 colors implies

How unprovable is Rabin's decidability theorem?

\mathsf{ACA}_0

The Polynomial and Linear Hierarchies in V0

. An emerging {\em leitmotiv} is that the known lower bounds for Hindmans Theorem and for its restriction to sums of at most 2 elements are already valid for a number of restricted versions which have simple proofs and better computability- and proof-theoretic upper bounds than the known upper bound for the full version of the theorem. We highlight the role of a sparsity-like condition on the solution set, which we call apartness.