## On the b 1 -bit-comprehension rule (2000)

Venue: | Logic Colloquium 98 |

Citations: | 5 - 0 self |

### BibTeX

@INPROCEEDINGS{Johannsen00onthe,

author = {Jan Johannsen and Chris Pollett},

title = {On the b 1 -bit-comprehension rule},

booktitle = {Logic Colloquium 98},

year = {2000},

pages = {262--279}

}

### OpenURL

### Abstract

Summary. The theory � b 1-CR of Bounded Arithmetic axiomatized by the � b 1-bit-comprehension rule is defined and shown to be strongly related to the complexity class TC 0. The � b 1-definable functions of � b 1-CR are those in uniform TC 0, and the � b 2-definable functions are computable by counterexample computations using TC 0-functions. The latter is used to show that a collapse of stronger theories to � b 1-CR implies that NP is contained in non-uniform TC 0. 1

### Citations

213 |
Bounded Arithmetic, Propositional Logic, and Complexity Theory, volume 60 of Encyclopedia of Mathematics and Its Applications
- Krajíček
- 1995
(Show Context)
Citation Context ...cation ·, #, |x|, Bit and which is closed under composition and CRN. 3 Theories of Bounded Arithmetic We briefly review the necessary background on Bounded Arithmetic, for more information see [4] or =-=[10]-=-. The language L2 of Bounded Arithmetic comprises . the usual signature of arithmetic 0, S, +, , ·, ≤, together with function symbols for ⌊ 1 2x⌋, MSP(x, i) := ⌊x/2i⌋, |x| and #. A quantifier of the f... |

123 |
Bounded Arithmetic. Bibliopolis
- Buss
- 1986
(Show Context)
Citation Context ...ultiplication ·, #, |x|, Bit and which is closed under composition and CRN. 3 Theories of Bounded Arithmetic We briefly review the necessary background on Bounded Arithmetic, for more information see =-=[4]-=- or [10]. The language L2 of Bounded Arithmetic comprises . the usual signature of arithmetic 0, S, +, , ·, ≤, together with function symbols for ⌊ 1 2x⌋, MSP(x, i) := ⌊x/2i⌋, |x| and #. A quantifier ... |

121 | On uniformity within NC1
- Mix-Barrington, Immerman, et al.
- 1990
(Show Context)
Citation Context ...o overcome this sometimes unwanted feature, circuit families are required to satisfy certain uniformity conditions. For TC 0 -circuits, the most suitable uniformity notion is DLogTime-uniformity, see =-=[3]-=- for the somewhat involved definition. DLogTime-uniform TC 0 is a fairly natural complexity class: it is characterized by first-order logic with majority quantifiers on ordered finite models [3] in De... |

58 |
Bounded Arithmetic and the Polynomial Hierarchy
- Krajíček, Pudlák, et al.
- 1991
(Show Context)
Citation Context ... Notes in Logic, 13 c○ 2000, Association for Symbolic Logic 262sOn the � b 1-Bit-Comprehension Rule 263 below) where the Student has the computational capabilities of TC 0 . Similar to the results of =-=[12]-=-, this will allow us to show that a collapse of stronger theories, S 1 2 or R1 2 , to �b 1-CR implies that every NP-predicate can be decided by non-uniform TC 0-circuits. 2 Uniform and Non-Uniform TC ... |

29 |
RSUV isomorphism
- Takeuti
- 1993
(Show Context)
Citation Context ...s a set of quantifier-free axioms specifying the interpretations of the function symbols of L2. It can most conveniently be taken as the set BASIC from [4] together with the axioms for MSP and . from =-=[14]-=-. For a class of formulas �, the axiom schema �-LIND is for each A(x) ∈ �, and �-LLIND is A(0) ∧ ∀x (A(x) → A(Sx)) → ∀x A(|x|) A(0) ∧ ∀x (A(x) → A(Sx)) → ∀x A(||x||) for A(x) ∈ �. In general, for m ≥ ... |

28 | The permanent requires large uniform threshold circuits. Submitted. A preliminary version of this paper appeared as [All96
- Allender
(Show Context)
Citation Context ...s characterized by first-order logic with majority quantifiers on ordered finite models [3] in Descriptive Complexity Theory, or by acceptance in time O(log(n)) on so-called Threshold Turing Machines =-=[2]-=-, or by the machine-independent characterization below, which is most convenient for our purposes. Whenever we speak of TC 0 in the following without further qualification, we mean DLogTime-uniform TC... |

20 |
On polynomial Size Frege Proofs of Certain Combinatorial Principles
- Clote
- 1993
(Show Context)
Citation Context ...X is equivalent to �b 1 -LIND and �b 1 -LLMAX is equivalent to �b 1 -LLIND over BASIC + open-LIND. 4 Proof of Conservativity The following two lemmas are well-known and easily proved by the method of =-=[6]-=-: Lemma 1. The � b 0 -predicates are computable in TC 0 . In particular, the L2-base functions are in TC 0 .s268 Jan Johannsen and Chris Pollett Lemma 2. Let f be a function in TC 0 . Then the functio... |

18 |
Arithmetizing uniform NC
- Allen
- 1989
(Show Context)
Citation Context ...and computational complexity. Theorem 1. – The �b i -definable functions in S i FP 2 are exactly those in �P i−1, for each i ≥ 1 [4]. – The �b 1 -definable functions in R1 – 2 are exactly those in NC =-=[1, 5]-=-. The �b 1 -definable functions in C 0 2 are exactly those in TC 0 [8, 9]. The comprehension axiom for formula A(x), denoted COMPA(a), is the formula ∃y <2 |a| ∀x <|a| � Bit(y, x) = 1 ↔ A(x) � . The �... |

16 |
First order bounded arithmetic and small Boolean circuit complexity classes
- Clote, Takeuti
- 1992
(Show Context)
Citation Context ... 0 , is ∀�b 1 -conservative over �b 1-CR. Theories of Bounded Arithmetic that correspond to the complexity class TC 0 have been described earlier by the authors [9, 8] as well as by Clote and Takeuti =-=[7]-=-. So why do we come up with yet another one? We think there are two reasons that make �b 1-CR more interesting than the previous theories for TC 0 . First, one can argue that it is the weakest natural... |

15 |
Some relations between subsystems of arithmetic and the complexity of computations
- Pudl'ak
- 1992
(Show Context)
Citation Context ...will show that �b 1-CR has a tighter connection to TC 0 than the previously considered theories: The �b 2 -theorems of �b 1-CR can be witnessed by counterexample computations (a concept introduced by =-=[13,11]-=- that we will define ⋆ Supported by DFG grant No. Jo 291/1-1. Logic Colloquium ’98 Edited by S. R. Buss, P. Hájek and P. Pudlák Lecture Notes in Logic, 13 c○ 2000, Association for Symbolic Logic 262sO... |

9 | On proofs about threshold circuits and counting hierarchies (extended abstract
- Johannsen, Pollett
- 1998
(Show Context)
Citation Context ... that has this rule as its main axiom. This theory is related to the computational complexity class TC 0 of functions computable by constant-depth threshold circuits. We show that the theory C 0 2 of =-=[9]-=-, whose �b 1 -definable functions are TC 0 , is ∀�b 1 -conservative over �b 1-CR. Theories of Bounded Arithmetic that correspond to the complexity class TC 0 have been described earlier by the authors... |

8 | A bounded arithmetic theory for constant depth threshold circuits
- Johannsen
- 1996
(Show Context)
Citation Context ... whose �b 1 -definable functions are TC 0 , is ∀�b 1 -conservative over �b 1-CR. Theories of Bounded Arithmetic that correspond to the complexity class TC 0 have been described earlier by the authors =-=[9, 8]-=- as well as by Clote and Takeuti [7]. So why do we come up with yet another one? We think there are two reasons that make �b 1-CR more interesting than the previous theories for TC 0 . First, one can ... |

5 |
A first order theory for the parallel complexity class NC
- Clote
- 1989
(Show Context)
Citation Context ...and computational complexity. Theorem 1. – The �b i -definable functions in S i FP 2 are exactly those in �P i−1, for each i ≥ 1 [4]. – The �b 1 -definable functions in R1 – 2 are exactly those in NC =-=[1, 5]-=-. The �b 1 -definable functions in C 0 2 are exactly those in TC 0 [8, 9]. The comprehension axiom for formula A(x), denoted COMPA(a), is the formula ∃y <2 |a| ∀x <|a| � Bit(y, x) = 1 ↔ A(x) � . The �... |

4 |
Interactive computations of optimal solutions
- Kraj'icek, Pudl'ak, et al.
- 1990
(Show Context)
Citation Context ...will show that �b 1-CR has a tighter connection to TC 0 than the previously considered theories: The �b 2 -theorems of �b 1-CR can be witnessed by counterexample computations (a concept introduced by =-=[13,11]-=- that we will define ⋆ Supported by DFG grant No. Jo 291/1-1. Logic Colloquium ’98 Edited by S. R. Buss, P. Hájek and P. Pudlák Lecture Notes in Logic, 13 c○ 2000, Association for Symbolic Logic 262sO... |