## Languages, Theory

### BibTeX

@MISC{Ghani_languages,theory,

author = {Neil Ghani and Tarmo Uustalu},

title = {Languages, Theory},

year = {}

}

### OpenURL

### Abstract

Recently there has been a great deal of interest in higherorder syntax which seeks to extend standard initial algebra semantics to cover languages with variable binding by using functor categories. The canonical example studied in the literature is that of the untyped λ-calculus which is handled as an instance of the general theory of binding algebras, cf. Fiore, Plotkin, Turi [8]. Another important syntactic construction is that of explicit substitutions. The syntax of a language with explicit substitutions does not form a binding algebra as an explicit substitution may bind an arbitrary number of variables. Nevertheless we show that the language given by a standard signature Σ and explicit substitutions is naturally modelled as the initial algebra of the endofunctor Id + FΣ ◦ + ◦ on a functor category. We also comment on the apparent lack of modularity in syntax with variable binding as compared to first-order languages. Categories and Subject Descriptors

### Citations

225 |
Locally presentable and accessible categories
- Adámek, Rosický
- 1994
(Show Context)
Citation Context ... categories and make passing use of coends and Kan extensions. We give a short summary of the important results concerning these concepts while referring the reader to the literature for more details =-=[17, 4]-=-. Alternatively, we provide significant intuition for all our constructions to allow the reader to follow the thrust of our argument for the category Set. Locally Presentable Categories One important ... |

100 | Semantical analysis of higher-order abstract syntax
- Hofmann
- 1999
(Show Context)
Citation Context ...fp categories [2, 20, 3, 10]. 4. UNTYPED λ-CALCULUS Can we extend the above analysis to languages with variable binding? Many proposals have been made, but one of the best is that of [8, 9] (cf. also =-=[12, 5]-=-) whose elegance relies in the fact that one uses exactly the same general program as for first-order languages but instantiated in functor categories. For example, if LX is the set of untyped λ-terms... |

81 | Monadic presentations of lambda-terms using generalized inductive types
- Altenkirch, Reus
- 1999
(Show Context)
Citation Context ...fp categories [2, 20, 3, 10]. 4. UNTYPED λ-CALCULUS Can we extend the above analysis to languages with variable binding? Many proposals have been made, but one of the best is that of [8, 9] (cf. also =-=[12, 5]-=-) whose elegance relies in the fact that one uses exactly the same general program as for first-order languages but instantiated in functor categories. For example, if LX is the set of untyped λ-terms... |

53 |
Infinite Trees and Completely Iterative Theories: A Coalgebraic View, Theoret
- Aczel, Adámek, et al.
(Show Context)
Citation Context ...rger arity can be handled by moving to locally κ-presentable categories [11]. Non-wellfounded syntax can be treated by replacing various initial algebra constructions by final coalgebra constructions =-=[21, 1, 11]-=-. Other forms of syntax, e.g., Σ-term graphs and Σ-rational terms can also be generalised to universal constructions involving FΣ-algebras on lfp categories [2, 20, 3, 10]. 4. UNTYPED λ-CALCULUS Can w... |

48 | Semantics of name and value passing
- Fiore, Turi
- 2001
(Show Context)
Citation Context ... FΣ-algebras on lfp categories [2, 20, 3, 10]. 4. UNTYPED λ-CALCULUS Can we extend the above analysis to languages with variable binding? Many proposals have been made, but one of the best is that of =-=[8, 9]-=- (cf. also [12, 5]) whose elegance relies in the fact that one uses exactly the same general program as for first-order languages but instantiated in functor categories. For example, if LX is the set ... |

24 | Composing Monads Using Coproducts
- Lüth, Ghani
- 2002
(Show Context)
Citation Context ...erms over X and substitution (in fact, one only needs variables, depth-1 F -terms and depth-1 G-terms). This and other applications of coproducts of monads as a modularity tool have been discussed in =-=[16, 15]-=-. Similar decomposition theorems for higher-order syntax fail, i.e., coproducts of monads do not make higher-order syntax modular, not alone at least. Specifically, one might maybe expect that EF ∼ = ... |

23 | Abstract syntax and variable binding (extended abstract
- Fiore, Plotkin, et al.
- 1999
(Show Context)
Citation Context ...nctor categories. The canonical example studied in the literature is that of the untyped λ-calculus which is handled as an instance of the general theory of binding algebras, cf. Fiore, Plotkin, Turi =-=[8]-=-. Another important syntactic construction is that of explicit substitutions. The syntax of a language with explicit substitutions does not form a binding algebra as an explicit substitution may bind ... |

20 |
2003a) Free iterative theories – a coalgebraic view
- Adámek, Milius, et al.
(Show Context)
Citation Context ...y final coalgebra constructions [21, 1, 11]. Other forms of syntax, e.g., Σ-term graphs and Σ-rational terms can also be generalised to universal constructions involving FΣ-algebras on lfp categories =-=[2, 20, 3, 10]-=-. 4. UNTYPED λ-CALCULUS Can we extend the above analysis to languages with variable binding? Many proposals have been made, but one of the best is that of [8, 9] (cf. also [12, 5]) whose elegance reli... |

20 | Monads and modular term rewriting
- Lüth, Ghani
- 1997
(Show Context)
Citation Context ...ns, e.g. the study of S-sorted algebraic theories as monads over Set S , the study of categories with structure using monads over Graph or Cat [6], the study of rewriting using monads over Pre or Cat =-=[14]-=-. It has long been a goal of theoreticians and language designers to incorporate datatypes involving variable binding into this framework. The classic example here is that of the untyped λ-calculus wh... |

19 |
A presentation of topoi as algebraic relative to categories or graphs
- Dubuc, Kelly
- 1983
(Show Context)
Citation Context ...ries other than Set have proved fruitful in many situations, e.g. the study of S-sorted algebraic theories as monads over Set S , the study of categories with structure using monads over Graph or Cat =-=[6]-=-, the study of rewriting using monads over Pre or Cat [14]. It has long been a goal of theoreticians and language designers to incorporate datatypes involving variable binding into this framework. The... |

19 | Combining computational effects: commutativity and sum
- Hyland, Plotkin, et al.
- 2002
(Show Context)
Citation Context ...Also untrue is L ∼ = Lapp + Labs where Lapp = µ(Id + × )and Labs = µ(Id + ◦ (1+)) are the languages of pure application terms and pure abstraction terms. To see this, we invoke the following theorem =-=[13]-=-. Theorem 2. Given an endofunctor F and a monad S on C, if the free monads over F and F ◦ S exist, then the coproduct of S and TF exists and TF + S ∼ = S ◦ TF ◦S ∼ = µ(S ◦ (Id + F ◦ )) From this theor... |

18 |
Semantic analysis of normalisation by evaluation for typed lambda calculus
- Fiore
- 2002
(Show Context)
Citation Context ...ired to make the theory work is a really compelling argument in favour of the (categorical formulation of) initial algebra semantics. Handling the syntax of simply typed λcalculus is possible as well =-=[5, 7, 19]-=- 5. EXPLICIT SUBSTITUTIONS The core of this paper asks whether the above treatment extends to explicit substitutions. This is certainly a reasonable idea since explicit substitutions involve variable ... |

12 |
Dualising initial algebras
- Ghani, Lüth, et al.
(Show Context)
Citation Context ...ot hold for higher-order syntax. There are various generalisations of this work which are worth pointing out. Operations with larger arity can be handled by moving to locally κ-presentable categories =-=[11]-=-. Non-wellfounded syntax can be treated by replacing various initial algebra constructions by final coalgebra constructions [21, 1, 11]. Other forms of syntax, e.g., Σ-term graphs and Σ-rational terms... |

7 | Coalgebraic monads
- Ghani, Luth, et al.
- 2002
(Show Context)
Citation Context ...y final coalgebra constructions [21, 1, 11]. Other forms of syntax, e.g., Σ-term graphs and Σ-rational terms can also be generalised to universal constructions involving FΣ-algebras on lfp categories =-=[2, 20, 3, 10]-=-. 4. UNTYPED λ-CALCULUS Can we extend the above analysis to languages with variable binding? Many proposals have been made, but one of the best is that of [8, 9] (cf. also [12, 5]) whose elegance reli... |

7 | Monads and modularity
- Lüth, Ghani
- 2002
(Show Context)
Citation Context ...erms over X and substitution (in fact, one only needs variables, depth-1 F -terms and depth-1 G-terms). This and other applications of coproducts of monads as a modularity tool have been discussed in =-=[16, 15]-=-. Similar decomposition theorems for higher-order syntax fail, i.e., coproducts of monads do not make higher-order syntax modular, not alone at least. Specifically, one might maybe expect that EF ∼ = ... |

6 |
2003b) On Rational Monads and Free Iterative Theories
- Adámek, Milius, et al.
(Show Context)
Citation Context ...y final coalgebra constructions [21, 1, 11]. Other forms of syntax, e.g., Σ-term graphs and Σ-rational terms can also be generalised to universal constructions involving FΣ-algebras on lfp categories =-=[2, 20, 3, 10]-=-. 4. UNTYPED λ-CALCULUS Can we extend the above analysis to languages with variable binding? Many proposals have been made, but one of the best is that of [8, 9] (cf. also [12, 5]) whose elegance reli... |