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## Notions of Computation and Monads (1991)

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Citations: | 867 - 15 self |

### Citations

688 |
Categories for the Working Mathematician.
- MacLane
- 1971
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Citation Context ... the literature on Category Theory and have the advantage of being dened only in terms of funtors and natural transformations, which make them more suitable for abstract manipulation. Denition 1.5 ([Mac71]) A monad over a category C is a triple (T ; ; ), where T : C ! C is a functor, : Id C : ! T and : T 2 : ! T are natural transformations and the following diagrams commute: T 3 A TA > T 2 A TA ... |

574 | Basic Concepts of Enriched Category Theory, London Math. Soc. Lecture Note (Cambridge Uni.
- Kelly
- 1982
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Citation Context ...sms r A : (1 A) ! A ; A;B;C : (A B) C ! A (B C) Remark 3.3 The diagrams above are taken from [Koc72], where a characterisation of strong monads is given in terms of C-enriched categories (see [Kel82=-=]-=-). Kocksxes a commutative monoidal closed category C (in particular a cartesian closed category), and in this setup he establishes a one-one correspondence between strengths st A;B : B A ! (TB) TA and... |

501 | Computational lambdacalculus and monads.
- Moggi
- 1989
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Citation Context ...e that every pccc can be viewed as a c -model. By analogy with p-exponentials, a T -exponential can be dened by giving an isomorphism C T (C A; B) = C(C; (TB) A ) natural in C 2 C. We refer to [Mog=-=89-=-c] for the interpretation of a call-by-value programming language in a c -model and the corresponding formal system, the c -calculus. 4 Strong monads over a topos In this section we show that, as fa... |

383 | Denotational Semantics: A Methodology for Language Development. Allyn and - Schmidt - 1986 |

251 | Toposes, Triples and Theories
- Barr, Wells
- 1985
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Citation Context ...tive extension of PL (or c PL). Lemma 4.5 If (T ; ; ) is a monad over a topos C satisfying the mono requirement, then it satises also the equalising requirement. Proof See Lemma 6 on page 110 of [BW85]. In other words, for any type the axiom (eqls.) is derivable in HML T from the set of axioms f(mono.)j type of HML T g. In general, when C is not a topos, the mono requirement does not entail t... |

244 |
Call-By-Value, and the λ-Calculus.
- Plotkin
- 1975
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Citation Context ...semantics, e.g. a partial function mapping every program (i.e. closed term) to its resulting value (if any), which induces a congruence relation on open terms called operational equivalence (see e.g. =-=[Plo75]-=-). Then the problem is to prove that two terms are operationally equivalent. • The denotational approach gives an interpretation of the (programming) language in a mathematical structure, the intended... |

229 |
Call-by-name, call-by-value and the -calculus
- Plotkin
- 1975
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Citation Context ...semantics, e.g. a partial function mapping every program (i.e. closed term) to its resulting value (if any), which induces a congruence relation on open terms called operational equivalence (see e.g. =-=[Plo75-=-]). Then the problem is to prove that two terms are operationally equivalent. The denotational approach gives an interpretation of the (programming) language in a mathematical structure, the intended... |

199 |
The formal theory of monads,
- Street
- 1972
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Citation Context ...(F; T op ; ^ TF )s ^ D( ^ TF; ^ TF ) _ ^ C(F; op ; ^ TF ) ^ C(F; S op ; ^ TF ) > ^ D( ^ SF; ^ TF ) Moreover, Y: : ! ^ is a 2-natural transformation. Since monads are a 2-categorical concept (see [St=-=r72-=-]), the 2-functor ^ maps monads in Cat to monads in CAT. Then, the statement of Theorem 4.9 about lifting of monads follows immediately from Proposition 4.11. It remains to dene the lifting ^ t of a t... |

196 |
The category-theoretic solution of recursive domain equations”.
- Smyth, Plotkin
- 1982
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Citation Context ...ory at the heart of Denotational Semantics, i.e. Domain Theory (see [GS89, Mos89]), has focused on the mathematical structures for giving semantics to recursive denitions of types and functions (see [=-=SP82]-=-), while other structures, that might be relevant to a better understanding of programming languages, have been overlooked. This paper identify one of such structures, i.e. monads , but probably there... |

180 |
An abstract view of programming languages
- Moggi
- 1989
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Citation Context ...9b] we give a categorical account of phase distinction and program modules, that could lead to the introduction of higher order modules in programming languages like ADA or ML (see [HMM90]), while in =-=[Mog89a] we p-=-ropose a \modular approach" to Denotational Semantics based on the idea of monad-constructor (i.e. an endofunctor on the category of monads over a category C). The metalanguage open also the poss... |

178 |
Introduction to Higher-Order Categorical Logic.
- Lambek, Scott
- 1988
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Citation Context ...A ` x: A f x: A ` e 1 : A 1 x: A ` f(e 1 ): A 2 f: A 1 ! A 2 eq x: A 1 ` e 1 : A 2 x: A 1 ` e 2 : A 2 x: A 1 ` e 1 =A2 e 2 2 The uniqueness of x s.t. e = [x] T follows from the mono requirement. 3 In =-=[LS86]-=- a stronger relation is sought between theories and categories with additional structure, namely an equivalence between the category of theories and translations and the category of small categories w... |

166 | Denotational Semantics - Mosses - 1990 |

165 | Semantic Domains, - Gunter, Scott - 1990 |

136 | Higher-order modules and the phase distinction.
- Harper, Mitchell, et al.
- 1990
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Citation Context ...e. Indeed, in [Mog89b] we give a categorical account of phase distinction and program modules, that could lead to the introduction of higher order modules in programming languages like ADA or ML (see =-=[HMM90]), wh-=-ile in [Mog89a] we propose a \modular approach" to Denotational Semantics based on the idea of monad-constructor (i.e. an endofunctor on the category of monads over a category C). The metalanguag... |

128 | Linear logic, *-autonomous categories, and cofree coalgebras, - Seely - 1989 |

92 | The linear abstract machine. - Lafont - 1988 |

71 | Edinburgh LCF: A Mechanized Logic - Gordon, Milner, et al. - 1979 |

65 | Programming with continuations. - Friedman, Haynes, et al. - 1984 |

62 |
Strong functors and monoidal monads.
- Kock
- 1972
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Citation Context ...B @ @ @ A T 2 B t A;TB > T (A TB) T t A;B > T 2 (A B) where r and are the natural isomorphisms r A : (1 A) ! A ; A;B;C : (A B) C ! A (B C) Remark 3.3 The diagrams above are taken from [Koc72], where a characterisation of strong monads is given in terms of C-enriched categories (see [Kel82]). Kocksxes a commutative monoidal closed category C (in particular a cartesian closed category), and... |

58 |
First order categorical logic
- Makkai, Reyes
- 1977
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Citation Context ...air hc; fi to c) and Colim A I : A I ! A (with I small category) is a functor mapping an I-diagram in A to its colimit. The following proposition is a 2-categorical reformulation of Theorem 1.3.10 of =-=[MR77]-=-. For the sake of simplicity, we use the strict notions of 2-functor and 2-natural transformation, although we should have used pseudo-functors and pseudo-natural transformations. Proposition 4.11 Let... |

55 | Identity and existence in intuitionistic logic, - Scott - 1979 |

52 |
Continuity and Effectiveness in Topoi.
- Rosolini
- 1986
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Citation Context ... the restriction of the functor T to objects and f ∗ = (T f); µB for f: A → T B. Remark 1.7 In general the categorical semantics of partial maps, based on a category C equipped with a dominion M (see =-=[Ros86]-=-), cannot be reformulated in terms of a Kleisli triple over C satisfying some additional properties, unless C has lifting, i.e. the inclusion functor from C into the category of partial maps P(C, M) h... |

49 |
The theory of constructions: categorical semantics and topostheoretic models. In: Categories in computer science and logic
- Hyland, Pitts
- 1989
(Show Context)
Citation Context .... Then a strong monad over a cartesian closed category C is just a monad over C in the 2-category of C-enriched categories. The second characterisation takes a class D of display maps over C (used in =-=[HP87-=-] to model dependent types), and denes a C-indexed category C=D . Then a strong monad over a category C withsnite products amounts to a monad over C=D in the 2-category of C-indexed categories, where ... |

45 |
Ultraproducts and categorical logic.
- Makkai
- 1985
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Citation Context ...tegory F(T ) with additional structure such that there is a one-one correspondence between models of T in a category C with additional structure and structure preserving functors from F(T ) to C (see =-=[KR77-=-]) 3 . This identication was originally proposed by Lawvere, who also showed that algebraic theories can be viewed as categories withsnite products. In Section 2.2 we give a class of theories that can... |

45 |
A type-theoretic alternative to
- Scott
- 1993
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Citation Context ...ased on the idea of monad-constructor (i.e. an endofunctor on the category of monads over a category C). The metalanguage open also the possibility to develop a new Logic of Computable Functions (see =-=[Sco69-=-]), based on an abstract semantic of computations rather than domain theory, for studying axiomatically dierent notions of computation and their relations. Some recent work by Crole and Pitts (see [CP... |

40 | A syntactic theory of sequential state - Felleisen, Friedman - 1989 |

36 | S.F.: Partial objects in constructive type theory. In: - Constable, Smith - 1987 |

34 | The partial lambda-calculus, - Moggi - 1988 |

32 |
Denotational semantics with partial functions
- Plotkin
- 1985
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Citation Context ...ations presented in this paper has been strongly in uenced by the reformulation of Denotational Semantics based on the category of cpos, possibly without bottom, and partial continuous functions (see =-=[Plo85]-=-) and the work on categories of partial morphisms in [Ros86, Mog86]. Our work generalises the categorical account of partiality to other notions of computations, indeed partial cartesian closed catego... |

30 | Programming, transforming, and proving with function abstractions and memories. - Mason, Talcott - 1989 |

26 |
Algebraic Theories. Volume 26 of Graduate Texts in Mathematics
- Manes
- 1976
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Citation Context ...ormal argumentations, that such a requirement amounts to say that T is part of a Kleisli triple (T ; ; ) and that the category of programs is the Kleisli category for such a triple. Denition 1.2 ([Man76]) A Kleisli triple over a category C is a triple (T ; ; ), where T : Obj(C) ! Obj(C), A : A ! TA for A 2 Obj(C), f : TA ! TB for f : A ! TB and the following equations hold: A = id TA A... |

25 | A category-theoretic account of program modules.” Category theory and computer science
- Moggi
- 1989
(Show Context)
Citation Context ...G. Plotkin, in terms of C-indexed categories (see [JP78]). Both characterisations are instances of a general methodological principle for studying programming languages (or logics) categorically (see =-=[Mog89b-=-]): when studying a complex language the 2-category Cat of small categories, functors and natural transformations may not be adequate; however, one may replace Cat with a dierent 2-category, whose obj... |

22 |
Relating theories of the λ-calculus
- Scott
- 1980
(Show Context)
Citation Context ...e λ-calculus. 4. We show that w.l.o.g. one may consider only (monads over) toposes, and we exploit this fact to establish conservative extension results. The methodology outlined above is inspired by =-=[Sco80]-=- 1 , and it is followed in [Ros86, Mog86] to obtain the λp-calculus. The view that “category theory comes, logically, before the λ-calculus”led us to consider a categorical semantics of computations f... |

21 | Computational foundations of basic recursive function theory - Constable, Smith - 1988 |

21 | The logic of topoi - Fourman - 1977 |

20 |
Relating theories of the � -calculus
- Scott
- 1980
(Show Context)
Citation Context ...e -calculus. 4. We show that w.l.o.g. one may consider only (monads over) toposes, and we exploit this fact to establish conservative extension results. The methodology outlined above is inspired by [=-=Sco80] 1 -=-, and it is followed in [Ros86, Mog86] to obtain the p -calculus. The view that \category theory comes, logically, before the -calculus"led us to consider a categorical semantics of computations... |

20 | Algebraic Theories, Graduate Texts - Manes - 1976 |

16 |
Continuity and E#ectivity in Topoi
- Rosolini
- 1986
(Show Context)
Citation Context ... the restriction of the functor T to objects and f = (Tf ); B for f : A ! TB. Remark 1.7 In general the categorical semantics of partial maps, based on a category C equipped with a dominion M (see [R=-=os86]-=-), cannot be reformulated in terms of a Kleisli triple over C satisfying some additional properties, unless C has lifting , i.e. the inclusion functor from C into the category of partial maps P(C; M) ... |

13 | New foundations for fixpoint computations
- Crole, Pitts
- 1990
(Show Context)
Citation Context ...9]), based on an abstract semantic of computations rather than domain theory, for studying axiomatically different notions of computation and their relations. Some recent work by Crole and Pitts (see =-=[CP90]-=-) has consider an extension of the metalanguage equipped with a logic for inductive predicates, which goes beyond equational reasoning. A more ambitious goal would be to try exploiting the capabilitie... |

12 | Verification of programs that destructively manipulate data - Mason - 1988 |

11 | 1989a] A sound and complete axiomatization of operational equivalence between programs with memory, Fourth annual symposium on logic in computer science - Mason, Talcott |

10 | Categories of partial morphisms and the partial lambda-calculus - MOGGI - 1986 |

7 | Strong Functors and Monoidal - Kock - 1972 |

6 |
Syntactic Aspects of the Non-deterministic Lambda Calculus
- Sharma
- 1984
(Show Context)
Citation Context ...o75] for call-by-value and call-by-name operational equivalence. This approach was later extended, following a similar methodology, to consider other features of computations like nondeterminism (see =-=[Sha8-=-4]), side-eects and continuations (see [FFKD86, FF89]). The calculi based only on operational considerations, like the v -calculus, are sound and complete w.r.t. the operational semantics, i.e. a pro... |

6 | Toposes, triples and theories, Springer-Verlag (1985), republished in - Wells |

6 | Identity and existence in intuitionistic logic, Applications of sheaves, - Scott - 1977 |

3 |
Indexed categories and their applications
- Johnstone, Paré
- 1978
(Show Context)
Citation Context ...re enriched in the obvious way (see Remark 1.4 in [Koc72]). There is another purely categorical characterisation of strong monads, suggested to us by G. Plotkin, in terms of C-indexed categories (see =-=[JP78]-=-). Both characterisations are instances of a general methodological principle for studying programming languages (or logics) categorically (see [Mog89b]): when studying a complex language the 2-catego... |

2 |
New foundations for computations
- Crole, Pitts
- 1990
(Show Context)
Citation Context ...69]), based on an abstract semantic of computations rather than domain theory, for studying axiomatically dierent notions of computation and their relations. Some recent work by Crole and Pitts (see [=-=CP90]-=-) has consider an extension of the metalanguage equipped with a logic for inductive predicates, which goes beyond equational reasoning. A more ambitious goal would be to try exploiting the capabilitie... |

1 | New foundations for tixpoint computations - CROLE, PITTS - 1990 |

1 | Higher-order modules and the phase distinction - J, MOGGI - 1990 |

1 | Basic Concepts of Enriched Category Theory,” Cambridge Univ - M - 1982 |

1 | The linear abstract machine, Theoret - unknown authors - 1988 |

1 | A category-theoretic account of program modules - MOGGt - 1989 |

1 | First Order Categorial Logic - MAKKAI, REYES - 1977 |

1 | Call-by-name, call-by-value and the I-calculus, Theoret - PLOTKIN - 1975 |

1 | Recating theories of the i-calculus - SCOTT - 1980 |