Skip to main content

Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic

  • Jan Krajíček (a1)

A proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: (1)

Feasible interpolation theorems for the following proof systems:




a subsystem of LK corresponding to the bounded arithmetic theory (α)


linear equational calculus


cutting planes.


New proofs of the exponential lower bounds (for new formulas)


for resolution ([15])


for the cutting planes proof system with coefficients written in unary ([4]).


An alternative proof of the independence result of [43] concerning the provability of circuit-size lower bounds in the bounded arithmetic theory (α).

In the other direction we show that a depth 2 subsystem of LK does not admit feasible monotone interpolation theorem (the so called Lyndon theorem), and that a feasible monotone interpolation theorem for the depth 1 subsystem of LK would yield new exponential lower bounds for resolution proofs of the weak pigeonhole principle.

Hide All
[1]Alon, N. and Boppana, R., The monotone circuit complexity of Boolean functions, Combinatorica, vol. 7 (1987), no. 1, pp. 122.
[2]Andreev, A. E., On a method for obtaining lower bounds for the complexity of individual monotone functions, Doklady ANSSSR, vol. 282 (1985), no. 5, pp. 10331037, in Russian.
[3]Beth, E. W., The foundations of mathematics, North-Holland, Amsterdam, 1959.
[4]Bonet, M. L., Pitassi, T., and Raz, R., Lower bounds for cutting planes proofs with small coefficients, preprint, 1994.
[5]Buss, S. R., Bounded arithmetic, Bibliopolis, Naples, 1986.
[6]Buss, S. R. and Turán, G., Resolution proofs of generalized pigeonhole principles, Theoretical Computer Science, vol. 62 (1988), pp. 311317.
[7]Chiari, M. and Krajíček, J., Witnessing functions in bounded arithmetic and search problems, submitted, 1994.
[8]Cook, S. A., Feasibly constructive proofs and the prepositional calculus, Proceedings of the 7th Annual ACM Symposium on Theory of Computing, ACM Press, 1975, pp. 8397.
[9]Cook, S. A. and Reckhow, A. R., The relative efficiency of prepositional proof systems, this Journal, vol. 44 (1979), no. 1, pp. 3650.
[10]Cook, W., Coullard, C. R., and Turán, G., On the complexity of cutting plane proofs, Discrete Applied Mathematics, vol. 18 (1987), pp. 2538.
[11]Craig, W., Linear reasoning: A new form of the Herbrand-Gentzen theorem, this Journal, vol. 22 (1957), no. 3, pp. 250268.
[12]Craig, W., Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory, this Journal, vol. 22 (1957), no. 3, pp. 269285.
[13]Friedman, H., The complexity of explicit definitions, Advances in Mathematics, vol. 20 (1976), pp. 1829.
[14]Gurevich, Y., Towards logic tailored for computational complexity, Proceedings of Logic Colloquium 1983 (Berlin), Springer Lecture Notes in Mathematics, no. 1104, Springer-Verlag, 1984, pp. 175216.
[15]Haken, A., The intractability of resolution, Theoretical Computer Science, vol. 39 (1985), pp. 297308.
[16]Karchmer, M. and Wigderson, A., Monotone circuits for connectivity require super-logarithmic depth, Proceedings of the 20th Annual ACM Symposium on Theory of Computing, ACM Press, 1988, pp. 539550.
[17]Krajíček, J., Exponentiation and second-order bounded arithmetic, Annals of Pure and Applied Logic, vol. 48 (1989), pp. 261276.
[18]Krajíček, J., No counter-example interpretation and interactive computation, Logic from Computer Science, Proceedings of a workshop held November 13–17, 1989, in Berkeley, Mathematical Sciences Research Institute Publication (Berlin) (Moschovakis, Y. N., editor), no. 21, Springer-Verlag, 1992, pp. 287293.
[19]Krajíček, J., Lower bounds to the size of constant-depth prepositional proofs, this Journal, vol. 59 (1994), no. 1, pp. 7386.
[20]Krajíček, J., Bounded arithmetic, prepositional logic and complexity theory, Cambridge University Press, 1995.
[21]Krajíček, J., On Frege and extended Frege proof systems, Feasible Mathematics II (Clote, P. and Remmel, J., editors), Birkhäuser, 1995, pp. 284319.
[22]Krajíček, J. and Pudlák, P., Prepositional proof systems, the consistency of first order theories and the complexity of computations, this Journal, vol. 54 (1989), no. 3, pp. 10631079.
[23]Krajíček, J., Prepositional provability in models of weak arithmetic, Computer Science Logic (Boerger, E., Kleine-Bunning, H., and Richter, M. M., editors), Lecture Notes in Computer Science, no. 440, Springer-Verlag, Berlin, 1990, Kaiserlautern, 10 1989, pp. 193210.
[24]Krajíček, J., Quantified prepositional calculi and fragments of bounded arithmetic, Zeitschrift für Mathematikal Logik und Grundlagen der Mathematik, vol. 36 (1990), pp. 2946.
[25]Krajíček, J., Some consequences of cryptographical conjectures for and EF, Proceedings of the meeting Logic and Computational Complexity (Leivant, D., editor), 1995, Indianapolis, 10 1994, to appear.
[26]Krajíček, J. and Takeuti, G., On bounded -polynomial induction, Feasible mathematics (Buss, S. R. and Scott, P. J., editors), Birkhäuser, 1990, pp. 259280.
[27]Krajíček, J., On induction-free provability, Annals of Mathematics and Artificial Intelligence, vol. 6 (1992), pp. 107126.
[28]Kreisel, G., Technical report nb. 3, Applied Mathematics and Statistics Labs, Stanford University, unpublished, 1961.
[29]Luby, M., Pseudo-randomness and applications, International Computer Science Institute, Berkeley, lecture notes, 1993.
[30]Mundici, D., A lower bound for the complexity of Craig's interpolants in sentential logic, Archiv für Mathematikal Logik, vol. 23 (1983), pp. 2736.
[31]Mundici, D., NP and Craig's interpolation theorem, Proceedings of Logic Colloquium 1982, North-Holland, 1984, pp. 345358.
[32]Mundici, D., Tautologies with a unique Craig interpolant, uniform vs. non-uniform complexity, Annals of Pure and Applied Logic, vol. 27 (1984), pp. 265273.
[33]Papadimitriou, A., Computational complexity, Addison-Wesley, 1994.
[34]Parikh, R., Existence and feasibility in arithmetic, this Journal, vol. 36 (1971), pp. 494508.
[35]Paris, J. and Wilkie, A. J., Counting problems in bounded arithmetic, Methods in Mathematical Logic, Springer Lecture Notes in Mathematics, no. 1130, Springer-Verlag, Berlin, 1985, pp. 317340.
[36]Paris, J., On the scheme of induction for bounded arithmetic formulas, Annals of Pure and Applied Logic, vol. 35 (1987), pp. 261302.
[37]Paris, J. B., Wilkie, A. J., and Woods, A. R., Provability of the pigeonhole principle and the existence of infinitely many primes, this Journal, vol. 53 (1988), pp. 12351244.
[38]Pratt, V. R., Every prime has a succinct certificate, SIAM Journal of Computing, vol. 4 (1975), pp. 214220.
[39]Razborov, A. A., Lower bounds on the monotone complexity of some Boolean functions, Soviet Mathem. Doklady, vol. 31 (1985), pp. 354357.
[40]Razborov, A. A., An equivalence between second order bounded domain bounded arithmetic and first order bounded arithmetic, Arithmetic, Proof Theory and Computational Complexity (Clote, P. and Krajíček, J., editors), Oxford University Press, 1993, pp. 247277.
[41]Razborov, A. A., On provably disjoint NP-pairs, preprint, 1994.
[42]Razborov, A. A., Bounded arithmetic and lower bounds in Boolean complexity, Feasible mathematics II (Clote, P. and Remmel, J., editors), Birkhäuser, 1995, pp. 344386.
[43]Razborov, A. A., Unprovability of lower bounds on the circuit size in certain fragments of bounded arithmetic, Izvestiya of the R. A. N., vol. 59 (1995), no. 1, pp. 201224.
[44]Razborov, A.A. and Rudich, S., Natural proofs, Proceedings of the 26th Annual ACM Symposium on Theory of Computing, ACM Press, 1994, pp. 204213.
[45]Takeuti, G., Proof theory, North-Holland, 1975.
[46]Takeuti, G., RSUV isomorphism, Arithmetic, Proof Theory and Computational Complexity (Clote, P. and Krajíček, J., editors), Oxford University Press, 1993, pp. 364386.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed