## On Interpolation and Automatization for Frege Systems (2000)

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Citations: | 45 - 5 self |

### BibTeX

@MISC{Bonet00oninterpolation,

author = {Maria Luisa Bonet and Toniann Pitassi and Ran Raz},

title = {On Interpolation and Automatization for Frege Systems},

year = {2000}

}

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### Abstract

The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexity-theoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC 0 -Frege systems. More specifically, we show that unless factoring (of Blum integers) is feasible, neither Frege nor TC 0 -Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomial size TC 0 -Frege. As a corollary, we obtain that TC 0 -Frege as well as any proof system that polynomially simulates it, is not automatizable (under the assumption that factoring of Blum integ...

### Citations

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Citation Context ...graphic primitive needed here is not a one way permutation as in [KP], but the more general structure of bit commitment. Our formulas A0, A1 are based on the Diffie–Hellman secret key exchange scheme =-=[DH]-=-. For simplicity, we state the formulas only for the least significant bit. (Our argument works for any bit.) Informally, our propositional statement DH will be DHn = A0(P, g, X, Y, a, b) ∧ A1(P, g, X... |

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Citation Context ... mean 2 i+j−2 if both xi and yj are true, and 0 otherwise. Last, we will describe our T C 0 -formula for computing the iterated product of m numbers. This formula is basically the original formula of =-=[BCH]-=- and is articulated as a T C 0 -formula in [M]. The iterative product P ROD[z1, . . . , zm] gives the product of z1, . . . , zm, where each zi is of length n, and we assume that m, n are both bounded ... |

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Citation Context ... zi by (sN+1 ∧ si) ∨ (¬sN+1 ∧ ti). 3.2. Iterated addition. We will now describe the T C0-formula SUM[x1, . . . , xm] that inputs m numbers, each n bits long, and outputs their sum x1+x2+· · ·+xm (see =-=[CSV]-=-). We assume that m ≤ N. The main idea is to reduce the addition of m numbers to the addition of two numbers. Let xi be xi,n, . . . , xi,1 (in binary representation). Let l = ⌈log2 N⌉. Let r = n 2l , ... |

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Citation Context ...perpolynomial lower bounds can be proven for a (sufficiently strong) proof system that admits feasible interpolation. This was presented by the sequence of papers [IPU, BPR, K1] and was first used in =-=[BPR]-=- to prove lower bounds for propositional proof systems. (The idea is also implicit in [Razb2].) In short, the statement F that is used is the clique interpolation formula, A0(g, x)∧ A1(g, y), where A0... |

69 | Some consequences of cryptographical conjectures for S1 2 and EF ”, in: Logic and Computational Complexity (Proc. of the meeting held in Indianapolis
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Citation Context ...ardness assumption. 1.2. Interpolation and one way functions. How can one prove that a certain propositional proof system S does not admit feasible interpolation? One idea, due to Krajíček and Pudlák =-=[KP]-=-, is to use one way permutations in the following way. Let h be a one way permutation and let A0(x, z), A1(y, z) be the following formulas. The formula A0: h(x) = z, AND the ith bit of x is 0. The for... |

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Citation Context ...ts feasible interpolation. This was presented by the sequence of papers [IPU, BPR, K1] and was first used in [BPR] to prove lower bounds for propositional proof systems. (The idea is also implicit in =-=[Razb2]-=-.) In short, the statement F that is used is the clique interpolation formula, A0(g, x)∧ A1(g, y), where A0 states that g is a graph containing a clique of size k (where the clique is described by the... |

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Citation Context ...system S admits feasible interpolation if, whenever S has a polynomial-sized refutation of a formula F (as above), an interpolation function associated with F has a polynomial-sized circuit. Krajíček =-=[K2]-=- was the first to make the connection between proof systems having feasible interpolation and circuit complexity. There is also a monotone version of the interpolation idea. Namely, for conjunctive no... |

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Citation Context ...r propositional logic was provided by Buss [B2], who proved the rather surprising fact that the k-symbol propositional provability problem is NP-complete for a particular Frege system. More recently, =-=[ABMP]-=- show that the k-symbol and k-line provability problems cannot be approximated to within linear factors for a variety of propositional proof systems, including resolution and all Frege systems, unless... |

26 | An exponential lower bound for the size of monotone real circuits - Cook, Haken - 1999 |

26 | On the complexity of propositional calculus
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Citation Context ..., Pud, CH], generalizations of cutting planes [BPR, K1, K3], relativized bounded arithmetic [Razb2], Hilbert’s Nullstellensatz [PS], the polynomial calculus [PS], and the Lovasz–Schriver proof system =-=[Pud3]-=-. 1.1. Automatizability and k-provability. As explained in the previous paragraphs, the existence of feasible interpolation for a particular proof system S gives rise to lower bounds for S. Feasible i... |

26 | The undecidability of k-provability - Buss - 1991 |

25 | Non-automatizability of bounded-depth Frege proofs - Bonet, Domingo, et al. - 1999 |

23 | Algebraic models of computation and interpolation for algebraic proof systems
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Citation Context ...sing the interpolation method: resolution [BPR], cutting planes [IPU, BPR, Pud, CH], generalizations of cutting planes [BPR, K1, K3], relativized bounded arithmetic [Razb2], Hilbert’s Nullstellensatz =-=[PS]-=-, the polynomial calculus [PS], and the Lovasz–Schriver proof system [Pud3]. 1.1. Automatizability and k-provability. As explained in the previous paragraphs, the existence of feasible interpolation f... |

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Citation Context ...ved that for P = p1 · p2, where p1, p2 are both primes such that p1 mod 4 = p2 mod 4 = 3 (i.e., P is a Blum integer), breaking the Diffie–Hellman cryptographic scheme is harder than factoring P! (See =-=[BBR]-=- and also [Sh, Mc]). It will require quite a bit of work to formalize the above statement and argument with a short T C 0 -Frege proof. Notice that we want the size of the propositional formula expres... |

18 |
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(Show Context)
Citation Context ... is a k-symbol S proof of f. The k-line provability problem for S is to determine whether or not there is a k-line S proof of f. The k-line provability is an undecidable problem for first-order logic =-=[B1]-=-; the first complexity result fors1942 MARIA LUISA BONET, TONIANN PITASSI, AND RAN RAZ the k-provability problem for propositional logic was provided by Buss [B2], who proved the rather surprising fac... |

18 | Cutting planes, connectivity, and threshold logic - Buss, Clote - 1996 |

14 | On Gödel’s Theorem on length of proofs II: lower bounds for recognizing k symbol provability
- Buss
- 1995
(Show Context)
Citation Context ...cidable problem for first-order logic [B1]; the first complexity result fors1942 MARIA LUISA BONET, TONIANN PITASSI, AND RAN RAZ the k-provability problem for propositional logic was provided by Buss =-=[B2]-=-, who proved the rather surprising fact that the k-symbol propositional provability problem is NP-complete for a particular Frege system. More recently, [ABMP] show that the k-symbol and k-line provab... |

13 | Discretely ordered modules as a first-order extension of the cutting planes proof system - Kraj'icek - 1998 |

13 | Tautologies with a unique Craig interpolant, uniform vs. nonuniform complexity - Mundici - 1984 |

8 | A lower bound for the complexity of Craig’s interpolants in sentential logic - Mundici - 1983 |

5 | Complexity of Craig's interpolation - Mundici - 1982 |

4 |
Threshold circuits of small majority depth
- Maciel
- 1995
(Show Context)
Citation Context ...otherwise. Last, we will describe our T C 0 -formula for computing the iterated product of m numbers. This formula is basically the original formula of [BCH] and is articulated as a T C 0 -formula in =-=[M]-=-. The iterative product P ROD[z1, . . . , zm] gives the product of z1, . . . , zm, where each zi is of length n, and we assume that m, n are both bounded by N. The basic idea is to compute the product... |

1 |
R.Gavaldá, A.Maciel, and T.Pitassi, Non-automatizability of bounded-depth Frege proofs
- Bonet, Domingo
- 1999
(Show Context)
Citation Context ...sentially due to the nonuniform nature of the iterated product formulas that we use. It would be interesting to know to what extent our result holds in the uniform T C 0 proof setting. A recent paper =-=[BDGMP]-=- extends our results to prove that bounded-depth Frege does not have feasible interpolation assuming factoring Blum integers is sufficiently hard (actually their assumptions are stronger than ours). A... |

1 |
On ACC 0 [p k ]-Frege proofs
- Maciel, Pitassi
- 1997
(Show Context)
Citation Context ..., ⊕0, and ⊕1. (Thk(x) is true if and only if the number of 1’s in x is at least k, and ⊕i(x) is true if and only if the number of 1’s in x is i mod 2.) Our system is essentially the one introduced in =-=[MP]-=-, (which is, in turn, an extension of the system PTK introduced by Buss and Clote [BC, section 10]). Intuitively, a family of formulas f1, f2, f3, . . . has polynomial-sized T C0-Frege proofs if each ... |