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NP search problems in low fragments of bounded arithmetic

Published online by Cambridge University Press:  12 March 2014

Jan Krajíček ,
Alan Skelley  and
Neil Thapen
Jan Krajíček
Affiliation:
Mathematical Institute, Academy of Sciences, Prague, Czech Republic Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic, E-mail: [email protected]
Alan Skelley
Affiliation:
Mathematical Institute, Academy of Sciences, Prague, Czech Republic, E-mail: [email protected]
Neil Thapen
Affiliation:
Mathematical Institute, Academy of Sciences, Prague, Czech Republic, E-mail: [email protected]
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Abstract

We give combinatorial and computational characterizations of the NP search problems definable in the bounded arithmetic theories and .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 2 , June 2007 , pp. 649 - 672
Copyright
Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

[1]Beame, Paul, Cook, Stephen, Edmonds, Jeff, Impagliazzo, Russell, and Pitassi, Toniann, The relative complexity of NP search problems, Journal of Computer and System Sciences, vol. 57 (1998), no. 1, pp. 3–19.CrossRefGoogle Scholar
[2]Buss, S., Bounded arithmetic, Bibliopolis, Naples, 1986.Google Scholar
[3]Buss, Samuel and Krajíček, Jan, An application of Boolean complexity to separation problems in bounded arithmetic, Proceedings of the London Mathematical Society, vol. 69 (1994), no. 3, pp. 1–21.Google Scholar
[4]Chiari, Mario and Krajíček, Jan, Witnessing functions in bounded arithmetic and search problems, this Journal, vol. 63 (1998), no. 3, pp. 1095–1115.Google Scholar
[5]Chiari, Mario and Krajíček, Jan, Lifting independence results in bounded arithmetic, Archive for Mathematical Logic, vol. 38 (1999), no. 2, pp. 123–138.CrossRefGoogle Scholar
[6]Cook, Stephen A., Feasibly constructive proofs and the propositional calculus (preliminary version), Conference record of seventh annual ACM symposium on theory of computing (Albuquerque, New Mexico), 05 5–7 1975, pp. 83–97.Google Scholar
[7]Ferreira, F., What are the -consequences of and ?, Annals of Pure and Applied Logic, vol. 75 (1995), no. 1. pp. 79–88.CrossRefGoogle Scholar
[8]Hanika, J., Herbrandizing search problems in bounded arithmetic, Mathematical Logic Quarterly, vol. 50 (2004), no. 6, pp. 577–586.CrossRefGoogle Scholar
[9]Hanika, J., Search problems and bounded arithmetic, Ph.D. thesis, Charles University, Prague, 2004, Available via the ECCC (http://eccc.hpi-web.de/eccc).Google Scholar
[10]Impagliazzo, R. and Krajíček, J., A note on conservativity relations among bounded arithmetic theories, Mathematical Logic Quaterly, vol. 48 (2002), no. 3, pp. 375–377.Google Scholar
[11]Johnson, David S., Papadimitriou, Christos H., and Yannakakis, Mihalis, How easy is local search?, Journal of Computer and System Sciences, vol. 37 (1988), no. 1, pp. 79–100.CrossRefGoogle Scholar
[12]Krajíček, Jan,.Google Scholar
[13]Krajíček, Jan, Lower bounds to the size of constant-depth propositional proofs, this Journal, vol. 59 (1994), no. 1, pp. 73–86.Google Scholar
[14]Krajíček, Jan, On the weak pigeonhole principle, Fundamenta Mathematicae, vol. 170 (2001), no. 1–3, pp. 123–140.CrossRefGoogle Scholar
[15]Krajíček, Jan and Pudlák, Pavel, Quantified propositional calculi and fragments of bounded arithmetic, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 36 (1990),no. 1, pp. 29–46.CrossRefGoogle Scholar
[16]Krajíček, Jan, Pudlák, Pavel, and Woods, Alan, An exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle, Random Structures and Algorithms, vol. 7 (1995), no. 1, pp. 15–39.CrossRefGoogle Scholar
[17]Krajíček, Jan and Takeuti, Gaisi, On induction-free provability, Annals of Mathematics and Artificial Intelligence, vol. 6 (1992), pp. 107–126.CrossRefGoogle Scholar
[18]Maciel, Alexis, Pitassi, Toniann, and Woods, Alan, A new proof of the weak pigeonhole principle, Journal of Computer and System Sciences, vol. 64 (2002), no. 4, pp. 843–872.CrossRefGoogle Scholar
[19]Morioka, Tsuyoshi, Logical approaches to the complexity of search problems: Proof complexity, quantified propositional calculus, and bounded arithmetic, Ph.D. thesis, University of Toronto, 2005, Available via the ECCC (http://eccc.hpi-web.de/eccc).Google Scholar
[20]Papadimitriou, Christos, On graph-theoretic lemmata and complexity classes (extended abstract), Proceedings of the 31st IEEE symposium on Foundations of Computer Science, vol. II, IEEE Computer Society, 1990, pp. 794–801.Google Scholar
[21]Papadimitriou, Christos and Yannakakis, Mihalis, Optimization, approximation and complexity classes, 20th annual ACM symposium on the Theory of Computing, ACM Press, 1988, pp. 229–234.Google Scholar
[22]Papadimitriou, Christos H., On the complexity of the parity argument and other inefficient proofs of existence, Journal of Computer and System Sciences, vol. 48 (1994), no. 3, pp. 498–532.CrossRefGoogle Scholar
[23]Paris, J. and Wilkie, A., Counting problems in bounded arithmetic, Methods in mathematical logic, Lecture Notes in Mathematics, vol. 1130, Springer-Verlag, 1985, pp. 317–40.CrossRefGoogle Scholar
[24]Paris, Jeff, Wilkie, Alex, and Woods, Alan, Provability of the pigeonhole principle and the existence of infinitely many primes, this Journal, vol. 53 (1988), pp. 1235–1244.Google Scholar
[25]Pitassi, Toniann, Beame, Paul, and Impagliazzo, Russell, Exponential lower bounds for the pigeonhole principle, Computational Complexity, vol. 3 (1993), pp. 97–308.CrossRefGoogle Scholar
[26]Pudlák, Pavel, Ramsey's theorem in bounded arithmetic, Proceedings of Computer Science Logic (Borger, E., Buning, H. Kleine, Richter, M. M., and Schonfeld, W., editors). Lecture Notes in Computer Science, vol. 553, Springer-Verlag, 1992, pp. 308–312.Google Scholar
[27]Pudlák, Pavel, On Σ1 sentences in bounded arithmetic, In preparation.Google Scholar