## Divide and Conquer in Parallel Complexity and Proof Theory (1992)

Citations: | 4 - 2 self |

### BibTeX

@TECHREPORT{Bloch92divideand,

author = {Stephen Austin Bloch},

title = {Divide and Conquer in Parallel Complexity and Proof Theory},

institution = {},

year = {1992}

}

### OpenURL

### Abstract

Copyright Stephen Austin Bloch, 1992 All rights reserved. The dissertation of Stephen Bloch is approved, and it is acceptable in quality and form for publication on micro-

### Citations

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Untersuchungen über das logische Schließen
- Gentzen
- 1935
(Show Context)
Citation Context ...formul appearing in a proof arbitrarily long, their logical complexity would be bounded by that of the formula to be proven. As it turns out, a principal theorem of Gentzen's work on sequent calculus =-=[Gen35]-=- states that in fact we can assume without loss of generality that there are only certain restricted kinds of cuts in a proof. This statement will be made more precise in Theorem 26 below. 6.5 Types o... |

510 |
The complexity of theorem proving procedures
- Cook
- 1971
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Citation Context ...n oracle-querying, polynomial-time machine such that for all instances of the problem, the correct answer is \yes" i there is a sequence of oracle answers that leads the machine to answer \yes". Cook =-=[Coo71]-=- described this class of problems, and showed one NP problem, the satis ability of Boolean formul with free propositional variables, \complete" in the sense that if it were in P, they all would be, th... |

220 |
A taxonomy of problems with fast parallel algorithms
- Cook
- 1985
(Show Context)
Citation Context ...ynomial-size, polylog-depth, bounded-fanin circuit families. It is trivial thatNC k AC k for allk, and not di cult to show thatAC k NC k+1 , and soAC = [kAC k =NC. A good introduction to this area is =-=[Coo85]-=-. 7sMost treatments of circuit complexity concentrate on decision problems, and assume that each circuit has only one output gate� such circuits can be coaxed to compute functions by the \bit-graph" m... |

215 | Bounded-width polynomial-size branching programs recognize exactly those languages in NC1
- Barrington
- 1989
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Citation Context ...n. A more elegant characterization, in [Clo92], replaces the treeevaluation function by Cobham-style recursion on notation with a constant (rather than polynomial) bound, relying on Barrington's work =-=[Bar89]-=- on bounded-width branching programs� both of these Clote papers, like [Bus86], rely on a language expressly designed to have polynomial growth rate. Arai [Ara92] characterizes uniform NC 1 with syste... |

191 | A new recursion-theoretic characterization of the poly-time functions
- Bellantoni, Cook
- 1992
(Show Context)
Citation Context ...tational complexity classes have depended on rather arti cial base functions, unnatural restrictions on recursion and other constructors, or large numbers of construction schemes. Bellantoni and Cook =-=[BC92]-=- developed the cleanest characterization to date of FP, the functions computable in polynomial time, replacing a number of unnatural bounds and base functions by the relatively simple notion of a two-... |

124 | On uniformity within NC1
- Barrington, Immerman, et al.
- 1990
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Citation Context ...cance 2 i , andZ(i) is interpreted asBIT(i�x). The classFO, de ned in terms of rst-order formul with random access over nite models, is a machine-independent description of a form of uniformAC 0 : as =-=[BIS90]-=- shows,FO problems are computable by constant-depth, unbounded-fanin circuit families whose direct connection languages are recognizable in deterministic log time. If we extend C with other predicates... |

118 |
Feasibly constructive proofs and the propositional calculus
- Cook
- 1975
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Citation Context ...der theory but rather as an in nite family of statements in a quanti er-free theory� we can then draw conclusions of computational interest from how fast the proofs of those statements grow (see e.g. =-=[Coo75]-=-, [Bus87], [Clo92], and [BIK + 92]). 1.4 Circuit Complexity The three most common models of parallel computation are parallel randomaccess machines (PRAMs), alternating Turing machines, and circuit fa... |

72 | Polynomial size proofs of the propositional pigeonhole principle - Buss - 1987 |

46 | Predicative Recursion and Computational Complexity
- Bellantoni
- 1992
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Citation Context ...the polynomial hierarchy. In recent personal communication he claims to have characterized log space, linear space, andNC in a similar manner, but the author has not seen the proofs.) Both Bellantoni =-=[Bel92b]-=- and the author [Blo92] are currently developing general logical analogues to the safe-parameters technique, which should yield prooftheoretic and model-theoretic characterizations of a variety of com... |

26 |
The intrinsic computational diculty of functions
- Cobham
- 1965
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Citation Context ...ations 19sChapter 2 Preliminaries to Part I 2.1 Introduction Much of the history of the study of computational complexity has been informed by Cobham's elegant recursion-theoretic characterization in =-=[Cob65]-=- of the polynomial-time functions as the closure of certain simple functions under composition and a form of recursion. Speci cally, Cobham's \bounded recursion on notation" iterates a number of times... |

20 |
On polynomial Size Frege Proofs of Certain Combinatorial Principles
- Clote
- 1993
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Citation Context ...her as an in nite family of statements in a quanti er-free theory� we can then draw conclusions of computational interest from how fast the proofs of those statements grow (see e.g. [Coo75], [Bus87], =-=[Clo92]-=-, and [BIK + 92]). 1.4 Circuit Complexity The three most common models of parallel computation are parallel randomaccess machines (PRAMs), alternating Turing machines, and circuit families. We shall n... |

11 |
machine-independent characterizations of the parallel complexity classes ALOGTIME, AC k , NC k and NC
- Sequential
- 1989
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Citation Context ...ir systems. Compton and LaFlamme [CL90] present recursion-theoretic and logical characterizations of uniformNC 1 , relying on a polynomial size bound implicit in their nite-models approach. Clote, in =-=[Clo89b]-=-, introduced a scheme of concatenation recursion on notation that corresponds to computing each bit of a function in parallel, and characterized uniform NC 1 ,AC 0 , and a variety of other parallel cl... |

7 |
Arithmetizing Uniform NC, Annals of pure and applied logic
- Allen
- 1991
(Show Context)
Citation Context ...ingly restricted axiomatization of number theory. This work has led to a number of similar characterizations of complexity classes in terms of the proof strength of such restricted theories (see e.g. =-=[All91]-=-, [Clo89a], [BKT92], [Ara92]), and thence to considerable interest in the di erences in proof strength among 6sthe theories themselves. Part II of the present work investigates several such axiomatiza... |

7 |
Frege system, ALOGTIME, and bounded arithmetic
- Arai
- 1992
(Show Context)
Citation Context ...e leading 1 bit of its binary form. (These operations will be formalized within the system by the functionsPADSTR andUNPADSTR, de ned in Section 6.10.) This representation is similar to those in e.g. =-=[Ara92]-=- and [Clo89a]. 79s6.3 Languages We work in several rst-order languages of arithmetic. De nition 33 The language L1 contains the following constant, function, and relation symbols: $ 0� 1 constants =� ... |

5 |
A first order theory for the parallel complexity class NC
- Clote
- 1989
(Show Context)
Citation Context ...bit of its binary form. (These operations will be formalized within the system by the functionsPADSTR andUNPADSTR, de ned in Section 6.10.) This representation is similar to those in e.g. [Ara92] and =-=[Clo89a]-=-. 79s6.3 Languages We work in several rst-order languages of arithmetic. De nition 33 The language L1 contains the following constant, function, and relation symbols: $ 0� 1 constants =� binary in x r... |

5 |
Clote and Gaisi Takeuti, Bounded arithmetic for NC, ALogTIME
- Peter
- 1992
(Show Context)
Citation Context ... hierarchy. In particular, thei=1case of this 77stheorem provides an extremely useful characterization of polynomial-time-computable functions. Various researchers (notably [All91], [CL90], [Clo89a], =-=[CT90]-=-, [CT92], and [Fer90]) have de ned similar axiomatic systems corresponding to other complexity classes. The ones that most concern us here are the systemsR i 2 of [Tak92],TNC of [Clo89a] and [CT92], a... |

3 |
Kraj cek, and Gaisi Takeuti. Provably total functions in bounded arithmetic theoriesR i 3,U i 2 andV i 2
- Buss
(Show Context)
Citation Context ...iomatization of number theory. This work has led to a number of similar characterizations of complexity classes in terms of the proof strength of such restricted theories (see e.g. [All91], [Clo89a], =-=[BKT92]-=-, [Ara92]), and thence to considerable interest in the di erences in proof strength among 6sthe theories themselves. Part II of the present work investigates several such axiomatizations. Such logical... |

2 |
A new recursion-theoretic characterization of the functional polytime hierarchy
- Bellantoni
- 1992
(Show Context)
Citation Context ...strated its power and generality. We speculate that further variants on this technique may yield equally elegant characterizations of log space,NC,NC k , and other complexity classes. (Bellantoni, in =-=[Bel92a]-=-, extends the characterization of P to the polynomial hierarchy. In recent personal communication he claims to have characterized log space, linear space, andNC in a similar manner, but the author has... |

2 | Kraj cek, Toniann Pitassi, and Pavel Pudl ak, Lower bounds on Hilbert's Nullstellensatz and propositional proofs - Beame, Impagliazzo |

1 |
and Jan Kraj cek. An application of Boolean complexity to separation problems
- Buss
- 1992
(Show Context)
Citation Context ...CI are all equivalent, and thus that T i kS i k Ri k fori>0. The converses of these inclusions are open, and seem unlikely: for example, T i 2S i 2 is false when relativized with a suitable predicate =-=[BK92]-=-. It is known that S i+1 kT i k [Bus86] andR i+1 kS i k (Takeuti), again fori>0. The converses of these inclusions, too, are open and seem unlikely:S i+1 2T i 2, for example, would imply the collapse ... |

1 |
Logical analogues to safe recursion
- Bloch
- 1992
(Show Context)
Citation Context .... In recent personal communication he claims to have characterized log space, linear space, andNC in a similar manner, but the author has not seen the proofs.) Both Bellantoni [Bel92b] and the author =-=[Blo92]-=- are currently developing general logical analogues to the safe-parameters technique, which should yield prooftheoretic and model-theoretic characterizations of a variety of complexity classes in the ... |

1 |
Bounded Arithmetic. Number 3 in Studies in Proof Theory. Bibliopolis
- Buss
- 1986
(Show Context)
Citation Context ...nti ers are disallowed and we count the alternation of bounded quanti ers in a suitable language, we can syntactically characterize the relations at any desired level of the polynomial hierarchy (see =-=[Bus86]-=-, chapter 1). A more philosophically interesting characterization, the main theorem of [Bus86], shows the functions at any desired level of the polynomial hierarchy to be exactly those which are not o... |

1 |
An algebra and a logic forNC 1
- Compton, LaFlamme
- 1990
(Show Context)
Citation Context ...pace, and Allen [All91] gave logical and recursion-theoretic characterizations ofNC, both (like Cobham) explicitly bounding the growth rates of functions de ned in their systems. Compton and LaFlamme =-=[CL90]-=- present recursion-theoretic and logical characterizations of uniformNC 1 , relying on a polynomial size bound implicit in their nite-models approach. Clote, in [Clo89b], introduced a scheme of concat... |

1 |
Polynomial time induction
- Ferreira
- 1990
(Show Context)
Citation Context ...cular, thei=1case of this 77stheorem provides an extremely useful characterization of polynomial-time-computable functions. Various researchers (notably [All91], [CL90], [Clo89a], [CT90], [CT92], and =-=[Fer90]-=-) have de ned similar axiomatic systems corresponding to other complexity classes. The ones that most concern us here are the systemsR i 2 of [Tak92],TNC of [Clo89a] and [CT92], andD i of [All91], whi... |