Hirst investigated a slight variant of Hindman's Finite Sums Theorem called Hilbert's Theorem and proved it equivalent over RCA0 to the In nite Pigeonhole Principle for all colors. This gave the rst example of a natural restriction of Hindman's Theorem provably much weaker than Hindman's Theorem itself. We here introduce another natural variant of Hindman's Theorem which we name the Adjacent Hi...

We prove three switching lemmas, for random restrictions for which variables are set independently; for random restrictions where variables are set in blocks (both due to H̊astad [3]); and for a distribution appropriate for the bijective pigeonhole principle [2, 4]. The proofs are based on Beame’s version [1] of Razborov’s proof of the switching lemma in [5], except using families of weighted re...

Let Ωn be the n -element set consisting of functions that have [n] as both domain and codomain. Since Ωn is finite, it is clear by the pigeonhole principle that, for any f ∈ Ωn, the sequence of compositional iterates f, f , f , f (4) . . . must eventually repeat. Let T(f) be the period of this eventually periodic sequence of functions, i.e. the least positive integer T such that, for all m ≥ n,...

We investigate the possibility to characterize (multi)functions that are b i-deenable with small i (i = 1; 2; 3) in fragments of bounded arithmetic T2 in terms of natural search problems deened over polynomial-time structures. We obtain the following results: 1. A reformulation of known characterizations of (multi)functions that are b 1-and b 2-deenable in the theories S 1 2 and T 1 2. 2. New c...

Binary code search is a technique that involves finding with similarity to given within database. It finds extensive application in scenarios such as vulnerability queries and defect analysis. While many existing methods employ advanced machine learning models for analysis, their lack of interpretability low efficiency dealing large-scale functions still remain challenges. To address these issu...

8 0. Preliminaries In this note we comment on the relationships among local versions of the parameter free induction, collection and pigeonhole schemas. We start by defining the versions that will interest us. We work in the usual language of arithmetic to which a new constant symbol a has been added. P-denotes a finite set of axioms such that if M C P-, then M is the nonnegative part of a comm...

Zusammenfassung: We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a Prover-Delayer game which models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bo...

Let Ωn be the n -element set consisting of functions that have [n] as both domain and codomain. Since Ωn is finite, it is clear by the pigeonhole principle that, for any f ∈ Ωn, the sequence of compositional iterates f, f , f , f (4) . . . must eventually repeat. Let T(f) be the period of this eventually periodic sequence of functions, i.e. the least positive integer T such that, for all m ≥ n,...