## An application of boolean complexity to separation problems in bounded arithmetic (1994)

Venue: | Proc. London Math. Society |

Citations: | 55 - 15 self |

### BibTeX

@INPROCEEDINGS{Buss94anapplication,

author = {Samuel R. Buss and Jan Krajlcek},

title = {An application of boolean complexity to separation problems in bounded arithmetic},

booktitle = {Proc. London Math. Society},

year = {1994},

pages = {1--21}

}

### Years of Citing Articles

### OpenURL

### Abstract

We develop a method for establishing the independence of some Zf(a)-formulas from S'2(a). In particular, we show that T'2(a) is not VZ*(a)-conservative over S'2(a). We characterize the Z^-definable functions of T2 as being precisely the functions definable as projections of polynomial local search (PLS) problems. Although it is still an open problem whether bounded arithmetic S2 is finitely axiomatizable, considerable progress on this question has been made: S2 +1 is V2f+1-conservative over T'2 [3], but it is not V2!f+2-conservative unless £f+2 = Ylf+2 [10], and in addition, T2 is not VZf+1-conservative over S'2 unless LogSpace s? = Af+1 [8]. In particular, S2 is not finitely axiomatizable provided that the polynomial-time hierarchy does not collapse [10]. For the theory S2(a) these results imply (with some additional arguments) absolute results: S'2 + (a) is V2f+,(a)-conservative but not VZf+2(a)-conservative over T'2(a), and T'2(a) is not VZf+i(c*)-conservative over S'2(a). Here a represents a new uninterpreted predicate symbol adjoined to the language of arithmetic which may be used in induction formulas; from a computer science perspective, a represents an oracle. In this paper we pursue this line of investigation further by showing that T'2(a) is also not V2f(a)-conservative over S'2(a). This was known for / = 1, 2 by [9,17] (see also [2]), and our present proof uses a version of the pigeonhole principle similar to the arguments in [2,9]. Perhaps more importantly, we formulate a general method (Theorem 2.6) which can be used to show the unprovability of other 2f(a)-formulas from S'2(a). Our methods are analogous in spirit to the proof strategy of [8]: prove a witnessing theorem to show that provability of a Zf+1(a)-formula A in S'2(a) implies that it is witnessed by a function of certain complexity and then employ techniques of boolean complexity to construct an oracle a such that the formula A cannot be witnessed by a function of the prescribed complexity. Our formula A shall be 2f(a) and thus we can use the original witnessing theorem of [2]. The boolean complexity used is the same as in [8], namely Hastad's switching lemmas [6].

### Citations

241 | Almost optimal lower bounds for small depth circuits
- H˚astad
- 1986
(Show Context)
Citation Context ...the prescribed complexity. Our formula A shall be 2f(a) and thus we can use the original witnessing theorem of [2]. The boolean complexity used is the same as in [8], namely Hastad's switching lemmas =-=[6]-=-. The work of the first author was supported in part by NSF grants DMS-8902480 and INT-8914569. Part of it was performed while he was visiting the Czechoslovakian Academy of Sciences. The work of the ... |

222 |
Parity, circuits, and the polynomial-time hierarchy
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- 1984
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Citation Context ...variables from BkJ(m), where S^2° ogf " y c large enough. and t = \og(S), for Thus we henceforth think of M as being a Zf+,-circuit oracle (with variables from Bki(m)) machine with 5, t, m constants. =-=(4)-=- Let Pj be randomly chosen restrictions from Uk[jim(qj), for j = i, i - 1, ..., 1, where ey is + if y is odd and - if j is even, and g, = ((2(1 - i +j)\og(m))/m)^. are interested in what the effect of... |

196 |
How easy is local search
- Johnson, Papadimitriou, et al.
- 1988
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Citation Context ...ly, if b^mSk-.x, let / have value 0 whenever fk is undefined, and if b=m,k_l, let f( m sk) ~ rr ^$k + 1 an d otherwise have value 0 whenever/, is undefined. 5. T\ and polynomial local search (5.1) In =-=[7]-=- a Polynomial Local Search problem (PLS-problem) L is defined to be a maximization problem satisfying the following conditions (we have made some inessential simplifications to the definition in [7]):... |

155 | On the complexity of the parity argument and other inefficient proofs of existence - Papadimitriou - 1994 |

126 |
Exponential lower bounds for the pigeonhole principle
- Pitassi, Beame, et al.
- 1993
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Citation Context ... g is further presumed to be the inverse of / then the pigeonhole principle for / is equivalent to the statement that if N has a node of indegree 0 then N must have a node of outdegree 0. However, by =-=[16,11,1]-=-, the pigeonhole principle for/is not provable even in T2(f). Thus T2(f, g) does not prove the existence of solutions for this PDLF-problem. Acknowledgement. We thank Mario Chiari for noticing an inac... |

122 |
Bounded Arithmetic; Bibliopolis
- Buss
- 1986
(Show Context)
Citation Context ... a represents an oracle. In this paper we pursue this line of investigation further by showing that T'2(a) is also not V2f(a)-conservative over S'2(a). This was known for / = 1, 2 by [9,17] (see also =-=[2]-=-), and our present proof uses a version of the pigeonhole principle similar to the arguments in [2,9]. Perhaps more importantly, we formulate a general method (Theorem 2.6) which can be used to show t... |

107 |
Metamathematics of First-Order Arithmetic
- Hájek, Pudlák
- 1998
(Show Context)
Citation Context ... reader is familiar with bounded arithmetic and with the basics of boolean complexity. A reference on boolean complexity is [6] and on bounded arithmetic is [2] or the broader survey in the monograph =-=[5]-=-. The boolean circuits used in this paper are always constructed with unbounded fanin ANDs and ORs in alternating levels; NOT gates are not used, instead input signals p may be negated (denoted p). 1.... |

74 |
Provability of the pigeonhole principle and the existence of infinitely many primes
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- 1988
(Show Context)
Citation Context ...ry predicate symbol and A' a (a, Xi,x2, JC3) be the nf(a)-formula defined in (2.5). Then we have: THEOREM (Paris, Wilkie, and Woods). For all i**0, WPHP(a, A 1 '") is provable by T2 +2 (a). Proof. In =-=[15]-=- it was shown that WPHP(a, /3) is provable in /A0(j3) + Q,, and thus also in r2(j3). Already, in [2], it has been observed that this proof can be carried out in Tl((3). This implies the theorem. (3.3)... |

68 | A Woods. Exponential lower bounds to the size of bounded depth Frege proofs of the pigeonhole principle. Random Structures and Algorithms
- Krajíček, Pudlák
- 1995
(Show Context)
Citation Context ... g is further presumed to be the inverse of / then the pigeonhole principle for / is equivalent to the statement that if N has a node of indegree 0 then N must have a node of outdegree 0. However, by =-=[16,11,1]-=-, the pigeonhole principle for/is not provable even in T2(f). Thus T2(f, g) does not prove the existence of solutions for this PDLF-problem. Acknowledgement. We thank Mario Chiari for noticing an inac... |

57 |
Bounded arithmetic and the polynomial hierarchy
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- 1991
(Show Context)
Citation Context ... bounded arithmetic S2 is finitely axiomatizable, considerable progress on this question has been made: S2 +1 is V2f+1-conservative over T'2 [3], but it is not V2!f+2-conservative unless £f+2 = Ylf+2 =-=[10]-=-, and in addition, T2 is not VZf+1-conservative over S'2 unless LogSpace s? = Af+1 [8]. In particular, S2 is not finitely axiomatizable provided that the polynomial-time hierarchy does not collapse [1... |

29 |
On graph-theoretic lemmata and complexity classes (extended abstract), in
- Papadimitriou
- 1990
(Show Context)
Citation Context ...xists a Zt-formula B such that T\ proves (3 b: ^t)B and B^>A1 These questions are not apparently equivalent since even if local optima are unique, they may not be provably unique in T\. Papadimitriou =-=[13]-=- has introduced two classes PLF and PLDF of search problems and shown that PLDF ^ PLF.t A PLDF search problem L has, for every input x, a directed graph Nx(c, c') on nodes c, c' <t(x) for some term of... |

28 | Axiomatizations and conservation results for fragments of bounded arithmetic; in Logic and Computation, proceedings of a Workshop held Carnegie-Mellon
- Buss
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Citation Context ...9. Part of it was performed while he was visiting the University of Illinois, Champaign-Urbana. 1991 Mathematics Subject Classification: primary 03F30; secondary 68Q15, 68Q25. Proc. London Math. Soc. =-=(3)-=- 69 (1994) 1-21.s2 SAMUEL R. BUSS AND JAN KRAJICEK Johnson, Papadimitriou, and Yannakakis [7] introduced a class of polynomial local search (PLS) problems. In the final section of this paper, we provi... |

23 |
Fragments of Bounded Arithmetic and Bounded Query Classes
- Krajíček
- 1993
(Show Context)
Citation Context ...n has been made: S2 +1 is V2f+1-conservative over T'2 [3], but it is not V2!f+2-conservative unless £f+2 = Ylf+2 [10], and in addition, T2 is not VZf+1-conservative over S'2 unless LogSpace s? = Af+1 =-=[8]-=-. In particular, S2 is not finitely axiomatizable provided that the polynomial-time hierarchy does not collapse [10]. For the theory S2(a) these results imply (with some additional arguments) absolute... |

20 |
Consequences of the provability
- Cook, Krajíček
(Show Context)
Citation Context ...(a) is not V2. l fta)-conservative over any S'j(a), for i^l, Proof. The corollary follows from Theorems 3.2 and 3.3. Use Remark (2.7) for the second part. The second part of Corollary 3.4 complements =-=[12]-=- where it was shown that T'j+i is not nt-conservative over T), for i,;^1. 4. The iteration principle (4.1) The previous section showed that T'2(a) is not V2f(a)-conservative over S'2(a) by reducing (v... |

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

16 |
No counter-example interpretation and interactive computation
- Krajíček
- 1992
(Show Context)
Citation Context ...ence perspective, a represents an oracle. In this paper we pursue this line of investigation further by showing that T'2(a) is also not V2f(a)-conservative over S'2(a). This was known for / = 1, 2 by =-=[9,17]-=- (see also [2]), and our present proof uses a version of the pigeonhole principle similar to the arguments in [2,9]. Perhaps more importantly, we formulate a general method (Theorem 2.6) which can be ... |

15 |
Some relations between subsystems of arithmetic and the complexity of computations
- Pudl'ak
- 1992
(Show Context)
Citation Context ...ence perspective, a represents an oracle. In this paper we pursue this line of investigation further by showing that T'2(a) is also not V2f(a)-conservative over S'2(a). This was known for / = 1, 2 by =-=[9,17]-=- (see also [2]), and our present proof uses a version of the pigeonhole principle similar to the arguments in [2,9]. Perhaps more importantly, we formulate a general method (Theorem 2.6) which can be ... |

11 | Bounded arithmetic and the polynomial hierarchy - cek, ak, et al. - 1991 |

10 | Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle, Random Structures and Algorithms - cek, ak, et al. - 1995 |

7 | On the Complexity of the Parity Argument and Other Inecient Proofs of Existence - Papadimitriou - 1994 |

6 | counter-example interpretation and interactive computation - No - 1992 |

4 | ak, Some relations between subsystems of arithmetic and the complexity of computations - Pudl - 1992 |

3 | ajek and P. Pudl' ak, Metamathematics of First-order Arithmetic - H' - 1993 |

1 | cek, Fragments of bounded arithmetic and bounded query classes - Kraj |

1 | i cek and G. Takeuti, On induction-free provability - Kraj' - 1992 |

1 | Some relations between subsystems of arithmetic and the complexity of computations - ak - 1992 |