## Markov’s principle for propositional type theory (2001)

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Venue: | Computer Science Logic, Proceedings of the 10 th Annual Conference of the EACSL |

Citations: | 7 - 5 self |

### BibTeX

@INPROCEEDINGS{Kopylov01markov’sprinciple,

author = {Alexei Kopylov and Aleksey Nogin},

title = {Markov’s principle for propositional type theory},

booktitle = {Computer Science Logic, Proceedings of the 10 th Annual Conference of the EACSL},

year = {2001},

pages = {570--584},

publisher = {Springer-Verlag}

}

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### Abstract

Abstract. In this paper we show how to extend a constructive type theory with a principle that captures the spirit of Markov’s principle from constructive recursive mathematics. Markov’s principle is especially useful for proving termination of specific computations. Allowing a limited form of classical reasoning we get more powerful resulting system which remains constructive and valid in the standard constructive semantics of a type theory. We also show that this principle can be formulated and used in a propositional fragment of a type theory.

### Citations

277 |
Constructive mathematics and computer programming. Logic, Methodology and Philosophy of Sciences 6:153–175
- Martin-Löf
- 1982
(Show Context)
Citation Context ...especially interested in the constructive recursive mathematics (CRM) approach developed by Markov [12,13] and in constructive type theories (especially those that are based on Martin-Löf type theory =-=[14]-=-) since we believe them to be highly relevant to Computer Science. In this paper we demonstrate how to apply the ideas of CRM to a constructive type theory thus creating a more powerful type theory th... |

274 |
Foundations of Constructive Mathematics
- Beeson
- 1985
(Show Context)
Citation Context ...ov and Aleksey Nogin 1.2 Markov’s Constructivism Constructive mathematics is interesting in Computer Science because of program correctness issues. There are several approaches to constructivism (see =-=[4,5,19]-=- for an overview). We are especially interested in the constructive recursive mathematics (CRM) approach developed by Markov [12,13] and in constructive type theories (especially those that are based ... |

102 | Type Theory and Functional Programming
- Thompson
- 1991
(Show Context)
Citation Context ...: ¬A ⊢ t ∈ T Γ ⊢ A Type Γ ⊢ t ∈ T (2.1) This rule formalizes exactly the philosophy of the recursive constructivism. 2 MetaPRL system [9,10] uses the unit element () or “it” as a •, NuPRL uses Ax and =-=[18]-=- uses Triv.Markov’s Principle for Propositional Type Theory 5 Remark 2.1. Note that in NuPRL–like type theories t ∈ T is well-formed only when t is in fact an element of T . Therefore the rule statin... |

76 |
Implementing Mathematics with the Nuprl Development System
- Constable, Allen, et al.
- 1986
(Show Context)
Citation Context ...h operator for this purpose. This operator can creates a proposition stating that a certain type is non-empty without providing an inhabitant, i.e. squash “forgets” proofs. It was first introduced in =-=[6]-=- and also used in [15] and MetaPRL system [9,10]. Using squash it is possible to define the notion of squash–stability, which is similar to self–realizability. The squash operator can be considered as... |

61 | Propositional lax logic
- Fairtlough, Mendler
- 1997
(Show Context)
Citation Context ...Squash operator as modality The squash operator can be considered as an intuitionistic modality. It turns out that it behaves like the lax modality (denoted by ○) in the Propositional Lax Logic (PLL) =-=[8]-=-. This logic was developed independently for several different purposes (see [8] for an overview). PLL is the extension of intuitionistic logic with the following rules (in Gentzen style): Γ ⊢ A Γ ⊢ [... |

59 |
Varieties of constructive mathematics
- Bridges, Richman
- 1987
(Show Context)
Citation Context ...ov and Aleksey Nogin 1.2 Markov’s Constructivism Constructive mathematics is interesting in Computer Science because of program correctness issues. There are several approaches to constructivism (see =-=[4,5,19]-=- for an overview). We are especially interested in the constructive recursive mathematics (CRM) approach developed by Markov [12,13] and in constructive type theories (especially those that are based ... |

56 |
A Non-Type-Theoretic Semantics for Type-Theoretic Language
- Allen
- 1987
(Show Context)
Citation Context ...is is true because of the rule (4.2). 5 Semantical consistency of Markov’s principle Theorem 5.1. The rule (4.3) (as well as its equivalents — (2.1), (4.1) and (4.2)) is valid in S. Allen’s semantics =-=[1,2]-=- if we consider it in a classical meta-theory. Proof. We need to show that Γ ⊢ [A ∨ ¬A] is true when A is a type. It is clear that [A ∨ ¬A] is a well–formed type. To prove that it is a true propositio... |

50 |
Extensional concepts in intensional type theory
- Hofmann
- 1995
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Citation Context ...y similar to the squash–stability defined in [9, Section 14.2] and to the notion of computational redundancy [3, Section 3.4]. The squash operator we use is similar to the notion of proof irrelevance =-=[11,17]-=-. Each object in a proof irrelevance type is considered to be equal to any other object of this type. In [17] proof irrelevance was expressed in terms of a certain modality △. If A is a type then △A i... |

47 |
proof irrelevance in modal type theory
- Intensionality
- 2001
(Show Context)
Citation Context ...y similar to the squash–stability defined in [9, Section 14.2] and to the notion of computational redundancy [3, Section 3.4]. The squash operator we use is similar to the notion of proof irrelevance =-=[11,17]-=-. Each object in a proof irrelevance type is considered to be equal to any other object of this type. In [17] proof irrelevance was expressed in terms of a certain modality △. If A is a type then △A i... |

33 |
A Non-type-theoretic Definition of Martin-Löf’s Types
- Allen
- 1987
(Show Context)
Citation Context ...is is true because of the rule (4.2). 5 Semantical consistency of Markov’s principle Theorem 5.1. The rule (4.3) (as well as its equivalents — (2.1), (4.1) and (4.2)) is valid in S. Allen’s semantics =-=[1,2]-=- if we consider it in a classical meta-theory. Proof. We need to show that Γ ⊢ [A ∨ ¬A] is true when A is a type. It is clear that [A ∨ ¬A] is a well–formed type. To prove that it is a true propositio... |

33 |
The MetaPRL Logical Programming Environment
- Hickey
- 2001
(Show Context)
Citation Context ...n creates a proposition stating that a certain type is non-empty without providing an inhabitant, i.e. squash “forgets” proofs. It was first introduced in [6] and also used in [15] and MetaPRL system =-=[9,10]-=-. Using squash it is possible to define the notion of squash–stability, which is similar to self–realizability. The squash operator can be considered as a modality. The propositional logic equipped wi... |

31 | Doit-yourself type theory - Backhouse, Chisholm, et al. - 1989 |

20 |
Constructive modal logics I
- Wijesekera
- 1990
(Show Context)
Citation Context ...ve [B] ↔ ¬¬B in PLL ∗ , we can derive A in PLL ∗ . Remark 6.2. It is possible to consider the lax modality in PLL + as the diamond modality in the natural intuitionistic analog of S4 (in the style of =-=[20]-=-) with an additional rule □A ↔ A. Note that since in intuitionistic logics □ and ♦ are not interdefinable, □A ↔ A does not imply ♦A ↔ A.Markov’s Principle for Propositional Type Theory 11 Example 6.3... |

12 | Computational complexity and induction for partial computable in type theory
- Constable, Crary
- 2001
(Show Context)
Citation Context ...eta-theory, because there is no uniform witness for for A ∨ ¬A. Corollary 5.2. The rule (4.3) (and its equivalents) is consistent with the NuPRL type theory containing the theory of partial functions =-=[7]-=-. Note however that the rule of excluded middle Γ ⊢ A ∨ ¬A is known to be inconsistent with the theory of [7]. In particular, in that theory we can prove that there exists an undecidable proposition. ... |

6 | Quotient types: A modular approach
- Nogin
- 2002
(Show Context)
Citation Context ...urpose. This operator can creates a proposition stating that a certain type is non-empty without providing an inhabitant, i.e. squash “forgets” proofs. It was first introduced in [6] and also used in =-=[15]-=- and MetaPRL system [9,10]. Using squash it is possible to define the notion of squash–stability, which is similar to self–realizability. The squash operator can be considered as a modality. The propo... |

5 | A Non-type-theoretic De of Martin-Lof's Types - Allen - 1987 |

4 |
On the continuity of constructive functions
- Markov
- 1954
(Show Context)
Citation Context ...ctness issues. There are several approaches to constructivism (see [4,5,19] for an overview). We are especially interested in the constructive recursive mathematics (CRM) approach developed by Markov =-=[12,13]-=- and in constructive type theories (especially those that are based on Martin-Löf type theory [14]) since we believe them to be highly relevant to Computer Science. In this paper we demonstrate how to... |

3 |
On constructive mathematics, Trudy Matematicheskogo Instituta Imeni V
- Markov
- 1962
(Show Context)
Citation Context ...ctness issues. There are several approaches to constructivism (see [4,5,19] for an overview). We are especially interested in the constructive recursive mathematics (CRM) approach developed by Markov =-=[12,13]-=- and in constructive type theories (especially those that are based on Martin-Löf type theory [14]) since we believe them to be highly relevant to Computer Science. In this paper we demonstrate how to... |

2 |
Aleksey Nogin, et al. MetaPRL home
- Hickey
(Show Context)
Citation Context ...n creates a proposition stating that a certain type is non-empty without providing an inhabitant, i.e. squash “forgets” proofs. It was first introduced in [6] and also used in [15] and MetaPRL system =-=[9,10]-=-. Using squash it is possible to define the notion of squash–stability, which is similar to self–realizability. The squash operator can be considered as a modality. The propositional logic equipped wi... |

1 |
The Friedman translation for Martin-Lof's type theory
- Palmgren
- 1995
(Show Context)
Citation Context ...not imply △A. However it seems that modal logic of △ modality is the same as logic of squash (i.e. PLL + ). As far as we know Markov’s principle in type theory was considered only by Erik Palmgren in =-=[16]-=-. He proved that a fragment of intentional Martin-Löf type theory is closed under Markov’s rule: Γ ⊢ ¬¬∃x : A.P [x] Γ ⊢ ∃x : A.P [x] where P [x] is an equality type (i.e. P [x] is t[x] = s[x] ∈ T . It... |

1 |
Propositional laxlogic
- Fairtlough, Mendler
- 1997
(Show Context)
Citation Context ...opositional Type Theory 579 The squash operator can be considered as an intuitionistic modality. It turns out that it behaves like the lax modality (denoted by ○) in the Propositional Lax Logic (PLL) =-=[8]-=-. This logic was developed independently for several different purposes (see [8] for an overview). PLL is the extension of intuitionistic logic with the following rules (in Gentzen style): Γ ⊢ A Γ ⊢ [... |

1 |
Quotient types –amodular approach
- Nogin
- 2001
(Show Context)
Citation Context ...urpose. This operator can creates a proposition stating that a certain type is non-empty without providing an inhabitant, i.e. squash “forgets” proofs. It was first introduced in [6] and also used in =-=[15]-=- and MetaPRL system [9,10]. Using squash it is possible to define the notion of squash–stability, which is similar to self–realizability. The squash operator can be considered as a modality. The propo... |

1 | On the continuity of constructive function. Uspekhi Matematicheskikh Nauk - Markov - 1954 |