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On Search Problems in Complexity Theory and in Logic (Abstract)

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Algorithms and Complexity (CIAC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3998))

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Abstract

A search problem is given by a binary relation B(x,y) in P, such that ∀ x ∃ y,|y| ≤ poly(|x|) B(x,y). The computational task is for given x find such a y. We believe that in general this is not possible in polynomial time and oracles are known for which this is the case.

Many-to-one and Turing reductions between search problems are defined in a natural way. We conjecture that there is no complete search problem.

Our aim is to classify search problems and show relations between the computational complexities of them and the proof complexities of the sentences ∀ x ∃ y,|y| ≤ poly(|x|) B(x,y). A typical example of a class of search problems is the class Polynomial Local Search defined as follows.

A PLS problem is given by a P-time relation R(p,x) and a P-time function F(p,y) such that R(n,n) holds for all n. The search problem is for every p and xp to find a yp such that

$$ R(p,y)\wedge (\neg F(p,y)<y\vee\neg R(p,F(p,y))). $$

A typical result relating proof complexity and computational complexity of search problems is the following theorem of Buss and Krajíček.

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© 2006 Springer-Verlag Berlin Heidelberg

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Pudlák, P. (2006). On Search Problems in Complexity Theory and in Logic (Abstract). In: Calamoneri, T., Finocchi, I., Italiano, G.F. (eds) Algorithms and Complexity. CIAC 2006. Lecture Notes in Computer Science, vol 3998. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11758471_3

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  • DOI: https://doi.org/10.1007/11758471_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-34375-2

  • Online ISBN: 978-3-540-34378-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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