G. D. Plotkin. Type theory and recursion. In Eighth Annual IEEE Symposium on Logic in Computer Science, page 374. IEEE Computer Society Press, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....system LU [Gir93] and related systems in which the (multi)sets of formulae occuring in sequents are split into di erent zones. Formulae in some zones are treated classically, whilst those in other zones are treated linearly. Intuitionistic logics inspired by LU have been proposed by Plotkin [Plo93] and Wadler [Wad93] It is desirable to study the proof and model theory of such systems directly, rather than treating them as syntactic sugar for, for example, ordinary linear logic (if only to verify that it is possible to treat them as such syntactic sugar) The logic of this paper should turn ....

....models compared with coproducts in linear categories. Beyond that, one could consider adding inductive or coinductive datatypes or second order quanti cation to the logic. This seems particularly worthwhile in the light of Plotkin s work on parametricity and recursion in a logic rather like ours [Plo93]. On the practical side, we should investigate whether or not the LNL term calculus lends itself more readily to ecient implementation than does the linear term calculus. The hope is that one can arrange an implementation with two memory spaces, corresponding to the two subsystems of LNL logic. ....

G. D. Plotkin. Type theory and recursion (abstract). In Proceedings of 8th Conference on Logic in Computer Science. IEEE Computer Society Press, 1993.

....the storage allocation properties of the resulting interpretation of linear logic. Section 5 describes related work. Section 6 offers some conclusions and ideas for further work. 2 Intuitionistic Linear Logic We use (a slight variant of) Plotkin s formulation of intuitionistic linear logic [Plo93] which, unlike many formulations of linear logic, has no explicit syntax for duplicating or discarding values. Such term constructs are important for models of reference counting implementations, but are unnecessary here since our operational semantics is formulated at a slightly higher level of ....

Gordon Plotkin. Type theory and recursion (abstract). In Logic in Computer Science. IEEE Press, 1993.

....also [6] Our work is complementary to theirs; roughly, their coercion interpretation reduces a calculus with subtyping to one with type equality, and our coercion interpretation reduces one with type equality to one with isomorphism. Another relevant syntactic investigation is that of Plotkin [19], who developed a second order intuitionistic linear type theory with a fixed point operator at the level of terms, showed how to encode recursive types, and derived some reasoning principles for them. While it would be interesting to build on Plotkin s theory, we have preferred to avoid type ....

G.D. Plotkin. Type theory and recursion (extended abstract). In Proceedings of the Eighth Annual page 374, 1993.

.... iteration type [ML83] and give a categorical characterisation of general recursion at higer types similar to the characterisation of primitive recursion at higher types in terms of Lawvere s concept of natural number object [LS86] A type theoretic approach to domain theory is that of [Plo93] There, rather than considering directly possible categorical structure, the idea is to work within a type theory pursuing the analogies: intuitionistic exponential = function space, and linear exponential = strict function space. More precisely, the basic setting is that of secondorder ....

G.D. Plotkin. Type theory and recursion (extended abstract). In 8 Conf., page 374. IEEE, Computer Society Press, 1993.

....Finally we want to detect immediately when a function is intuitionistic. Hence it is more appropriate to have both and Gammaffi as primitive operations and disregard . This leads to consideration of the mixed intuitionistic and linear type theory (henceforth named ILT) described by Plotkin [Plo93] and Wadler [Wad90] obtained from DILL by (i) adding intuitionistic implication, and (ii) removing the modality from the type operators. The syntactic behaviour of ILT is very similar to that of DILL. But when it comes to semantics, the situation is a little more complicated. It is not obvious ....

G.D. Plotkin. Type theory and recursion. In Proc. of Logic in Computer Science, 1993.

....general result about relational parametricity in linear type theory. In suitably parametric models of (intuitionistic) polymorphic calculus types of the form 8ff: T ff ff) ff denote initial T algebras, and this paves the way for a characterization of all second order types in prenex form. Plotkin [1993] in lectures has indicated that the corresponding property in linear polymorphic type theory is that 8ff: T ff Gammaffi ff) ff denotes an initial T algebra, for covariant functors T on a linear category where morphisms correspond to terms of Gammaffi type. It is not immediately obvious ....

Plotkin, G. D. 1993. Type theory and recursion (extended abstract). In Proceedings, Eighth Annual IEEE Symposium on Logic in Computer Science (Montreal, Canada, 19--23 June 1993), pp. 374. IEEE Computer Society Press.

....also [6] Our work is complementary to theirs; roughly, their coercion interpretation reduces a calculus with subtyping to one with type equality, and our coercion interpretation reduces one with type equality to one with isomorphism. Another relevant syntactic investigation is that of Plotkin [19], who developed a second order intuitionistic linear type theory with a fixed point operator at the level of terms, showed how to encode recursive types, and derived some reasoning principles for them. While it would be interesting to build on Plotkin s theory, we have preferred to avoid type ....

G.D. Plotkin. Type theory and recursion (extended abstract). In Proceedings of the Eighth Annual IEEE Symposium on Logic in Computer Science, page 374, 1993.

....to the Kleisli category of . Thus, Oles s strictness condition arises naturally if we take Cpo together with the lifting comonad as fundamental, and look for a model of intuitionistic linear logic based on functors into Cpo rather than looking directly for a model of intuitionistic logic (cf. [27]) 4.2 Semantic Model The semantics is given in MOb . For the types, define [ comm] Ob( comm) aint] Ob( aint) s Theta t] s] Theta [ t] s t] s] t] The defined type var gets the interpretation [ var] aint comm] Theta [ aint] Variables of this ....

G. D. Plotkin. Type theory and recursion. In Proceedings, Symposium on Logic in Computer Science, Montreal, 1993. IEEE Computer Society Press.

....system LU [Gir93] and related systems in which the (multi)sets of formulae occuring in sequents are split into different zones. Formulae in some zones are treated classically, whilst those in other zones are treated linearly. Intuitionistic logics inspired by LU have been proposed by Plotkin [Plo93] and Wadler [Wad93] It is desirable to study the proof and model theory of such systems directly, rather than treating them as syntactic sugar for, for example, ordinary linear logic (if only to verify that it is possible to treat them as such syntactic sugar) The logic of this paper should turn ....

....models compared with coproducts in linear categories. Beyond that, one could consider adding inductive or coinductive datatypes or second order quantification to the logic. This seems particularly worthwhile in the light of Plotkin s work on parametricity and recursion in a logic rather like ours [Plo93]. On the practical side, we should investigate whether or not the LNL term calculus lends itself more readily to efficient implementation than does the linear term calculus. The hope is that one can arrange an implementation with two memory spaces, corresponding to the two subsystems of LNL logic. ....

G. D. Plotkin. Type theory and recursion (abstract). In Proceedings of 8th Conference on Logic in Computer Science. IEEE Computer Society Press, 1993.

....are [Girard, 1986, Coquand et al. 1987, Taylor, 1990, Ehrhard, 1993] Berry s motivation in introducing stable functions was actually to try to capture the notion of sequentially computable function at higher types. For the theory of sequential functions on concrete domains, we refer to [Kahn and Plotkin, 1993, Curien, 1993] 8.3 Reformulations of Domain Theory At various points in our development of Domain Theory (see e.g. Section 3.2) we have referred to the need to switch between different versions C, C , C of some category of domains, depending on whether bottom elements are required, and if ....

....strong use of the monadic approach. This work really belongs to Axiomatic Domain Theory, to which we will return in subsection 4 below. 8.3. 3 Linear Types Another proposal by Gordon Plotkin is to use Linear Types (in the sense of Linear Logic [Girard, 1987] as a metalanguage for Domain Theory [Plotkin, 1993]. This is based on the following observation. Consider a category C of domains with bottom elements and strict continuous functions. This category has products and coproducts, given by cartesian products and coalesced sums. It also has a monoidal closed structure given by smash product and ....

G. D. Plotkin. Type theory and recursion. In Eigth Annual IEEE Symposium on Logic in Computer Science, page 374. IEEE Computer Society Press, 1993.

....into the linear calculus. Semantically this has proved a very useful viewpoint rather than devising a model of the calculus one can, in its stead, produce a model of the linear calculus [6] This approach has been utilised, for example, by Plotkin in studying recursion and parametricity [21] and by Abramsky et al. 2] to produce fully abstract models of PCF. A more operational perspective is to consider the linear calculus as some sort of intermediate language to which the calculus is compiled. 1 This has an obvious practical advantage in that the linear calculus is explicit ....

G.D. Plotkin. Type theory and recursion (extended abstract). In Proceedings of Symposium on Logic in Computer Science, page 374, 1993.

....are [Girard, 1986, Coquand et al. 1987, Taylor, 1990, Ehrhard, 1993] Berry s motivation in introducing stable functions was actually to try to capture the notion of sequentially computable function at higher types. For the theory of sequential functions on concrete domains, we refer to [Kahn and Plotkin, 1993, Curien, 1993] 8.3 Reformulations of Domain Theory At various points in our development of Domain Theory (see e.g. Section 3.2) we have referred to the need to switch between different versions C, C , C of some category of domains, depending on whether bottom elements are required, and if ....

....strong use of the monadic approach. This work really belongs to Axiomatic Domain Theory, to which we will return in subsection 4 below. 8.3. 3 Linear Types Another proposal by Gordon Plotkin is to use Linear Types (in the sense of Linear Logic [Girard, 1987] as a metalanguage for Domain Theory [Plotkin, 1993]. This is based on the following observation. Consider a category C of domains with bottom elements and strict continuous functions. This category has products and coproducts, given by cartesian products and coalesced sums. It also has a monoidal closed structure given by smash product and ....

G. D. Plotkin. Type theory and recursion. In Eigth Annual IEEE Symposium on Logic in Computer Science, page 374. IEEE Computer Society Press, 1993.

.... iteration type [ML83] and give a categorical characterisation of general recursion at higher types similar to the characterisation of primitive recursion at higher types in terms of Lawvere s concept of natural number object [LS86] A type theoretic approach to domain theory is that of [Plo93] There, rather than considering directly possible categorical structure, the idea is to work within a type theory pursuing the analogies: intuitionistic exponential = function space, and linear exponential = strict function space. More precisely, the basic setting is that of second order ....

....intuitionistic, or Cartesian closed, type theory and recursion) and most remarkably, one obtains solutions of arbitrary domain equations, and not just covariant ones. This uses Freyd s reduction of recursive to inductive types [Fre90] These results have been presented by Plotkin in lectures [Plo93] though they have not been published yet. But the point that linear type theory allows for a better treatment of parametricity with recursion rings clear, and there are likely to be further applications of linear type theory as far as parametricity is concerned. One example already is in work on ....

G. D. Plotkin. Type theory and recursion. In Eighth Annual IEEE Symposium on Logic in Computer Science, page 374. IEEE Computer Society Press, 1993.

....with finite sums and a linear fixpoint operator; so the presence of a linear fixpoint operator in a linear category is consistent with the presence of finite sums. Thus, the inconsistency of recursion with this standard construct vanish when we go to a linear context, which is in accordance with [Plo93]. In [MRA93] a different approach to recursion in a linear context is taken. The (I; Omega ; fragment of the linear calculus is extended with natural numbers, corresponding to a weak natural numbers object in the categorical model. The discussion above implies that this approach is ....

G. D. Plotkin. Type theory and recursion (extended abstract). In 8th LICS Conference. IEEE, 1993.

....show that xLIN refines Krivine s Abstract Machine (KAM) Then we introduce a call by need machine and investigate update in place. 2 Dual Intuitionistic Linear Logic The linear language we propose here as input to our abstract machine is based on Plotkin s version of intuitionistic linear logic [13], also known in the literature under the name of Dual Intuitionistic Linear Logic (DILL) as coined in Barber s work [2] 2.1 DILL syntax DILL borrows from Girard s Logic of Unity [7] the elegant idea of separating assumptions into two classes: intuitionistic, which can be freely contracted ....

G. Plotkin (1993). Type theory and recursion (Extended Abstract). In Logic and Computer Science, IEEE Press.

....the storage allocation properties of the resulting interpretation of linear logic. Section 5 describes related work. Section 6 offers some conclusions and ideas for further work. 2 Intuitionistic Linear Logic We use (a slight variant of) Plotkin s formulation of intuitionistic linear logic [Plo93] which, unlike many formulations of linear logic, has no explicit syntax for duplicating or discarding values. Such term constructs are important for models of reference counting implementations, but are unnecessary here since our operational semantics is formulated at a slightly higher level of ....

Gordon Plotkin. Type theory and recursion (abstract). In Logic in Computer Science. IEEE Press, 1993.

....Girard s system LU [8] and related systems in which the (multi)sets of formulae occuring in sequents are split into different zones. Formulae in some zones are treated classically, whilst those in other zones are treated linearly. Intuitionistic logics inspired by LU have been proposed by Plotkin [12] and by Wadler [15] It is desirable to study the proof and model theory of such systems directly, rather than treating them as syntactic sugar for, for example, ordinary linear logic (if only to verify that it is possible to treat them as such syntactic sugar) The logic of this paper should turn ....

....should investigate further how to treat the additives. Beyond that, one could consider adding inductive or coinductive datatypes or second order quantification to the logic. This seems particularly worthwhile in the light of Plotkin s work on parametricity and recursion in a logic rather like ours [12]. On the practical side, we should investigate whether or not the LNL term calculus lends itself more readily to efficient implementation than does the linear term calculus. The hope is that one can arrange an implementation with two memory spaces, corresponding to the two subsystems of LNL logic. ....

G. D. Plotkin. Type theory and recursion (abstract). In Proceedings of 8th Conference on Logic in Computer Science. IEEE Computer Society Press, 1993.

....pCpo is equivalent to Cppo they are conceptually different. The first seems computationally more natural, fitting with standard formulations of recursion theory that emphasise partial functions. The latter has a natural generalisation to models of intuitionistic linear type theory with recursion [Plo93]; the connection with operational notions is unclear. It would be interesting to make abstract categorical studies of other notions of computation, for example nondeterminism or probabilistic computation. There behaviour concerns more than termination and semantics more than existence and one ....

G.D. Plotkin. Type theory and recursion (extended abstract). In 8 th LICS Conf., page 374. IEEE, Computer Society Press, 1993.

No context found.

G. D. Plotkin. Type theory and recursion. In Eighth Annual IEEE Symposium on Logic in Computer Science, page 374. IEEE Computer Society Press, 1993.

No context found.

Plotkin, G. (1993) Type theory and recursion (extended abstract). In Proc. Logic in Computer Science (LICS'93), pp. 374.

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