## On provably disjoint NP-pairs (1994)

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Venue: | ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY |

Citations: | 44 - 2 self |

### BibTeX

@TECHREPORT{Razborov94onprovably,

author = {Alexander A. Razborov},

title = {On provably disjoint NP-pairs},

institution = {ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY},

year = {1994}

}

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### Abstract

In this paper we study the pairs (U; V ) of disjoint NP-sets representable in a theory T of Bounded Arithmetic in the sense that T proves U " V = ;. For a large variety of theories T we exhibit a natural disjoint NP-pair which is complete for the class of disjoint NP-pairs representable in T . This allows us to clarify the approach to showing independence of central open questions in Boolean complexity from theories of Bounded Arithmetic initiated in [11]. Namely, in order to prove the independence result from a theory T , it is sufficient to separate the corresponding complete NP-pair by a (quasi)poly-time computable set. We remark that such a separation is obvious for the theory S(S 2 ) + S \Sigma 2 \Gamma PIND considered in [11], and this gives an alternative proof of the main result from that paper.

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Citation Context ...ary Frege system. At this point it is no longer important that we work in the sequential calculus, but we prefer to stick to this for the sake of uniformity. ffl EF is the extended Frege proof system =-=[18, 4]-=-. It additionally allows us to use extension axioms of the form p j A, where p is a new propositional variable (called extension atom) which did not appear earlier in the proof. 5 For an unsatisfiable... |

145 | Complexity measures for public-key cryptosystems
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Citation Context ... least two good reasons to be interested in this issue. Firstly, the question of existence of such a pair not separable by a set in P is closely connected to the existence of public-key cryptosystems =-=[5]-=-. Part of this work was done while the author was visiting BRICS, Basic Research in Computer Science, Centre of the Danish National Research Foundation. Supported by the grant # 93-011-16015 of the Ru... |

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Bounded Arithmetic. Bibliopolis
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Citation Context ...c to purely complexity questions. The paper is concluded by a brief discussion of their status in Section 7. 2. Background from Logic 2.1. Systems of Bounded Arithmetic We assume the familiarity with =-=[1]-=-, and use the now-standard notation for denoting various hierarchies and fragments of Bounded Arithmetic from that book. L 1 is the first order language which consists of the constant 0, function symb... |

116 |
Feasibly constructive proofs and the propositional calculus (preliminary version
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Citation Context ...fi 1 (0) : : : fi 1 (a 1 \Gamma 1). The following two propositions are slight modifications of [7, Theorem 2.4] and [7, Theorem 2.5] respectively (the latter also follows from earlier results of Cook =-=[3]-=- via the correspondence between PV and S 1 2 [1, Chapter 6] and RSUV -isomorphism [14, 15, 9]): 6 Proposition 2.4. U 1 1 ` Ref F (a 0 ; a 1 ; fi 0 ; fi 1 ) oe :T r 2 (a 1 ; ff; fi 1 ). Proposition 2.5... |

83 |
Counting problems in bounded arithmetic
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Citation Context ..., 9]): 6 Proposition 2.4. U 1 1 ` Ref F (a 0 ; a 1 ; fi 0 ; fi 1 ) oe :T r 2 (a 1 ; ff; fi 1 ). Proposition 2.5. V 1 1 ` RefEF (a 0 ; a 1 ; fi 0 ; fi 1 ) oe :T r 2 (a 1 ; ff; fi 1 ). Paris and Wilkie =-=[8]-=- showed that I \Delta 0 (ff; fi) ` Ref F d (a 0 ; a 1 ; fi 0 ; fi 1 ) oe :T r 2 (a 1 ; ff; fi 1 ) for any fixed ds0. We will need the following refinement of their result: Lemma 2.6. For any fixed ds0... |

60 |
The gap between monotone and non-monotone circuit complexity is exponential
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Citation Context ...L 2 ) is representable in ST 3 2 and, thus, (CHR;CL 2 )sqp m (SAT ; REF (F 1 )). On the other hand, the known polytime computable separator for (CHR;CL 2 ) is based upon very deep combinatorial ideas =-=[16]-=-. I do not know of any evidence of this sort that (SAT ; REF (R)) is hard. This could be the next accessible question. 8. Acknowledgement I am indebted to Jan Kraj ' icek for his initial suggestion to... |

56 | Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic
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- 1995
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Citation Context ... disjoint NP-pairs representable in T . This allows us to clarify the approach to showing independence of central open questions in Boolean complexity from theories of Bounded Arithmetic initiated in =-=[11]-=-. Namely, in order to prove the independence result from a theory T , it is sufficient to separate the corresponding complete NP-pair by a (quasi)poly-time computable set. We remark that such a separa... |

49 | Bounded arithmetic and lower bounds in Boolean complexity
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Citation Context ... Research, and by an AMS-FSU grant. 1 The second motivation comes from the attempts to understand on the formal level the machinery existing in non-uniform Boolean complexity for proving lower bounds =-=[10, 12, 11]. Of the m-=-ain importance for this approach is the following observation. Let U consist of truth-tables of all "simple" Boolean functions, and let V * ) ff \Phi s j f 2 U g ; where s is a supposedly co... |

29 |
Oracles for structural properties: The isomorphism problem and public-key cryptography
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Citation Context ...s together, we obtain the independence result modulo the hardness assumption. The question if there exist disjoint NP-pairs which can not be separated by a set in P is open. Moreover, it was shown in =-=[6]-=- that there exists an oracle relative to which P 6= NP, and still such pairs do not exist. Thus, the assumption of the existence of P-inseparable disjoint NP-pairs seems to be stronger than merely P 6... |

29 |
RSUV isomorphism
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Citation Context ...ations of [7, Theorem 2.4] and [7, Theorem 2.5] respectively (the latter also follows from earlier results of Cook [3] via the correspondence between PV and S 1 2 [1, Chapter 6] and RSUV -isomorphism =-=[14, 15, 9]-=-): 6 Proposition 2.4. U 1 1 ` Ref F (a 0 ; a 1 ; fi 0 ; fi 1 ) oe :T r 2 (a 1 ; ff; fi 1 ). Proposition 2.5. V 1 1 ` RefEF (a 0 ; a 1 ; fi 0 ; fi 1 ) oe :T r 2 (a 1 ; ff; fi 1 ). Paris and Wilkie [8] ... |

28 | Axiomatizations and conservation results for fragments of bounded arithmetic; in Logic and Computation, proceedings of a Workshop held Carnegie-Mellon - Buss - 1987 |

28 | An Equivalence between Second Order Bounded Domain Bounded Arithmetic and First Order Bounded Arithmetic
- Razborov
- 1993
(Show Context)
Citation Context ...ations of [7, Theorem 2.4] and [7, Theorem 2.5] respectively (the latter also follows from earlier results of Cook [3] via the correspondence between PV and S 1 2 [1, Chapter 6] and RSUV -isomorphism =-=[14, 15, 9]-=-): 6 Proposition 2.4. U 1 1 ` Ref F (a 0 ; a 1 ; fi 0 ; fi 1 ) oe :T r 2 (a 1 ; ff; fi 1 ). Proposition 2.5. V 1 1 ` RefEF (a 0 ; a 1 ; fi 0 ; fi 1 ) oe :T r 2 (a 1 ; ff; fi 1 ). Paris and Wilkie [8] ... |

21 | On Frege and Extended Frege Proof Systems
- Krajíček
- 1995
(Show Context)
Citation Context ...e important in the sequel. Let Tr(a,α,β) be the predicate asserting that the truth assignment α makes the Boolean formula encoded by the string β(0)...β(a−1) true (truth definition). Proposition 2.2 (=-=[7]-=-). Tr(a,α,β) has a ∆ 1,b 1 -definition in U 1 1 about which U 1 1 proves the usual Tarski’s conditions. Let Trd(a,α,β)bethevariantofTr(a,α,β)inwhichβ(0)...β(a−1) encodes a Boolean formula from Σd ∪ Πd... |

7 |
Bounded existential induction
- Wilmers
- 1985
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Citation Context ...ain either only occurrences of ff or only occurrences of fi. The inductive definition of S \Sigma b i+1 ; S \Pi b i+1 is the same as for \Sigma b i+1 ; \Pi b i+1 . 3 The hierarchy E i ; U i (see e.g. =-=[17]-=-) was defined as the ordinary hierarchy of bounded formulae in the language of Peano Arithmetic (where we do not have the notion of a sharply bounded quantifier at all). A bounded formula is D i in a ... |

4 |
cek, On Frege and extended Frege proof systems
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Citation Context ...sequel. Let T r(a; ff; fi) be the predicate asserting that the truth assignment ff makes the Boolean formula encoded by the string fi(0) : : : fi(a \Gamma 1) true (truth definition). Proposition 2.2 (=-=[7]-=-). T r(a; ff; fi) has a \Delta 1;b 1 -definition in U 1 1 about which U 1 1 proves the usual Tarski's conditions. Let T r d (a; ff; fi) be the variant of T r(a; ff; fi) in which fi(0) : : : fi(a\Gamma... |

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S i 3 and ffi V i 2
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Citation Context ...ations of [7, Theorem 2.4] and [7, Theorem 2.5] respectively (the latter also follows from earlier results of Cook [3] via the correspondence between PV and S 1 2 [1, Chapter 6] and RSUV -isomorphism =-=[14, 15, 9]-=-): 6 Proposition 2.4. U 1 1 ` Ref F (a 0 ; a 1 ; fi 0 ; fi 1 ) oe :T r 2 (a 1 ; ff; fi 1 ). Proposition 2.5. V 1 1 ` RefEF (a 0 ; a 1 ; fi 0 ; fi 1 ) oe :T r 2 (a 1 ; ff; fi 1 ). Paris and Wilkie [8] ... |

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Citation Context ...ary Frege system. At this point it is no longer important that we work in the sequential calculus, but we prefer to stick to this for the sake of uniformity. ffl EF is the extended Frege proof system =-=[18, 4]-=-. It additionally allows us to use extension axioms of the form p j A, where p is a new propositional variable (called extension atom) which did not appear earlier in the proof. 5 For an unsatisfiable... |

1 |
Natural proofs. Preliminary version appeared
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- 1994
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Citation Context ...MPLE t 2 NP. It turns out that the computational hardness of this language to a certain extent captures the hardness of proving lower bounds on the circuit size of explicit functions. For example, in =-=[12]-=- Razborov and Rudich introduced the notion of a natural proof justified by a careful analysis of existing proofs for restricted models. This notion can be reformulated in terms of purely structural pr... |