## The Logic of the Partial λ-Calculus With Equality

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@MISC{Schröder_thelogic,

author = {Lutz Schröder},

title = {The Logic of the Partial λ-Calculus With Equality},

year = {}

}

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### Abstract

We investigate the logical aspects of the partial λ-calculus with equality, exploiting an equivalence between partial λ-theories and partial cartesian closed categories (pcccs) established here. The partial λ-calculus with equality provides a full-blown intuitionistic higher order logic, which in a precise sense turns out to be almost the logic of toposes, the distinctive feature of the latter being unique choice. We give a linguistic proof of the generalization of the fundamental theorem of toposes to pcccs with equality; type theoretically, one thus obtains that the partial λ-calculus with equality encompasses a Martin-Löf-style dependent type theory. This work forms part of the semantical foundations for the higher order algebraic specification language HasCasl.

### Citations

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(Show Context)
Citation Context ...ategories and quasitoposes. In terms of logic, locally cartesian closed categories correspond to MartinLöf style dependent type theory [27], and toposes to intuitionistic type theory with power types =-=[11]-=- (the precise logical counterpart of quasitoposes is, to our knowledge, open). In particular, this means that the partial λ-calculus with equality encodes a dependent type theory; more precisely, one ... |

149 |
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(Show Context)
Citation Context ...theory; categorically, this means that every topos E is locally cartesian closed [9, 27], i.e. every slice E/A is cartesian closed — this is the non-trivial part of the fundamental theorem of toposes =-=[10]-=-. It is known that this theorem holds also for quasitoposes [16, 28], and a proof that the statement generalizes to pcccs with equality can be extracted from [28]. This implies that the partial λ-calc... |

56 |
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Citation Context ...7-1)s2 pcccs with equality fit between locally cartesian closed categories and quasitoposes. In terms of logic, locally cartesian closed categories correspond to MartinLöf style dependent type theory =-=[27]-=-, and toposes to intuitionistic type theory with power types [11] (the precise logical counterpart of quasitoposes is, to our knowledge, open). In particular, this means that the partial λ-calculus wi... |

55 |
Categories of partial maps
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Citation Context ...phism and mg ∈ M, then g ∈ M. In particular, M contains all isomorphisms. For a dominional category (C, M), the partial morphisms form a category P(C, M), which contains C as a (non-full) subcategory =-=[17, 18]-=-. As usual, we call a category (functor, subcategory) cartesian if it has (preserves, is closed under) finite products; the terminal object is denoted by 1. In a cartesian dominional category (C, M), ... |

49 |
A Model Theoretic Oriented Approach to Partial Algebras (Introduction to Theory and Application of Partial Algebras
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(Show Context)
Citation Context ...ions of the form α e = α just state that α is defined; they are abbreviated as def α (and e.g. def(α, β) codes the same set of equations as def α ∧ def β). An existentially conditioned equation (ece) =-=[5]-=- in context Γ is a sentence of the form Γ ⊲ def ¯α ⇒ ψ, where ¯α is a multi-term and ψ is an existential equation in context Γ . The axioms of a partial λ-theory are given as eces. The deduction syste... |

43 |
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Citation Context ...ise by exponentiation in the slice category. The point here is that local cartesian closedness is equivalent to the existence of right adjoints Πf for all pullback functors f ∗ : C/A → C/B, f : B → A =-=[7]-=-. Intuitively, for types B(x), D(x) depending on x : A, i.e. bundles f : B → A, h : D → A, the fibre over x : A of the function space f → h in C/A is the function space B(x) → D(x). For g : C → B and ... |

39 | On the interpretation of type theory in locally cartesian closed categories
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Citation Context ...ms’ (functions).s6 Dependent Types An important aspect of toposes is that they admit a Martin-Löf style dependent type theory; categorically, this means that every topos E is locally cartesian closed =-=[9, 27]-=-, i.e. every slice E/A is cartesian closed — this is the non-trivial part of the fundamental theorem of toposes [10]. It is known that this theorem holds also for quasitoposes [16, 28], and a proof th... |

36 | The Requirement and Design Specification Language Spectrum: an Informal Introduction
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Citation Context ...lay an important role in modern algebraic specification, serving to model both non-termination and irregular termination; specification languages featuring partial functions include RSL [8], SPECTRUM =-=[3]-=-, and Casl [2, 15]. The natural generalization of the simply typed λ-calculus to partial functions is the partial λ-calculus [13, 14, 18], which forms the basis for the recently introduced wide-spectr... |

32 |
Continuous lattices, Toposes, Algebraic Geometry and Logic
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(Show Context)
Citation Context .... In particular, every quasi-topos is a pccc with equality (but not conversely [1]). A typical example of a pccc without equality is the category of cpos and continuous functions with Scott open sets =-=[26]-=- as admissible subobjects.sDefinition 5. A cartesian dominional functor between two pcccs is called partial cartesian closed (pcc) if it preserves partial function spaces. This defines the category PC... |

24 |
Abstract and concrete categories, Wiley Interscience
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Citation Context ... (A −→◦ B) called its abstraction. For a pccc (C, M), the embedding C ↩→ P(C, M) is left adjoint, being isomorphic to ×1. Spelling this out yields that M-partial morphisms in (C, M) are representable =-=[1, 18]-=-, with the partial morphisms into A represented by 1 −→◦ A. In particular, Ω = 1 −→◦ 1 classifies M-subobjects. By consequence, every map in M is a regular monomorphism. Of particular interest is the ... |

15 | The RAISE Specification Language
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(Show Context)
Citation Context ...al functions play an important role in modern algebraic specification, serving to model both non-termination and irregular termination; specification languages featuring partial functions include RSL =-=[8]-=-, SPECTRUM [3], and Casl [2, 15]. The natural generalization of the simply typed λ-calculus to partial functions is the partial λ-calculus [13, 14, 18], which forms the basis for the recently introduc... |

14 |
Casl user manual
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(Show Context)
Citation Context ...nt role in modern algebraic specification, serving to model both non-termination and irregular termination; specification languages featuring partial functions include RSL [8], SPECTRUM [3], and Casl =-=[2, 15]-=-. The natural generalization of the simply typed λ-calculus to partial functions is the partial λ-calculus [13, 14, 18], which forms the basis for the recently introduced wide-spectrum language HasCas... |

12 | Monad-independent dynamic logic in HasCasl, J. Logic Comput., to appear. Earlier version
- Schröder, Mossakowski
- 2003
(Show Context)
Citation Context ...rs a setting for both specification and implementation of higher order functional programs; moreover, it has served as a background formalism for the development of monad-generic computational logics =-=[22, 24]-=-. A central role in all this is played by the fact that the partial λ-calculus with equality induces a full intuitionistic higher order logic, corresponding to HasCasl’s internal logic [23]. Here, we ... |

10 |
Partiality, cartesian closedness and toposes
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(Show Context)
Citation Context ...-theory has equality if there exists, for each type s, a (defined) closed term eqs : ss −→◦ 1 (i.e. a binary predicate on s) such that x, y : s ⊲ eqs (x, y) ⇒ x e = y and x : s ⊲ eqs (x, x) (See also =-=[6, 13]-=-.) Note that the axioms for eqs are eces. When there is no danger of confusion, we shall write α e = β in place of eq (α, β). Equality gives rise to a full-fledged intuitionistic logic, along much the... |

8 | HasCasl – Integrated functional specification and programming. Language summary, available at http://www. informatik.uni-bremen.de/agbkb/forschung/formal_methods/CoFI/HasCASL
- Schröder, Mossakowski, et al.
(Show Context)
Citation Context ...The natural generalization of the simply typed λ-calculus to partial functions is the partial λ-calculus [13, 14, 18], which forms the basis for the recently introduced wide-spectrum language HasCasl =-=[23, 25]-=-. HasCasl offers a setting for both specification and implementation of higher order functional programs; moreover, it has served as a background formalism for the development of monad-generic computa... |

6 |
Boolean topoi and the theory of sets
- Mitchell
- 1972
(Show Context)
Citation Context ...ntradict the mentioned conservativity result. However, this is resolved by noting that the type theory of [11] (like most other versions of topos logic including the original MitchellBénabou language =-=[12]-=-) in fact cannot express unique choice, since it does not have actual function types. In other words, the logic of pcccs with internal equality differs from topos logic in that it takes functions rath... |

6 |
Continuity and effectiveness
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(Show Context)
Citation Context ...ication languages featuring partial functions include RSL [8], SPECTRUM [3], and Casl [2, 15]. The natural generalization of the simply typed λ-calculus to partial functions is the partial λ-calculus =-=[13, 14, 18]-=-, which forms the basis for the recently introduced wide-spectrum language HasCasl [23, 25]. HasCasl offers a setting for both specification and implementation of higher order functional programs; mor... |

6 | The HasCasl prologue: categorical syntax and semantics of the partial λ-calculus, available as http://www.informatik.uni-bremen.de/~lschrode/ hascasl/plam.ps
- Schröder
(Show Context)
Citation Context ... no longer possible to define partial function spaces (Γ. φ) −→◦ (∆. ψ) as subspaces of Γ −→◦ ∆. (The obvious idea of using the Yoneda extension is probably not the right one, for reasons laid out in =-=[19]-=-.) In [19], this problem is solved by moving to an extended theory with equality and a dominance [18]; the classifying category of the original theory is then obtained as a subcategory of the classify... |

5 |
Categories of partial morphisms and the λp-calculus, Category Theory and
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- 1986
(Show Context)
Citation Context ...ication languages featuring partial functions include RSL [8], SPECTRUM [3], and Casl [2, 15]. The natural generalization of the simply typed λ-calculus to partial functions is the partial λ-calculus =-=[13, 14, 18]-=-, which forms the basis for the recently introduced wide-spectrum language HasCasl [23, 25]. HasCasl offers a setting for both specification and implementation of higher order functional programs; mor... |

1 |
Equational lifting monads, Category Theory and
- Bucalo, Führmann, et al.
- 1999
(Show Context)
Citation Context ...category (C, M) or, in the higher order case equivalently, of Kleisli categories arising from representations of partial morphisms, i.e. from the adjunction between (C, M) and P(C, M) for some (C, M) =-=[4, 6, 17]-=-. In these approaches, categories of the form P(C, M) are typically distinguished by a splitting condition for subfunctions of the identity which ensures that domains of partial functions are actually... |

1 |
models of the partial λ-calculus
- Henkin
- 2003
(Show Context)
Citation Context ...tion for subfunctions of the identity which ensures that domains of partial functions are actually objects. For the purposes of this paper, as well as the (logically posterior) paper on Henkin models =-=[21]-=-, it appears to be more convenient to work directly with the underlying dominional categories. In particular, this makes the relation of pcccs with toposes and locally cartesian closed categories more... |

1 |
Towards integrated specification and development of functional programs, Algebraic Methodology
- HasCasl
- 2002
(Show Context)
Citation Context ...The natural generalization of the simply typed λ-calculus to partial functions is the partial λ-calculus [13, 14, 18], which forms the basis for the recently introduced wide-spectrum language HasCasl =-=[23, 25]-=-. HasCasl offers a setting for both specification and implementation of higher order functional programs; moreover, it has served as a background formalism for the development of monad-generic computa... |

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1 | Locally cartesian closed categories and type theory - Springer - 1972 |