## On Feasible Numbers (1995)

Venue: | Logic and Computational Complexity, LNCS Vol. 960 |

Citations: | 5 - 1 self |

### BibTeX

@INPROCEEDINGS{Sazonov95onfeasible,

author = {Vladimir Yu. Sazonov},

title = {On Feasible Numbers},

booktitle = {Logic and Computational Complexity, LNCS Vol. 960},

year = {1995},

pages = {30--51},

publisher = {Springer}

}

### OpenURL

### Abstract

. A formal approach to feasible numbers, as well as to middle and small numbers, is introduced, based on ideas of Parikh (1971) and improving his formalization. The "vague" set F of feasible numbers intuitively satisfies the axioms 0 2 F , F + 1 ` F and 2 1000 62 F , where the latter is stronger than a condition considered by Parikh, and seems to be treated rigorously here for the first time. Our technical considerations, though quite simple, have some unusual consequences. A discussion of methodological questions and of relevance to the foundations of mathematics and of computer science is an essential part of the paper. 1 Introduction How to formalize the intuitive notion of feasible numbers? To see what feasible numbers are, let us start by counting: 0,1,2,3, and so on. At this point, A.S. Yesenin-Volpin (in his "Analysis of potential feasibility", 1959) asks: "What does this `and so on' mean?" "Up to what extent `and so on'?" And he answers: "Up to exhaustion!" Note that by cos...

### Citations

270 | Relational queries computable in polynomial time
- Immerman
- 1986
(Show Context)
Citation Context ... + 1))] 4 S(n) [:S(n + 1)] 1 9x(S(x)&:S(x + 1)) ? S(n + 1) 1 : Using successively these inferences for n = 0; 1; : : : ; 8, together with the evident proof of S(0), and the natural deduction . . . . S=-=(9)-=- [S(10)] 3 [10slog 2 log 2 y] 2 8y(log 2 log 2 y ! 10) log 2 log 2 y ! 10 ? ? 2 :S(10) 3 9x(S(x)&:S(x + 1)) [:9x(S(x)&:S(x + 1))] 4 ? 9x(S(x)&:S(x + 1)) 4 we obtain the required normal proof with feas... |

115 | Feasibly constructive proofs and the propositional calculus - Cook - 1975 |

109 | Metamathematics of first-order arithmetic - Hájek, Pudlák - 1993 |

87 | Functional interpretations of feasibly constructive arithmetic - Cook, Urquhart - 1993 |

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

75 | Natural Deduction - Prawitz - 1971 |

68 | Bounded Arithmetic
- Buss
- 1986
(Show Context)
Citation Context ...M (x)) and 8x(S(x) )M (x)). Note that provability of M (0); :M (1024) and 8x(M (x) )M (x + 1)) gives no contradiction here. Indeed, the reader may see that the corresponding deduction M (0); M (0) )M =-=(1)-=-; M (1); M (1) )M (2); M (2); : : : ; M (1024) by multiple application of modus ponens rule is not normal one because M (x) ) M (x + 1) is actually deduced by introduction of implication rule (see the... |

50 | Algebras of feasible functions - Gurevich - 1983 |

32 | Predicative Arithmetic - Nelson - 1986 |

21 | Polynomial computability and recursivity in finite domains, Elektronische Informationsverarbeitung und Kybernetik 60 - Sazonov - 1980 |

17 |
The polynomial hierarchy and intuitionistic bounded arithmetic, Structure in Complexity Theory
- Buss
- 1986
(Show Context)
Citation Context ... (x)). Note that provability of M (0); :M (1024) and 8x(M (x) )M (x + 1)) gives no contradiction here. Indeed, the reader may see that the corresponding deduction M (0); M (0) )M (1); M (1); M (1) )M =-=(2)-=-; M (2); : : : ; M (1024) by multiple application of modus ponens rule is not normal one because M (x) ) M (x + 1) is actually deduced by introduction of implication rule (see the proof below), so mod... |

8 | Correctness of inconsistent theories with notions of feasibility - Dragalin - 1985 |

8 | Limitations to mathematical knowledge - Gandy - 1982 |

2 |
Automata and life (in Russian), Kibernetika -- neogranichennye vozmozhnosti i vozmozhnye ogranichenija. Itogi razvitija
- Kolmogorov
- 1979
(Show Context)
Citation Context ...xslog 2 y) (here y 6= 0 is an inessential technical restriction to simplify one formal proof below) and S(x) := "x is a small number " := 9y(xslog 2 log 2 y). Then, we have Fact 4. FEAS ` n =-=f S(0); :S(10)-=-; 9x(S(x)&:S(x + 1)); M (0); :M (1024), 8x(M (x) ) M (x + 1)); 8xsy(S(y) ) S(x)); 8xsy(M (y) ) M (x)) and 8x(S(x) )M (x)). Note that provability of M (0); :M (1024) and 8x(M (x) )M (x + 1)) gives no c... |

1 | Bounded Arithmetic, Propositional Logic and Complexity Theory - icek - 1995 |

1 | The lower bounds of complexity the deductions increasing after cut elimination - Orevkov - 1979 |

1 | A logical approach to the problem "P - Sazonov - 1980 |