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Examining Fragments of the Quantified Propositional Calculus

Published online by Cambridge University Press:  12 March 2014

Steven Perron
Steven Perron*
Affiliation:
University of Toronto, Department of Computer Science, M5S 3G4, Toronto, Ontario, Canada, E-mail: [email protected]
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Abstract

When restricted to proving formulas, the quantified propositional proof system is closely related to the theorems of Buss's theory . Namely, has polynomial-size proofs of the translations of theorems of , and proves that is sound. However, little is known about when proving more complex formulas. In this paper, we prove a witnessing theorem for similar in style to the KPT witnessing theorem for . This witnessing theorem is then used to show that proves is sound with respect to formulas. Note that unless the polynomial-time hierarchy collapses is the weakest theory in the S2 hierarchy for which this is true. The witnessing theorem is also used to show that is p-equivalent to a quantified version of extended-Frege for prenex formulas. This is followed by a proof that Gi, p-simulates with respect to all quantified propositional formulas. We finish by proving that S2 can be axiomatized by plus axioms stating that the cut-free version of is sound. All together this shows that the connection between and does not extend to more complex formulas.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 3 , September 2008 , pp. 1051 - 1080
Copyright
Copyright © Association for Symbolic Logic 2008

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References

REFERENCES

[1]Buss, Samuel R., Bounded arithmetic, Ph.D. thesis, Princeton University, 1986.Google Scholar
[2]Buss, Samuel R., On Herbrand's theorem, Lecture Notes in Computer Science, vol. 960, 1995.Google Scholar
[3]Cook, Stephen, Feasibly constructive proofs and the propositional calculus, Proceedings of the 7th ACM symposium on the theory of computation, 1975, pp. 83–97.Google Scholar
[4]Cook, Stephen, Theories for complexity classes and their propositional translations, (Krajicek, Jan, editor), Quadernidi Matematica, 2003, pp. 175–227.Google Scholar
[5]Cook, Stephen and Morioka, Tsuyoshi, Quantified propositional calculus and a second-order theory for NC1, Archive for Mathematical Logic, vol. 44 (2005), no. 6, pp. 711–749.CrossRefGoogle Scholar
[6]Cook, Stephen and Nguyen, Phuong, Foundations of proof complexity: Bounded arithmetic and propositional translations, Available from http://www.cs.toronto.edu/~sacook/csc2429h/book, 2006.Google Scholar
[7]Cook, Stephen and Thapen, Neil, The strength of replacement in weak arithmetic, ACM Transactions on Computational Logic, vol. 7 (2006), no. 4, pp. 749–764.CrossRefGoogle Scholar
[8]Krajícek, Jan, Lower bounds to the size of constant-depth propositional proofs, this Journal, vol. 59 (1994), no. 1, pp. 73–86.Google Scholar
[9]Krajícek, Jan, Bounded arithmetic, propositional logic, and complexity theory, Cambridge University Press, 1995.CrossRefGoogle Scholar
[10]Krajícek, Jan and Pudlák, Pavel, Quantified propostitional calculi and fragments of bounded arithmetic, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 36 (1990), pp. 29–46.CrossRefGoogle Scholar
[11]Krajícek, Jan, Pudlák, Pavel, and Takeuti, Gaisi, Bounded arithmetic and the polynomial hierarchy, Annals of Pure and Applied Logic, vol. 52 (1991), no. 1–2, pp. 143–153.CrossRefGoogle Scholar
[12]Krajícek, Jan, Skelley, Alan, and Thapen, Neil, NP search problems in low fragments of bounded arithmetic, this Journal, vol. 72 (2007), no. 2, pp. 649–672.Google Scholar
[13]Krajícek, Jan and Takeuti, Gauisi, On induction-free provability. Annals of Mathematics and Artificial Intelligence, vol. 6 (1992), pp. 107–126.CrossRefGoogle Scholar
[14]Maciel, Alexis and Pitassi, Toniann, Conditional lower bound for a system of constant-depth proofs with modular connectives, LICS '06, IEEE Computer Society, 2006, pp. 189–200.Google Scholar
[15]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.Google Scholar
[16]Nguyen, Phuong, Separating dag-like and tree-like proof systems, LICS '07, IEEE Computer Society, 2007, pp. 235–244.Google Scholar
[17]Perron, Steven James, Examining the fragments of G, LICS '07, IEEE Computer Society, 2007, pp. 225–234.Google Scholar
[18]Pudlák, Pavel, Some relations between subsystems of arithmetic and complexity of computations, Logic from computer science (Moschovakis, Y. N., editor), Springer, New York, 1992, pp. 499–519.CrossRefGoogle Scholar