We study search problems and reducibilities between them with known or potential relevance to bounded arithmetic theories. Our primary objective is to understand the sets of low complexity consequences (esp. Σ b 1 or Σ b 2 ) of theories S i 2 and T i 2 for a small i, ideally in a rather strong sense
A sorting network in bounded arithmetic
✍ Scribed by Emil Jeřábek
- Publisher
- Elsevier Science
- Year
- 2011
- Tongue
- English
- Weight
- 300 KB
- Volume
- 162
- Category
- Article
- ISSN
- 0168-0072
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In this paper we introduce a system AID (alogtime inductive deÿnitions) of bounded arithmetic. The main feature of AID is to allow a form of inductive deÿnitions, which was extracted from Buss' propositional consistency proof of Frege systems F in Buss (Ann. Pure Appl. Logic 52 (1991) 3-29). We show
Let G(k, r) denote the smallest positive integer g such that if 1=a 1 , a 2 , ..., a g is a strictly increasing sequence of integers with bounded gaps a j+1 &a j r, 1 j g&1, then [a 1 , a 2 , ..., a g ] contains a k-term arithmetic progression. It is shown that G(k, 2) > -(k & 1)Â2 ( 43 ) (k&1)Â2 ,
The theories S i 1 (α) and T i 1 (α) are the analogues of Buss' relativized bounded arithmetic theories in the language where every term is bounded by a polynomial, and thus all definable functions grow linearly in length. For every i, a Σ b i+1 (α)-formula TOP i (a), which expresses a form of the t