Abstract
It is known that constant-depth Frege proofs of some tautologies require exponential size. No such lower bound result is known for more general proof systems. We consider tree-like sequent calculus proofs in which formulas can contain modular connectives and only the cut formulas are restricted to be of constant depth. Under a plausible hardness assumption concerning small-depth Boolean circuits, we prove exponential lower bounds for such proofs. We prove these lower bounds directly from the computational hardness assumption. We start with a lower bound for cut-free proofs and “lift” it so it applies to proofs with constant-depth cuts. By using the same approach, we obtain the following additional results. We provide a much simpler proof of a known unconditional lower bound in the case where modular connectives are not used. We establish a conditional exponential separation between the power of constant-depth proofs that use different modular connectives. We show that these tree-like proofs with constant-depth cuts cannot polynomially simulate similar dag-like proofs, even when the dag-like proofs are cut-free. We present a new proof of the non-finite axiomatizability of the theory of bounded arithmetic I Δ0(R). Finally, under a plausible hardness assumption concerning the polynomial-time hierarchy, we show that the hierarchy \({G_i^*}\) of quantified propositional proof systems does not collapse.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Ajtai (1988). The complexity of the pigeonhole principle. In Proceedings of the 29th IEEE Symposium on Foundations of Computer Science, 346–355.
Paul Beame, Russell Impagliazzo, Jan Krajíček , Toniann Pitassi & Pudlák Pavel (1996). Lower bounds on Hilbert’s Null-stellensatz and propositional proofs. Proceedings of the London Mathematical Society 73(3), 346–355.
E. Ben-Sasson & A. Wigderson (1999). Short proofs are narrow – resolution made simple. In Proceedings of the 31st ACM Symposium on Theory of Computing.
M. Bonet, C. Domingo, R. Gavalda, A. Maciel & T. Pitassi(2004). Non-automatizability of bounded-depth Frege proofs. Computational Complexity 13, 47–68.
S. Buss (1988). Weak Formal Systems and Connections to Computational Complexity. Lecture notes, University of California, Berkeley.
P. Clote & E. Kranakis (2002). Boolean Functions and Computation Models. Springer.
Cook S., Morioka T. (2005) Quantified Propositional Calculus and a Second-Order Theory for NC1. Archive for Mathematical Logic 44(6): 711–749
S. Cook & P. Nguyen (2010). Logical Foundations of Proof Complexity. ASL Perspectives in Logic. Cambridge University Press.
M. Furst, J.B. Saxe & M. Sipser (1984). Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory 17, 13–27.
Haken A. (1985) The intractability of Resolution. Theoretical Computer Science 39: 297–308
A. Haken & S.A. Cook (1999). An exponential lower bound for the size of monotone real circuits. Journal of Computer and System Sciences 58, 326–335.
J. Håstad (1986). Computational Limitations of Small Depth Circuits. MIT Press, Cambridge, MA, U.S.A.
J. Krajíček (1994). Lower bounds to the size of constant-depth propositional proofs. Journal of Symbolic Logic 59(1), 73–86.
J. Krajíček (1995). Bounded Arithmetic, Propositional Logic and Computational Complexity. Cambridge University Press.
J. Krajíček & P. Pudlák (1990). Quantified Propositional Calculi and Fragments of Bounded Arithmetic. Zeitschrift f. Mathematkal Logik u. Grundlagen d. Mathematik 36, 29–46.
J. Krajíček , P. Pudlák & G. Takeuti (1991). Bounded arithmetic and the polynomial hierarchy. Annals of Pure and Applied Logic 52, 143–153.
J. Krajíček , P. Pudlák & A. Woods (1995). Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle. Random Structures and Algorithms 7, 15–39.
A. Maciel & T. Pitassi (2006). A Conditional Lower Bound for a System of Constant-Depth Proofs with Modular Connectives. In Proceedings of the 21st Annual IEEE Symposium on Logic in Computer Science.
T. Morioka (2005). Logical Approaches to the Complexity of Search Problems: Proof Complexity, Quantified Propositional Calculus, and Bounded Arithmetic. Ph.D. thesis, University of Toronto.
P. Nguyen (2007). Separating DAG-Like and Tree-Like Proof Systems. In Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science.
S. Perron (2008). Power of Non-Uniformity in Proof Complexity. Ph.D. thesis, University of Toronto.
T. Pitassi, P. Beame & R. Impagliazzo (1993). Exponential lower bounds for the pigeonhole principle. Computational Complexity 97–140.
Pudlák P. (1997) Lower bounds for resolution and cutting planes proofs and monotone computations. Journal of Symbolic Logic 62(3): 981–998
R. Smolensky (1987). Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proceedings of the 19th ACM Symposium on Theory of Computing, 77–82.
R. Smolensky (1993). On Representations by Low-Degree Polynomials. In Proceedings of the 34th IEEE Symposium on Foundations of Computer Science.
Statman R. (1978) Bounds for Proof-Search and Speed-up in the Predicate Calculus. Annals of Mathematical Logic 15: 225–287
A.C.-C. Yao (1985). Separating the polynomial-time hierarchy by oracles. In Proceedings of the 26th IEEE Symposium on Foundations of Computer Science, 1–10.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Maciel, A., Nguyen, P. & Pitassi, T. Lifting lower bounds for tree-like proofs. comput. complex. 23, 585–636 (2014). https://doi.org/10.1007/s00037-013-0064-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00037-013-0064-x