## A Categorical Model for Higher Order Imperative Programming (1993)

Venue: | Mathematical Structures in Computer Science |

Citations: | 13 - 12 self |

### BibTeX

@INPROCEEDINGS{Naumann93acategorical,

author = {David A. Naumann},

title = {A Categorical Model for Higher Order Imperative Programming},

booktitle = {Mathematical Structures in Computer Science},

year = {1993},

pages = {8--4}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper gives the first complete axiomatization for higher types in the refinement calculus of predicate transformers.

### Citations

461 | Computational lambda-calculus and monads - Moggi - 1989 |

108 | Parametricity and local variables - O’Hearn, Tennent - 1995 |

98 | Laws of programming
- Hoare, Hayes, et al.
- 1987
(Show Context)
Citation Context ...hism (theorems 8.5, 9.4). This categorical account can guide development of operators and laws for an axiomatic presentation, augmenting the laws of first order imperative programming (Morgan, 1994b; =-=Hoare et al., 1987-=-). With axiomatics in mind, most of our results are proved by inequational reasoning from the defining properties of the power adjunction, lifted exponent, etc. The remaining results, e.g., lemma 7.1 ... |

88 | Cartesian bicategories I - Carboni, Walters - 1987 |

74 | Accessible categories: the foundations of categorical model theory - Makkai, Paré - 1989 |

74 | Powerdomains and predicate transformers: a topological view - Smyth - 1983 |

73 |
Methods and Logics for Proving Programs
- Cousot
- 1990
(Show Context)
Citation Context ...tal correctness of imperative programs remain unsatisfactory for a number of reasons. (a) There are inherent limitations even for relative completeness of proof systems, even for partial correctness (=-=Cousot, 1990-=-). (b) There are semantic difficulties involving interference and interactions between local variables and higher order procedures. Fully abstract semantics has proved to be surprisingly elusive (O'He... |

47 | Programming from Specifications: Second Edition - Morgan - 1994 |

44 | Categories Allegories - Freyd, Scedrov - 1991 |

31 | O.: Categories, relations and dynamic programming - Moor - 1992 |

30 |
Domain theory in logical form, Annals of Pure and Applied Logic 51
- Abramsky
- 1991
(Show Context)
Citation Context ...asible programs, the semantics of specifications involves technical complications such as predicates that are not Scott-open sets. (It is perspicuous to view Scott-open sets as observable predicates (=-=Abramsky, 1991-=-), but an accurate theory of effectiveness requires more restricted predicates, and specifications require less restricted ones.) Methodologists have prefered models like binary relations or predicate... |

29 |
Handbook of Categorical Algebra. Basic Category Theory
- Borceux
- 1994
(Show Context)
Citation Context ...hroughout the paper. To make the paper more accessible, some definitions are less general than usual (Street, A Categorical Model for Higher Order Imperative Programming 9 1980; Makkai & Par'e, 1989; =-=Borceux, 1994-=-). When patience wears thin, you may wish to to skip to x3.2 and use the index as needed. The application of function g to argument a is denoted by ga, even in cases like Gra where the function has a ... |

22 | Predicate transformer semantics of a higher order imperative language with record subtyping
- Naumann
- 2001
(Show Context)
Citation Context ...rictions imposed on global variables for reasons of efficiency. The location-free predicate transformer semantics of local variables is nicely suited to such languages (Morgan, 1994b; Naumann, 1994a; =-=Naumann, 1996-=-). Turning to (c) and (d), there is no shortage of alternative models for first order imperative programs. Operational semantics is needed for low-level correctness proofs (e.g., for compilers). Domai... |

21 |
Predicate transformer semantics of an Oberon-like language
- Naumann
- 1994
(Show Context)
Citation Context ...n thanks to restrictions imposed on global variables for reasons of efficiency. The location-free predicate transformer semantics of local variables is nicely suited to such languages (Morgan, 1994b; =-=Naumann, 1994-=-a; Naumann, 1996). Turning to (c) and (d), there is no shortage of alternative models for first order imperative programs. Operational semantics is needed for low-level correctness proofs (e.g., for c... |

21 | A model for syntactic control of interference
- O’Hearn
- 1993
(Show Context)
Citation Context ...he fixed rules of a calculus; mature engineering disciplines freely mix formal calculations with reasoning about a standard model. As for (b), I prefer to deal with interference-controlled languages (=-=O'Hearn, 1993-=-), at least until (c) and (d) are addressed adequately. Also, much of the difficulty is absent in languages like C and Oberon thanks to restrictions imposed on global variables for reasons of efficien... |

20 | O.: An algebraic construction of predicate transformers - Gardiner, Martin, et al. - 1994 |

19 |
Preordered Categories and Predicate Transformers
- Martin
- 1991
(Show Context)
Citation Context ...ducts and lax coproducts in Tran, which are quite similar to those in ITran. The lax coexponent in Tran is discussed in order to motivate the move to ITran. Except where noted, the results appear in (=-=Martin, 1991; Naumann,-=- 1995b). Proposition 4.4. Each coproduct A + B of sets is a categorical product in Tran, equipped with the Stone duals ['] of the injection functions ' as "projections". The associated funct... |

15 | Full abstraction for the second order subset of an Algol-like language
- Sieber, Aug
- 1994
(Show Context)
Citation Context ...ntic difficulties involving interference and interactions between local variables and higher order procedures. Fully abstract semantics has proved to be surprisingly elusive (O'Hearn & Tennant, 1995; =-=Sieber, 1995-=-). (c) Domain-theoretic semantics of imperative programs is given using secondary categorical structure (?-lifting, powerdomains) rather than being intrinsic to the semantic category. This complicates... |

10 |
The Cuppest Capjunctive Capping, and Galois
- Morgan
- 1994
(Show Context)
Citation Context ...ike C and Oberon thanks to restrictions imposed on global variables for reasons of efficiency. The location-free predicate transformer semantics of local variables is nicely suited to such languages (=-=Morgan, 1994-=-b; Naumann, 1994a; Naumann, 1996). Turning to (c) and (d), there is no shortage of alternative models for first order imperative programs. Operational semantics is needed for low-level correctness pro... |

9 |
Predicate transformers and higher order programs
- Naumann
- 1995
(Show Context)
Citation Context ...that the full homset is not appropriate as a data type: Programs are strict, positively conjunctive, and continuous, so those morphisms are the only ones that would be stored or passed as parameters (=-=Naumann, 1995b). Taking-=- for A/B just those "healthy" morphisms makes coap healthy, but it does not give an adjunction in the subprocat of such transformers, and the problem with id / B remains. Moreover, for purpo... |

8 | Pre-adjunctions in order enriched categories - Martin, Hoare, et al. - 1991 |

6 | Algebraic proofs of consistency and completeness - Gardiner - 1995 |

4 | Isomorphisms between predicate and state transformers - Bonsangue, Kok - 1993 |

4 | A recursion theorem for predicate transformers on inductive data types - Naumann - 1994 |

4 | Data refinement, call by value, and higher order programs. unpublished draft
- Naumann
- 1995
(Show Context)
Citation Context ...gory, not of the span category. For transformer semantics of higher order commands in refinement calculus the full homset is the requisite coexponent (Naumann, 1994a) but it satisfies very weak laws (=-=Naumann, 1995-=-b). My contribution here is to use skew spans starting from posets instead of sets. The resulting category ITran is comprised of transformers between lattices of upward closed sets. Because predicates... |

3 | Order-enrichment for categories of partial maps
- Fiore
- 1995
(Show Context)
Citation Context ... contrast, some notions of "categories of partial maps" use only equations, but many equations are needed to replace the usual ones (other categorical treatments of partiality make more use =-=of order (Fiore, 1995-=-)). Inequational weakenings of categorical notions are used by He and Hoare (1989) to classify language features and phenomena. There has been extensive work on calculi of transformers for programs in... |

2 | Order ideals in categories. Pacific journal of mathematics - Carboni - 1986 |

1 | Relational catamorphisms. Pages 287--318 of: IFIP TC2/WG2.1 working conference on constructing programs - Backhouse, Bruin, et al. - 1991 |

1 | A Categorical Model for Higher Order Imperative Programming 49 - Carboni, Kelly - 1991 |

1 | Inductive data types for predicate transformers. Information processing letters - Moor - 1992 |

1 | Categorical semantics for programming languages - He - 1989 |