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## A Logical Calculus for Polynomial-time Realizability (1991)

Venue: | Journal of Methods of Logic in Computer Science |

Citations: | 3 - 0 self |

### Citations

828 | Light Linear Logic.
- Girard
- 1998
(Show Context)
Citation Context ...mma A means "A is realizable"; similarly e k\Gamma A means "e realizes A".) Second, among the structure rules we keep thinning, but drop exchange and contraction, (roughly as in Gi=-=rard's linear logic [7]-=-). Again, it was shown in [4] that one could have k\Gamma A oe (B oe C) without having k\Gamma B oe (A oe C). For example, in [4] there is an example that occurs frequently. In the notation of this pa... |

306 | Investigations into logical deductions. - Gentzen - 1969 |

87 | Functional interpretations of feasibly constructive arithmetic,
- Cook, Urquhart
- 1988
(Show Context)
Citation Context ...= 0sx 6= 0) (Z3) (x :); x = 0 \Gamma! :(x 6= 0) (Z4) (x :); x 6= 0 \Gamma! :(x = 0) We also have the following versions of Buss's [1] axioms of arithmetic, or equivalently, those of Cook and Urquhart =-=[3]-=-: (1) (x; y :); x ! ysx = y \Gamma! x ! Sy (2) (x :); x = Sx \Gamma!? (3) B (x :) \Gamma! 0sx (4) (x; y :); xsy \Gamma! x = ysSxsy (5) (x; y :); x = Sy \Gamma! 2 \Delta x = SS(2 \Delta y) (6) (x; y :)... |

34 | Deductive systems and categories - Lambek - 1968 |

16 |
Multicategories revisited
- Lambek
- 1987
(Show Context)
Citation Context ... as part of the class of polynomially curried functions. We expect to present more details in a separate paper. An aside for the category-theorist: our structure is essentially a closed multicategory =-=[9, 10]-=- with finite products and coproducts: oe gives the internal hom but this is not a hom for the product structure given by . We do not have a tensor\Omega ---the comma in the sequent notation takes that... |

16 |
Hyperdoctrines, natural deduction and the Beck condition. Zeitschrift für mathematische Logik und Grundlagen der
- Seely
- 1983
(Show Context)
Citation Context ...e done without loss in expressive power by a liberal use of "dummy free variables"; perhaps the simplest technical way to do this is to add new function symbols to L corresponding to project=-=ions (see [12]-=- for example, where first order logic with equality is handled this way). Note that in quantifying a formula, exactly one free variable disappears---viz. the one quantified. Definition 3 A sequent \Ga... |

7 |
Deductive systems and categories II
- Lambek
- 1969
(Show Context)
Citation Context ... as part of the class of polynomially curried functions. We expect to present more details in a separate paper. An aside for the category-theorist: our structure is essentially a closed multicategory =-=[9, 10]-=- with finite products and coproducts: oe gives the internal hom but this is not a hom for the product structure given by . We do not have a tensor\Omega ---the comma in the sequent notation takes that... |

3 | Polynomially graded logic - Nerode, Remmel, et al. - 1989 |

2 |
Bounded Arithmetic, Doctoral dissertation
- Buss
- 1985
(Show Context)
Citation Context ...1 Review of polynomial-time realizability We review informally the basic definitions from [4]---for full details, the reader should refer to that paper. We assume a formal language L based on that in =-=[1]-=-, with constants including 0, function letters including +, \Delta , exp (exponentiation), S (successor), ] (smash product), b c (greatest integer), and j j (length), and predicate letters including =... |

2 | The polynomial time hierarchy and intuitionistic bounded arithmetic - Buss - 1986 |

2 |
Proofs, programs and run-times
- Crossley
- 1989
(Show Context)
Citation Context ...nd for polynomial-time realizability as defined by Crossley and Remmel. The sequent calculus admits cut elimination, thus giving a decision procedure for the propositional fragment. 0 Introduction In =-=[4]-=-, a restricted notion of realizability is introduced, a special case of which is polynomial-time realizability: this is like Kleene's original realizability, save for three features. First, closed ato... |

1 | Graded multicategories of polynomial-time realizers", pp. 182 -- 197
- Seely
- 1989
(Show Context)
Citation Context ...ere \Gamma = \Gamma 1 \Gamma 2 .) However, it is stronger than ordinary conjunction (which is, of course, symmetric): k\Gamma A; B \Gamma! AsB is valid (take C = AsB in the result quoted above). (See =-=[13] for some -=-details concerning categorical structure for a related system.) Third, we extend the usual notion of "sequent" by the addition of variable declarations. A variable declaration (x :) delimits... |