E.S. Bainbridge, P.J. Freyd, A. Scedrov and P.J. Scott. Functorial polymorphism. Theoretical Computer Science, Volume 70, pages 35--64, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....deduction, augmented with inference rules for relation symbols in the obvious way. There are standard axioms for equational reasoning implying extensionality for arrow and universal types. Parametric polymorphism requires all instances of a polymorphic functional to exhibit a uniform behaviour [Str67,BFSS90,Rey83]. We adopt relational parametricity [Rey83,MR91] A polymorphic functional instantiated at two related domains should give related instances. This is asserted by the schema Param : ##.#[#, #] f (##.#[#, eq ] f The logic with Param is sound; we have, e.g. the parametric per model of ....

....We adopt relational parametricity [Rey83,MR91] A polymorphic functional instantiated at two related domains should give related instances. This is asserted by the schema Param : ##.#[#, #] f (##. #[#, eq ] f The logic with Param is sound; we have, e.g. the parametric per model of [BFSS90] and the syntactic models of [Has91] In order to prove the existence of a model, one has to show that Param holds for all closed f . If one then expands the statement, one obtains a syntactic analogue of the Basic Lemma for logical relations, but here involving universal types. Lemma 4.1 (Basic ....

E. Bainbridge, P. Freyd, A. Scedrov, and P. Scott. Functorial polymorphism. Theoretical Computer Science 70:35--64 (1990).

....between syntax and semantics. We are not aware of any previously published results of this type; however, the idea is related to representation theorems in category theory [FS91] to full abstraction theorems in programming language semantics [Mil75, Plo77] to studies of parametric polymorphism [BFSS90, HRR89]; and to the completeness conjecture in [Gir91a] 3 We now make a first statement in broad terms of our results. We have refined Blass game semantics for Linear Logic. This refinement is not a complication; on the contrary, it makes the definitions smoother and more symmetric. Thus, we get a ....

S. Bainbridge, P. J. Freyd, A. Scedrov, and P. Scott. Functorial polymorphism. Theoretical Computer Science, 70:35--64, 1990.

....for equiv instead of a function, extending models to interpret predicates in such a way that predicates are not required to be undecidable. language imposes, for example functions in a polymorphic language like SML might be required to be parametric, i.e. behave uniformly for all type instances [BFSS90] Imposing such restrictions, whatever they are, gives rise to examples like the one above. Whether or not such an example illustrates a problem that needs to be solved is a di#erent question. There is also an e#ect on the meaning of quantification: #f : # # . # is more likely to hold if the ....

E. Bainbridge, P. Freyd, A. Scedrov and P. Scott. Functorial polymorphism. Theoretical Computer Science, 70:35--64, 1990.

....as sets A equipped with a mapping : A FA where F : Set Set. Our aim is to show the potential of de ning coalgebras by means of operations and equations (Section 2) Our equations for coalgebras dualise equations for algebras and generalise previous concepts of coalgebraic coequations (cf. [24,8,12,10]) to situations without cofree coalgebras. We prove a general co Birkho theorem showing that covarieties of coalgebras are always de nable by coequations (Section 3) and we present a full explanation of Davis s characterisation of coequational categories (Section 4) Finally, we show that the ....

....es the coequation 1 = 2 i every morphism h : K RX is coequalised by 1 and 2 , i.e. i h factors through an equaliser RY This notion of a coequation as a subobject S of a cofree object RX is a special case of Manes [17] Theorem 3.4, page 227. It was further investigated in [24,8,22,12,10]. Conversely, for any subobject S RX, take the cokernel pair f; g : RX A and compose it with A : A RUA given by the unit of the adjunction U a R. Then the pair A f; A g produces the pair of natural transformations X in our sense. Thus, in the presence of cofree coalgebras, our ....

H. Peter Gumm. Equational and implicational classes of colgebras. Theoretical Computer Science, 260, 2001.

.... precursors, as noted in [AJ92b] including representation theorems in category theory [FS91] full abstraction results in programming language semantics [Mil75, Plo77] Plotkin s characterization of de nability in the calculus using logical relations [Plo80] studies of parametric polymorphism [BFSS90, HRR89], and the completeness conjecture in [Gir91] However, the contribution of [AJ92b] was to clearly identify and formulate this issue as a precise and interesting research programme, and to prove the rst in what has become quite a rich sequence of results. 28 Geometric proofs via proof nets. ....

S. Bainbridge, P. J. Freyd, A. Scedrov and P. Scott. Functorial Polymorphism. Theoretical Computer Science 70:35-64, 1990. 31

....the coequation oe 1 = oe 2 iff every morphism h : K RX is coequalised by oe 1 and oe 2 , i.e. iff h factors through an equaliser S RX s 1 s 2 RY This notion of a coequation as a subobject S of a cofree object RX is a special case of Manes [14]3.4. It was further investigated in [19, 6, 17, 11, 7]. Conversely, for any subobject S RX, take the cokernel pair f; g : RX A and compose it with j A : A RUA given by the unit j of the adjunction U a R. Then the pair j A ffi f; j A ffi g produces the pair of natural transformations X U Y U in our sense. Thus, in the presence of cofree ....

H. Peter Gumm. Equational and implicational classes of colgebras. Theoretical Computer Science, 260, 2001.

.... precursors, as noted in [AJ92b] including representation theorems in category theory [FS91] full abstraction results in programming language semantics [Mil75, Plo77] Plotkin s characterization of de nability in the calculus using logical relations [Plo80] studies of parametric polymorphism [BFSS90, HRR89], and the completeness conjecture in [Gir91] However, the contribution of [AJ92b] was to clearly identify and formulate this issue as a precise and interesting research programme, and to prove the rst in what has become quite a rich sequence of results. 28 Geometric proofs via proof nets. ....

S. Bainbridge, P. J. Freyd, A. Scedrov and P. Scott. Functorial Polymorphism. Theoretical Computer Science 70:35-64, 1990. 31

....are given by identity functions. Let 2 N be a code of the total identity function on N. Note that also realizes every identity and every inclusion. Composition is given set theoretically. We only need to note that the composition of realizable functions is realizable. It is well known (see [1]) that PER is a Cartesian closed category and can be used to model 1 the simply typed lambda calculus. Exponent pers are denoted by A ) B and product pers are denoted by A Theta B. The definitions of these can be found in [1] 2.1 The Functor Q : PER SET Above we described Q taking a per A to ....

....of realizable functions is realizable. It is well known (see [1] that PER is a Cartesian closed category and can be used to model 1 the simply typed lambda calculus. Exponent pers are denoted by A ) B and product pers are denoted by A Theta B. The definitions of these can be found in [1]. 2.1 The Functor Q : PER SET Above we described Q taking a per A to the quotient set Q(A) In other words, Q is a function from the objects of PER to the objects of SET. We can extend Q to act on arrows in a functorial way as follows. For any arrow f : A B in PER, let Q(f) f . This makes ....

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E. S. Bainbridge, P. J. Freyd, A. Scedrov, P. J. Scott. Functorial Polymorphism. Theoretical Computer Science 70 (1990), 35-64.

....we refer to [75, 43, 51, 61, 23, 89] In the literature there exists a variety of definitions of term graphs. Besides hypergraphs, directed graphs, terms with labels, and recursion equations have been used as underlying structures. Acyclic graphs have been dealt with in [34, 95, 96, 97] while [83, 92, 59, 15, 37, 63, 32] also consider cyclic graphs. By equipping function symbols with additional labels, sharing of different occurrences of a subterm in a term can be expressed through identical labels. Such labelled terms correspond to acyclic term graphs and have been studied in [76, 74, 82] In [36, 4, 2, 67] ....

Peter Padawitz. Graph grammars and operational semantics. Theoretical Computer Science, 19:117--141, 1982. 61

....x = Y , where YA : A ) A) lift ## [A; A] hid;u A i ## [A; A] A; A] p ## A: Theorem 3. Over any category of hyperdomains, the operator Y is dinatural. For de nition of dinaturality and the fact that all dinatual operators of type (A ) A) A are necessarily xpoint operators we refer to [BFSS90] and [SP00] The equation (3) follows by combining Theorem 3 with a theorem of Simpson [Sim93] implying that the least xpoint operator in standard categories of domains is the only dinatural xpoint operator. Another consequence of Theorem 3 is that the existence of hyperfunction coalgebras does ....

E. Bainbridge, P.J. Freyd, A. Scedrov, and P. Scott. Functorial polymorphism. Theoretical Computer Science, 70:35-64, 1990.

....of datatypes, frame models which assign sets to types might be of greater use. Because we are using categorical constructions, it seems to be better to consider categorical models as described in [See87] and [Pit87] based on indexed categories. A new proposal is Functorial Polymorphism (see [BFSS90]) where the semantic of polymorphic terms is explained by dinatural transformations. This concept seems to be particularly useful in the context of inductive types, however it is not clear to me, how the functorial models relate to the more conventional set theoretic and categorical model ....

....categorical constructions in terms of set theoretic models. So it might be better to use Seely s and Pitt s categorical model definitions ( See87] Pit87] 2. In the light of functorial polymorphism it might be possible to prove the conjecture as a simple corollary of the theorems proven in [BFSS90]. However to do this, a better understanding of this construction is required. It might also be preferably to prove an equivalent theorem which does not require so much sophisticated category theory. 27 Giving rise to a 4 dimensional cube which is slightly difficult to draw, even with the most ....

E.S. Bainbridge, P.J. Freyd, A. Scedrov, and P.J. Scott. Functorial polymorphism. Theoretical Computer Science, 70:35 -- 64, 1990.

....jungles, we refer to [76,43,51,62,23,91] In the literature there exists a variety of definitions of term graphs. Besides hypergraphs, directed graphs, terms with labels, and recursion equations have been used as underlying structures. Acyclic graphs have been dealt with in [34,97,98,99] while [85,94,60,15,37,64, 32] also consider cyclic graphs. By equipping function symbols with additional labels, sharing of different occurrences of a subterm in a term can be expressed through identical labels. Such labelled terms correspond to acyclic term graphs and have been studied in [77,75,83,84] In [36,4,2,68] ....

Peter Padawitz. Graph grammars and operational semantics. Theoretical Computer Science, 19:117--141, 1982.

.... categories has been found for this hierarchy [Weh99] The next steps will be the development of general structural recursion for the hierarchy in style of this paper, and the development of higher dimensional parametricity to derive program transformations similar to the analysis for system F [Wad93, BFSS90, PA93]. I am currently working on applications exploring the expressive power of dimension three. Preliminary results deal with simply typed lambda calculus, internalisation of algebraic data specifications and examples from Okasakis purely functional data types [Oka98] A challenge is to internalise ....

E. S. Bainbridge, P. J. Freyd, A. Scedrov, and P. J. Scott. Functorial polymorphism. Theoretical Computer Science, 70:35--64, 1990.

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E.S. Bainbridge, P.J. Freyd, A. Scedrov, and P.J. Scott, Functorial polymorphism. Theoretical Computer Science 70, 35-64.

....Parametric PER Model in the Reynolds Ma Framework Chad E. Brown January 21, 1999 1 Introduction In Reynolds and Ma [2] semantics for the polymorphic lambda calculus are given using PL categories. A notion of parametricity is also defined. In Bainbridge, Freyd, Scedrov, and Scott [1], a parametric per model is constructed by interpreting abstraction types as an apparently smaller per than the intersection. In this paper, I will unify these two works by placing the parametric per model of [1] into the framework of [2] by defining two PL categories and showing that the ....

....A notion of parametricity is also defined. In Bainbridge, Freyd, Scedrov, and Scott [1] a parametric per model is constructed by interpreting abstraction types as an apparently smaller per than the intersection. In this paper, I will unify these two works by placing the parametric per model of [1] into the framework of [2] by defining two PL categories and showing that the parametricity hypothesis is satisfied. 2 The PL categories K and R In the standard per model, we could define a PL category K with K 0 = PER and then define a PL category R with R 0 = SR depending on both PER and ....

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E. S. Bainbridge, P. J. Freyd, A. Scedrov, P. J. Scott. Functorial Polymorphism. Theoretical Computer Science 70 (1990), 35-64.

....C. In this section we compare this categorical generalization with three forms of logical relations that have already appeared in the literature, Kripke logical relations [Plo80, MM91] cpo logical relations [MS76, Rey74, CP92] and the relational setting over PER models discussed in Section 4 of [BFS90]. Kripke logical relations over ordinary Henkin models were first used in [Plo80] in a characterization of lambda definability. Kripke logical relations were then adapted to Kripke lambda models in [MM91] Inductive relations (as well as strict inductive relations) are widely 10 used in relating ....

....models in [MM91] Inductive relations (as well as strict inductive relations) are widely 10 used in relating denotational semantics and in proofs by fixed point induction. Relations over PER models may be used to derive parametricity properties of lambda definable functions, see Section 4 of [BFS90]. These cases of generalized logical relations are still concrete enough so that it makes sense to consider an analogue of a subscone, a certain subcategory of a comma category consisting of those objects for which relevant mappings are given by inclusions. Hence in these cases we will be able ....

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E.S. Bainbridge, P.J. Freyd, A. Scedrov, and P.J. Scott. Functorial Polymorphism. Theoretical Computer Science, 70:35--64, 1990. Corrigendum ibid., 71:431, 1990.

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E.S. Bainbridge, P.J. Freyd, A. Scedrov and P.J. Scott. Functorial polymorphism. Theoretical Computer Science, Volume 70, pages 35--64, 1990.

No context found.

E.S. Bainbridge, P.J. Freyd, A. Scedrov, and P.J. Scott. Functorial polymorphism. Theoretical Computer Science, 70:35--64, 1990.

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E. S. Bainbridge and P. J. Freyd and A. Scedrov and P. J. Scott. Functorial Polymorphism. Theoretical Computer Science 70,1:35--64, 1988.

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E. S. Bainbridge, Peter J. Freyd, Andre Scedrov, and Philip J. Scott. Functorial polymorphism. Theoretical Computer Science, 70(1):35--64, January 15 1990. Corrigendum in (3) 71, 10 April 1990, p. 431.

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E. S. Bainbridge, Peter J. Freyd, Andre Scedrov, and Philip J. Scott. Functorial polymorphism. Theoretical Computer Science, 70(1):35--64, January 15 1990. Corrigendum in (3) 71, 10 April 1990, p. 431.

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Peter Freyd. Cartesian logic. Theoretical Computer Science, 278:3--21, 2002.

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E. S. Bainbridge, P. J. Freyd, A. Scedrov, and P. J. Scott. Functorial polymorphism. Theoretical Computer Science, 70, 1990.

No context found.

Peter Freyd. Cartesian logic. Theoretical Computer Science, 278:3--21, 2002.

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E. S. Bainbridge, P. J. Freyd, A. Scedrov, and P. J. Scott. Functorial polymorphism. Theoretical Computer Science, 70, 1990.

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