## Relating the Bounded Arithmetic and Polynomial Time Hierarchies (1994)

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Venue: | Annals of Pure and Applied Logic |

Citations: | 28 - 1 self |

### BibTeX

@ARTICLE{Buss94relatingthe,

author = {Samuel R. Buss and Then T},

title = {Relating the Bounded Arithmetic and Polynomial Time Hierarchies},

journal = {Annals of Pure and Applied Logic},

year = {1994},

volume = {75},

pages = {75--67}

}

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### Abstract

The bounded arithmetic theory S 2 is finitely axiomatized if and only if the polynomial hierarchy provably collapses. If T 2 equals S then T 2 is equal to S 2 and proves that the polynomial time hierarchy collapses to # , and, in fact, to the Boolean hierarchy over # and to # i+1 /poly .

### Citations

122 |
Bounded Arithmetic; Bibliopolis
- Buss
- 1986
(Show Context)
Citation Context ...b i -IND. The following inclusions are known for these theories: S 0 2 # T 0 2 # S 1 2 # T 1 2 # S 2 2 # T 2 2 #s# Supported in part by NSF grant DMS-9205181 1 and their union is the theory S 2 = T 2 =-=[1]-=-. However, with the exception of S 0 2 #= T 0 2 (see [13]), it is not known whether the rest of the theories of bounded arithmetic are distinct. It is a well-known fact that S i 2 and T i 2 are finite... |

81 | Existence and feasibility in arithmetic - Parikh - 1971 |

58 | Notes on polynomially bounded arithmetic
- Zambella
- 1996
(Show Context)
Citation Context ... will review the necessary background material needed from [10]. After preparing the first draft of this paper, we learned that D. Zambella has independently discovered the main results of this paper =-=[15]-=-. 3 2 The KPT Witnessing Theorem for T i 2 There are two important witnessing theorems for T i 2 . The first follows from the `Main Theorem' for S i+1 2 and the fact that S i+1 2 is # b i+1 -conservat... |

57 |
Bounded arithmetic and the polynomial hierarchy
- Krajíček, Pudlák, et al.
- 1991
(Show Context)
Citation Context ...on between the possible collapses of the hierarchy of bounded arithmetic theories and the polynomial hierarchy. This question has already been partially answered by the work of KrajíčekPudlák-Takeuti =-=[10]-=- who showed that if T i 2 = S i+1 2 for any i ≥ 1, then the polynomial hierarchy collapses with Σ p i+2 =Π p i+2 (in fact, they show that in this case, Σ p i+1 ⊂ ∆ p i /poly). The main results of this... |

55 | kn application of Boolean complexity to separation problems in bounded arithmetic
- Buss, Krajicek
- 1994
(Show Context)
Citation Context ... to # p i is equal to the set # p T (# p i ) of predicates polynomial time, Turing reducible to # p i . However, it is open whether this last condition implies the polynomial hierarchy collapses. See =-=[5, 4, 6]-=- for more on this connection. The prerequisites for reading this paper are a basic knowledge of bounded arithmetic theories as contained in [1]. The reader would also benefit from knowledge of [10] an... |

50 | On truth-table reducibility to SAT
- Buss, Hay
- 1991
(Show Context)
Citation Context ... to # p i is equal to the set # p T (# p i ) of predicates polynomial time, Turing reducible to # p i . However, it is open whether this last condition implies the polynomial hierarchy collapses. See =-=[5, 4, 6]-=- for more on this connection. The prerequisites for reading this paper are a basic knowledge of bounded arithmetic theories as contained in [1]. The reader would also benefit from knowledge of [10] an... |

50 | The Boolean hierarchy and the polynomial hierarchy: A closer connection
- Chang, Kadin
- 1996
(Show Context)
Citation Context ... to # p i is equal to the set # p T (# p i ) of predicates polynomial time, Turing reducible to # p i . However, it is open whether this last condition implies the polynomial hierarchy collapses. See =-=[5, 4, 6]-=- for more on this connection. The prerequisites for reading this paper are a basic knowledge of bounded arithmetic theories as contained in [1]. The reader would also benefit from knowledge of [10] an... |

50 |
Turing machines that take advice. L’Enseignement Mathematique 28
- Karp, Lipton
- 1982
(Show Context)
Citation Context ...he condition that T i 2 = S i+1 2 . We have shown above that if T i 2 = S i+1 2 , then T i 2 proves that the # p i+2 - predicates are contained in # p i+1 /poly . From this, the methods of KarpLipton =-=[7]-=- imply that the entire polynomial time hierarchy is contained in # p i+1 /poly and in # p i+1 /poly ; furthermore, the proof of this containment can be formalized in T i 2 . The methods of Karp-Lipton... |

45 |
On the scheme of induction for bounded arithmetic formulas
- Wilkie, Paris
- 1987
(Show Context)
Citation Context ...ate that this hierarchy of theories collapses if and only if S 2 is finitely axiomatized. This latter condition is equivalent to I# 0 +# 1 being finitely axiomatized (see Parikh [11] and Wilkie-Paris =-=[14]-=- for this alternate, and original, approach to bounded arithmetic). There are close connections between theories of bounded arithmetic and the polynomial hierarchy. First, the class of predicates defi... |

20 |
conservation results for fragments of bounded arithmetic, in Logic and Computation, proceedings of a Workshop held Carnegie-Mellon
- Axiomatizations
- 1987
(Show Context)
Citation Context ...lish that the collapse of the polynomial time hierarchy is equivalent to a natural # 0 1 -condition like (#) . It is known that S i+1 2 is conservative over T i 2 with respect to ## b i+1 - sentences =-=[2]-=-. On the other hand, the axioms of S i+1 2 can be expressed as ## b i+2 -sentences (in this formulation, an induction axiom of S i+1 2 will become a ## b i+2 -formula with a sharply bounded existentia... |

16 |
No counter-example interpretation and interactive computation
- Krajíček
- 1992
(Show Context)
Citation Context ... all polynomial time functions. Note that since the functions fm are Σ b i+1 -definable by T i 2 , they must be p i+1 -functions. Theorem 1 is due to [10]; some later, related results can be found in =-=[8, 12, 3]-=-. We do not include a proof here. We next use Theorem 1 to establish a consequence of the condition T i 2 = S i+1 2 . We assume that i ≥ 0 and work with the theory T i 2 ; when i = 0 our results are i... |

15 |
Some relations between subsystems of arithmetic and the complexity of computations
- Pudl'ak
- 1992
(Show Context)
Citation Context ... all polynomial time functions. Note that since the functions fm are Σ b i+1 -definable by T i 2 , they must be p i+1 -functions. Theorem 1 is due to [10]; some later, related results can be found in =-=[8, 12, 3]-=-. We do not include a proof here. We next use Theorem 1 to establish a consequence of the condition T i 2 = S i+1 2 . We assume that i ≥ 0 and work with the theory T i 2 ; when i = 0 our results are i... |

11 |
Bounded arithmetic and the polynomial hierarchy
- cek, ak, et al.
- 1991
(Show Context)
Citation Context ...on between the possible collapses of the hierarchy of bounded arithmetic theories and the polynomial hierarchy. This question has already been partially answered by the work of KrajcekPudl ak-Takeuti =-=[10]-=- who showed that if T i 2 = S i+1 2 for any i # 1 , then the polynomial hierarchy collapses with # p i+2 = # p i+2 (in fact, they show that in this case, # p i+1 # # p i /poly ). The main results of t... |

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

5 |
of bounded arithmetic and bounded query classes
- Fragments
- 1993
(Show Context)
Citation Context ...2 -conservative over T i 2 is equivalent to saying that S i+1 2 = T i 2 . An open problem is to try to relate the condition S i 2 = T i 2 to the possible collapse of the polynomial hierarchy. Krajcek =-=[9]-=- shows that if S i 2 = T i 2 , then the set # p tt (# p i ) of predicates logspace, Turing reducible to # p i is equal to the set # p T (# p i ) of predicates polynomial time, Turing reducible to # p ... |

4 |
ak, Some relations between subsystems of arithmetic and the complexity of computations
- Pudl
- 1992
(Show Context)
Citation Context ...all polynomial time functions. Note that since the functions f m are # b i+1 -definable by T i 2 , they must be p i+1 -functions. Theorem 1 is due to [10]; some later, related results can be found in =-=[8, 12, 3]-=-. We do not include a proof here. We next use Theorem 1 to establish a consequence of the condition T i 2 = S i+1 2 . We assume that i # 0 and work with the theory T i 2 ; when i = 0 our results are i... |

3 |
cek, No counter-example interpretation and interactive computation
- Kraj
- 1992
(Show Context)
Citation Context ...all polynomial time functions. Note that since the functions f m are # b i+1 -definable by T i 2 , they must be p i+1 -functions. Theorem 1 is due to [10]; some later, related results can be found in =-=[8, 12, 3]-=-. We do not include a proof here. We next use Theorem 1 to establish a consequence of the condition T i 2 = S i+1 2 . We assume that i # 0 and work with the theory T i 2 ; when i = 0 our results are i... |

3 |
Sharply Bounded Arithmetic and the Function a ˙−1, Logic and computation
- Takeuti
- 1987
(Show Context)
Citation Context ...heories: S 0 2 # T 0 2 # S 1 2 # T 1 2 # S 2 2 # T 2 2 #s# Supported in part by NSF grant DMS-9205181 1 and their union is the theory S 2 = T 2 [1]. However, with the exception of S 0 2 #= T 0 2 (see =-=[13]-=-), it is not known whether the rest of the theories of bounded arithmetic are distinct. It is a well-known fact that S i 2 and T i 2 are finitely axiomatized for i > 0 , and thus it is immediate that ... |

1 | On truth-table reducibility to SAT, Information and Computation - Buss, HAY - 1991 |

1 | of bounded arithmetic and bounded query classes - F'agmcnts - 1993 |

1 | Bounded arithmetic and the polynomial hierarchy - KRAJf0EK, PUDLXK, et al. - 1991 |

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