Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-19T05:49:25.902Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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