M. L. Bonet, T. Pitassi, and R. Raz, No feasible interpolation for TC 0 -Frege proofs, in Proceedings of the 38th Annual Symposium on Foundations of Computer Science, Piscataway, New Jersey, 1997, IEEE Computer Society, pp. 264--263.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....than polynomial speed up between classical and intuitionistic propositional calculus, i.e. there are intuitionistic tautologies that have polynomial size proofs in the classical sequent calculus, but no polynomial size proofs in the intuitionistic sequent calculus. Proof Bonnet, Pitassi and Raz [1] constructed tautologies which have polynomial size proofs and which cannot have such proofs in any system admitting feasible interpolation, provided that factoring is hard. Let us note that such a speed up follows also from the assumption that PSpace 6 NP=poly , but the last corollary gives ....

M. L. Bonet, T. Pitassi, and R. Raz, No feasible interpolation for TC 0 -Frege proofs, in Proceedings of the 38th Annual Symposium on Foundations of Computer Science, Piscataway, New Jersey, 1997, IEEE Computer Society, pp. 264--263.

....than polynomial speed up between classical and intuitionistic propositional calculus, i.e. there are intuitionistic tautologies that have polynomial size proofs in the classical sequent calculus, but no polynomial size proofs in the intuitionistic sequent calculus. Proof Bonnet, Pitassi and Raz [1] constructed tautologies which have polynomial size proofs and which cannot have such proofs in any system admitting feasible interpolation, provided that factoring is hard. Let us note that such a speed up follows also from the assumption that PSpace 6 NP=poly , but the last corollary gives ....

M. L. Bonet, T. Pitassi, and R. Raz, No feasible interpolation for TC 0 -Frege proofs, in Proceedings of the 38th Annual Symposium on Foundations of Computer Science, Piscataway, New Jersey, 1997, IEEE Computer Society, pp. 264--263.

.... T . In this case, feasible interpolation for T implies feasible b 1 interpolation for the theory T . In this way, the rst mentioned result of [11] implies that Extended Frege systems do not enjoy feasible interpolation, unless RSA is insecure. This was improved upon by Bonet et al. [2], who showed that the TC 0 Frege system, in which every line is represented by a constant depth threshold circuit, does not have feasible interpolation if the DieHellman problem is hard. Our result can be seen as the uniform version of this result, which was posed as a problem in [2] and the ....

....et al. 2] who showed that the TC 0 Frege system, in which every line is represented by a constant depth threshold circuit, does not have feasible interpolation if the DieHellman problem is hard. Our result can be seen as the uniform version of this result, which was posed as a problem in [2], and the idea of our proof is taken from that paper. The natural candidate for a bounded arithmetic theory corresponding to TC 0 Frege proofs is of course C 0 2 . Although the simulation of the b 1 consequences of C 0 2 by polynomial size TC 0 Frege proofs has never been worked out ....

[Article contains additional citation context not shown here]

M. L. Bonet, T. Pitassi, and R. Raz. No feasible interpolation for TC 0 -Frege proofs. In Proc. 38th Symposium on Foundations of Computer Science, pages 254-263, 1997.

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M. L. Bonet, T. Pitassi, and R. Raz, No feasible interpolation for TC -Frege proofs, Proceedings of the 38th annual symposium on foundations of computer science, IEEE Computer Society, Piscataway, New Jersey, 1997, pp. 264--263.

....to fixing the inherent inefficiency of propositional proof systems, is to use the more efficient ones. Then we are faced with the second problem. How hard is it then to find proofs It seems that the more efficient a proof system is, the harder it is to find proofs in it. In Bonet,Pitassi and Raz [7] a notion of automatizability is defined. We say a propositional proof system is automatizable if and only if there is a deterministic procedure to find proofs in that system in polynomial time with respect to the smallest proof in that system. In the sequence of papers [19, 6, 3] it is proved ....

M.L. Bonet, T. Pitassi, and R. Raz. No feasible interpolation for tc 0 - frege proofs. In Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), pages 254--265, 1997.

....false. One approach to fixing the inherent inefficiency of propositional proof systems, is to use the more efficient ones. Then we are faced with the second problem. How hard is it then to find proofs It seems that the more efficient a proof system is, the harder it is to find proofs in it. In [BPR97] a notion of automatizability is defined. We say a propositional proof system is automatizable if and only if there is a deterministic procedure to find proofs in that system in polynomial time with respect to the smallest proof in that system. In the sequence of papers [KP95, BPR97, BDGMP99] it ....

....in it. In [BPR97] a notion of automatizability is defined. We say a propositional proof system is automatizable if and only if there is a deterministic procedure to find proofs in that system in polynomial time with respect to the smallest proof in that system. In the sequence of papers [KP95, BPR97, BDGMP99] it is proved that any propositional proof system that simulates Authors Address: Universitat Politecnica de Catalunya. Departamento de L.S.I. emails:fbonet,galesig lsi.upc.es y Partly supported by Project CICYT, TIC 98 0410 C02 01 z Partly Supported by an EC grant under the TMR ....

M. Bonet, T. Pitassi, R. Raz. No Feasible Interpolation for TC 0 Frege Proofs. Proceedings of FOCS 97 pp. 254-263.

....the shortest proof of F in S. The only propositional proof systems that are known to be automatizable are algebraic proof systems like Hilbert s Nullstellensatz [2] and Polynomial Calculus [12] On the other hand bounded depth Frege proof systems are not automatizable, assuming factoring is hard [29, 10, 7]. Since Frege systems and Extended Frege systems polynomially simulate bounded depth Frege systems, they are also not automatizable under the same assumptions. Note that automatizability is equivalent to the approximability to within a polynomial factor of the following optimization problem: ....

M. L. Bonet, T. Pitassi, and R. Raz. No feasible interpolation for TC 0 -Frege proofs. In Proc. 38th Symposium on Foundations of Computer Science, pages 254-263, 1997. Also accepted for publication in SIAM Journal of Computing.

....for approximating the Minimum Step Length Frege Proof problem to within a polynomial. Another closely related prior result is the connection between the (non)automatizability of Frege systems and the (non)feasibility of factoring integers that was recently discovered by Bonet Pitassi Raz [9]. A proof system T is said be automatizable provided there is an algorithm M and a polynomial p such that whenever T n holds, M( produces some T proof of in time p(n) see [12] Obviously the automatizability of Frege systems is closely related to the solution of the Minimum Length Frege ....

....the minimum length proof problem concerns the associated decision problem. Our theorems give a linear or quasi linear lower bound on the automatizability of the Minimum Proof Length problem based on the assumption that P 6= NP or that NP 6 QP. It has recently been shown by Bonet Pitassi Raz [9] that Frege systems are not automatizable unless Integer Factorization is in P , and more recently, that bounded depth Frege systems are also not automatizable under a similar hardness assumption [5] These results give stronger non approximability conclusions, but require assuming a much stronger ....

M. L. Bonet, T. Pitassi, and R. Raz, No feasible interpolation for TC 0 -Frege proofs, in Proceedings of the 38th Annual Symposium on Foundations of Computer Science, Piscataway, New Jersey, 1997, IEEE Computer Society, pp. 264--263.

....to fixing the inherent inefficiency of propositionalproof systems, is to use the more efficient ones. Then we are faced with the second problem. How hard is it then to find proofs It seems that the more efficient a proof system is, the harder it is to find proofs in it. In Bonet,Pitassi and Raz [5] a notion of automatizability is defined. We say a propositional proof system is automatizable if and only if there is a deterministic procedure to find proofs in that system in polynomial time with respect to the smallest proof in that system. In the sequence of papers [15, 5, 6] it is proved ....

....In Bonet,Pitassi and Raz [5] a notion of automatizability is defined. We say a propositional proof system is automatizable if and only if there is a deterministic procedure to find proofs in that system in polynomial time with respect to the smallest proof in that system. In the sequence of papers [15, 5, 6] it is proved that any propositional proof system that simulates bounded depth Frege is not automatizable, unless some widely accepted cryptographic conjectures are violated. There are some algorithms to find proofs in some proof systems. For instance, 2, 1] gave algorithms for resolution, and ....

M. Bonet, T. Pitassi, R. Raz. No Feasible Interpolation for TC 0 Frege Proofs. Proceedings of FOCS 97. pp. 254-263.

.... 13, 4] generalizations of Cutting Planes [2, 8, 7] relativized bounded arithmetic [15] Hilbert s Nullstellensatz [14] the polynomial calculus [14] and the Lovasz Schriver proof system [12] On the other hand, in a separate sequence of papers beginning with a key idea due to Krajcek and Pudlak [9, 3], it has been shown that under sufficiently strong cryptographic assumptions, many stronger proof systems do not have feasible interpolation. The main ideas are as follows. Suppose that H is a permutation that is generally believed to be one way. Formulate A 0 ( x; z) as saying H(x) z and ....

....assumptions: Extended Frege, Frege and even TC 0 Frege systems. These negative results are important not only as a guide for searching for lower bound techniques, but also because they imply that the proof system in question cannot be automatized. This connection was first made explicit by [3] and takes us back to the second motivation for studying propositional proof systems. A proof system S is automatizable if there exists a deterministic procedure D that takes as input a formula f and returns an S refutation of f (if one exists) in time polynomial in the size of the shortest S ....

[Article contains additional citation context not shown here]

M. L. Bonet, T. Pitassi, and R. Raz. No feasible interpolation for TC 0 -Frege proofs. In Proceedings of the 38th IEEE Symposium on Foundations of Computer Science, pages 254--263, 1997.

....of the shortest proof of F in S. The only propositional proof systems that we know are automatizable are algebraic proof systems like Hilbert s Nullstellensatz [2] and Polynomial Calculus [9] On the other hand bounded depth Frege proof systems are not automatizable, assuming factoring is hard [24, 7, 5]. Since Frege systems and Extended Frege systems polynomiallysimulates bounded depth Frege systems, they are also not automatizable under the same assumptions. A commonly used strategy for finding proofs is to reduce the search space by defining restricted versions of resolution that are still ....

M. L. Bonet, T. Pitassi, and R. Raz. No feasible interpolation for TC 0 -Frege proofs. In Proc. 38th FOCS, pages 254--263, 1997.

....of the shortest proof of F in S. The only propositional proof systems that we know are automatizable are algebraic proof systems like Hilbert s Nullstellensatz [2] and Polynomial Calculus [10] On the other hand bounded depth Frege proof systems are not automatizable, assuming factoring is hard [27, 8, 5]. Since Frege systems and Extended Frege systems polynomially simulate bounded depth Frege systems, they are also not automatizable under the same assumptions. A commonly used strategy for finding proofs is to reduce the search space by defining restricted versions of resolution that are still ....

M. L. Bonet, T. Pitassi, and R. Raz. No feasible interpolation for TC 0 -Frege proofs. In Proc. 38th Symposium on Foundations of Computer Science, pages 254--263, 1997.

No context found.

M. L. Bonet, T. Pitassi, and R. Raz, No feasible interpolation for TC 0 -Frege proofs, in Proceedings of the 38th Annual Symposium on Foundations of Computer Science, Piscataway, New Jersey, 1997, IEEE Computer Society, pp. 264--263.

No context found.

M. L. Bonet, T. Pitassi, and R. Raz, No feasible interpolation for TC - Frege proofs, in Proceedings of the 38th Annual Symposium on Foundations of Computer Science, Piscataway, New Jersey, 1997, IEEE Computer Society, pp. 264--263.

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