G.D. Plotkin. Denotational semantics with partial functions. Unpublished lecture notes from CSLI summer school, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... the fixpoint type in FIX= we introduce a constructive logic [Bee85] called FIX, of properties of terms over FIX= There are similarities between FIX and the traditional axiomatic domain theory of LCF [Pau87] and to Plotkin s approach to denotational semantics using partial continuous functions [Plo85]. 4.2 The Predicate Logic FIX The FIX propositions constitute part of a predicate logic with equality. The rules for equality, conjunction and universal quantification (over elements of a given type) form a fragment of first order intuitionistic predicate calculus [Dum77] Additionally there are ....

....intention of this chapter is just to investigate how well suited the FIX logic is for interpreting and reasoning about two quite standard languages. The FIX logic can be viewed as a metalogic in which we interpret both QL and HPCF; for an account of this style of programming language analysis see [Plo85]. 6.2 The Language QL We define the language QL by specifying the basic syntax of types and raw expressions; this syntax will then be given a static and dynamic semantics. 99 The Types and Expressions of QL The types of QL are given by the grammar: The (raw) expressions of QL are given by the ....

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G.D. Plotkin. Denotational semantics with partial functions. Unpublished lecture notes from CSLI summer school, 1985.

....more complicated, because of the need to incorporate algebra structures on objects. On the other hand, Lemma 9.7 disappears, because P is small complete. 10 The language FPC In this section, we give a brief overview of Plotkin s call by value recursively typed calculus, FPC, introduced in [33]. For full details see [3] We use X;Y; to range over type variables, and ; to range over types, which : X j j j j X: Here the pre x X binds X. We use ; to range over nite sequences of distinct type variables. We write to mean that all ....

....here is that ( t] is de ned by an external induction on the structure of terms t, so it is apparently not possible to formulate an induction hypotheses that can be established by an internal induction on derivations of t v. For the proof of Proposition 12.2(2) we adapt the approach of [33, 3] to our setting. The strategy is to de ne binary relations relating closed terms to their internal denotations. A closed term t : has a denotation ( t] pP(1; However, values v : enjoy the extra property that ( v] #, i.e. that ( v] P(1; using the hom set inclusion given ....

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G.D. Plotkin. Denotational semantics with partial functions. Lecture notes, C.S.L.I. Summer School, 1985.

....with repect to functors defined by type expressions. The above identifies the structure required by a model of FPC, but does not indicate where to find examples of models. Nevertheless, several sources of such models are known. Domain theory provides the classical example of the category of #cpos [24]. More generally, axiomatic domain theory has successfully abstracted the idiosyncracies of domains to provide a host of neo classical models [2, 4] A quite different type of model is given by gametheoretic semantics [18] Finally, while the structure has not previously been exhibited in the ....

....6.6 pP has bilimits of I bichains. 7. An internal interpretation of FPC In this section, we apply Theorem 1 to obtain an interpretation of Plotkin s call by value recursively typed # calculus, FPC, in the internal category pP. We give a brief summary of the language FPC, introduced in [24]. For full details see [2] We use X,Y, to range over type variables, and #, #, to range over types, which are given by: # : X # # X.#. Here the prefix X binds X . We use #, to range over finite sequences of distinct type variables. We write # to mean that ....

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G.D. Plotkin. Denotational semantics with partial functions. Lecture notes, C.S.L.I. Summer School, 1985.

....Our explanation of type expressions as functors, although based on semantic ideas, is purely syntactic. It is perhaps more elementary and compelling than its semantic counterparts. We also introduce a simple logic for the first order calculus with recursive types. It is a logic of partial terms [4, 18]; because of partiality, we avoid the inconsistencies connected with recursive types. Alternatively, we could have reasoned in the models of the logic, possibly doing without the logic altogether. We 1 have preferred to use the logic because of the syntactic character of our enterprise. The ....

....y y = z oe x = z V 1in (s i = t i ) oe e[s 1 ; s n ] e[t 1 ; t n ] V. For products, we have: s; t) # oe s # t # fst(e) # oe e # snd(e) # oe e # fst(x; y) x snd(x; y) y (fst(z) snd(z) z VI. For exponentials, the axioms are those of the typed partial calculus [18, 14, 15]: x: e) # s(t) # oe s # t # x: e[x] y: e[y] x: e[x] y) e[y] x: fx = f s[x] t[x] x: s[x] x: t[x] x not free in assumptions above s[x] t[x] Since the substitutivity axiom applies only to terms that are known to exist, the law for fi equality is not equational but conditionally ....

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G.D. Plotkin. Denotational semantics with partial functions. Lecture at C.S.L.I. Summer School, 1985.

....compactness (which yields zero objects) is inconsistent with cartesian closure. This, in principle, precludes a unified treatment of sums, products, exponentials and recursive types via the usual universal properties. However, it was via a direct semantic analysis of non terminating computations [Plo85] involving categories of partial maps [RR88] and, in particular, via the notion of partial cartesian closure [LM84] that an appropriate categorical setting emerged. With this background it was possible, for the first time, to consider categorical models for a rich type theory with recursive ....

....setting emerged. With this background it was possible, for the first time, to consider categorical models for a rich type theory with recursive types. In [Fio94a, FP94] a notion of categorical model for the metalanguage FPC a type theory with sums, products, exponentials and recursive types [Plo85, Gun92, Win93] was defined. Very roughly, categorical models of FPC are algebraically compact partial cartesian closed categories with binary coproducts. Impact of axiomatic domain theory In relating operational and denotational semantics. The investigation of the relation between ....

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G.D. Plotkin. Denotational semantics with partial functions. Lecture at C.S.L.I. Summer School, 1985.

....M. M is based on the equational fragment of Crole and Pitts FIX logic [5] but contains a single parameterised recursive datatype which is used to model computations engaged in I O, and does not (explicitly) contain a fixpoint type. Following Plotkin s use of a metalogic to study object languages [24] we equip the programs (closed expressions) of M with an operational semantics. Our second theorem shows the good fit between the domain theoretic semantics of M and its operational semantics: we prove that the denotational semantics is sound and adequate with respect to the operational ....

....ae e j ( if i (e( e j ( a j ) if i (e( in j (a j ) To prove Theorem 2, we shall show that there is a type indexed family of relations oe [ oe] Theta Prog oe satisfying certain conditions. Such formal approximation relations are fairly standard (see for example [7, 22, 24]) so we simply give these conditions at lifted and recursive types: e oe P iff 9d 2 [ oe] e = lift (d) implies 9P 1 :P Lift(P 1 ) and d oe P 1 , r U(oe) P iff r = or 9P j :P c j (P j ) and 9d j :r = in j (d j ) and d j oe j [ oe =X0 ] P j . We shall need the following ....

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Gordon D. Plotkin. Denotational semantics with partial functions. Stanford CSLI 1985.

.... in an otherwise classical predicate logic (perhaps with identity) as chronicled in the work of some of the eld s pioneers [18, 20] The relevance of free logic to some branches of mathematics and related areas of mathematical computer science was explicitly argued beginning in the 1960 s [29, 30, 24, 23]. As noted elsewhere [15] free logic has been implicit from the beginning in a central area of computer science, program speci cation and veri cation [17] The term free logic has not been current in the mathematics and computer science literature, however, because some authors object to the ....

Gordon D. Plotkin. Denotational semantics with partial functions. Lecture Notes, CSLI Summer School, Stanford University, July 1985.

....by inequality. For example, essentially the same proof that shows equalisers to be monos shows inserters to be monos as well. For a cpo D, define D to be the subcpo of D Theta D with underlying set f(x; y) j x yg. In Cpo and in pCpo, the category of cpos and partial continuous functions [Plo85] (i.e. partial functions whose domain of definition is Scott open and such that its restriction to the domain of definition is continuous) Dae D Theta D is the inserter of ( 1 ; 2 ) This motivates the abstract definition of inequality: Definition 6.2 In a Poset category, the inequality ....

G.D. Plotkin. Denotational semantics with partial functions. Lecture at C.S.L.I. Summer School, 1985.

....A) Gamma F (A; A) for endomorphisms g : A Gamma A of C. We assume that the reader is able to verify, or believe, various routine calculations in models, especially those just involving products, sums, monads and exponentials. 1. 2 Call by Value FPC We describe the version of Plotkin s FPC [Plo85] that we use. We follow the presentation of [Fio94] as far as is convenient. Our notation is slightly terser, and we include in the calculus a type 0 and constants A of each type. These additions are useful for our arguments, but as they are definable in the unextended calculus, they make no ....

Gordon D. Plotkin. Denotational semantics with partial functions. Lecture at C.S.L.I. summer school, 1985.

.... P ( Top; M open ) is by a partial map from X to Y s.t. the inverse image g 1 (A) of an open subset A of Y is an open subset of X. A full sub category of Top is the category CPO of partial orders complete w.r.t. lubs of chains and monotonic maps preserving lubs of chains (see [Plo85] We don t assume the existence of a least element in a cpo, because we want an open subset of a cpo (with the induced partial order) to be a cpo, so that M open restricted to CPO is a domain structure over CPO. 1.2 Data types and partial morphisms Having introduced partial morphisms, ....

G. D. Plotkin. Denotational semantics with partial functions. Lecture at C.S.L.I. Summer School, 1985.

...., but probably there are others just waiting to be discovered. The categorical semantic of computations 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 categories turn out to be a special case of c models (see De nition 3.9) A type theoretic approach ....

G.D. Plotkin. Denotational semantics with partial functions. Lecture Notes at C.S.L.I. Summer School, 1985.

.... coincidence theorem [32] Models of ADT are domain theoretic models of recursive types admitting a rich type structure (e.g. suitable for interpreting products, higher types, and sums) Canonical examples of models of ADT are pCpo (the category of small cpos and partial continuous functions [31]) and Cppo (the category of small cpos with bottom element and strict continuous functions) with respect to the domain theoretic enrichment base ( Cpo Gamma Gamma Cppo . 2 Constructing models of ADT We explore the construction of models of ADT, first from domain theoretic ....

G.D. Plotkin. Denotational semantics with partial functions. Lecture at C.S.L.I. Summer School, 1985.

....metalanguage, we introduce a constructive logic, called FIX, of properties of terms over the metalanguage. Thus there are strong connections between FIX and the traditional axiomatic domain theory of LCF [13] and to Plotkin s approach to denotational semantics using partial continuous functions [15]. However, our logic is inherently more constructive, since it is based on the notion of evaluation of a (possibly non terminating) computation to a value, rather than on non termination and on information ordering between (possibly partial) computations. Definition of the FIX logic: The FIX ....

G.D.Plotkin, Denotational semantics with partial functions, unpublished lecture notes from CSLI Summer School (1985).

....Our explanation of type expressions as functors, although based on semantic ideas, is purely syntactic. It is perhaps more elementary and compelling than its semantic counterparts. We also introduce a simple logic for the first order calculus with recursive types. It is a logic of partial terms [4, 18]; because of partiality, we avoid the inconsistencies connected with recursive types. Alternatively, we could have reasoned in the models of the logic, possibly doing without the logic altogether. We have preferred to use the logic because of the syntactic character of our enterprise. The logic ....

....= y y = z oe x = z V 1in (s i = t i ) oe e[s 1 ; s n ] e[t 1 ; t n ] V. For products, we have: s; t) # oe s # t # fst(e) # oe e # snd(e) # oe e # fst(x; y) x snd(x; y) y (fst(z) snd(z) z VI. For exponentials, the axioms are those of the typed partial calculus [18, 14, 15]: x: e) # s(t) # oe s # t # x: e[x] y: e[y] x: e[x] y) e[y] x: fx = f s[x] t[x] x: s[x] x: t[x] x not free in assumptions above s[x] t[x] Since the substitutivity axiom applies only to terms that are known to exist, the law for fi equality is not equational but conditionally ....

[Article contains additional citation context not shown here]

G.D. Plotkin. Denotational semantics with partial functions. Lecture at C.S.L.I. Summer School, 1985.

....M. M is based on the equational fragment of Crole and Pitts FIX logic [5] but contains a single parameterised recursive datatype which is used to model computations engaged in I O, and does not (explicitly) contain a fixpoint type. Following Plotkin s use of a metalogic to study object languages [24] we equip the programs (closed expressions) of M with an operational semantics. Our second theorem shows the good fit between the domain theoretic semantics of M and its operational semantics: we prove that the denotational semantics is sound and adequate with respect to the operational ....

....i Gamma1 (e( e j ( a j ) if i Gamma1 (e( in j (a j ) To prove Theorem 2, we shall show that there is a type indexed family of relations oe [ oe] Theta Prog M oe satisfying certain conditions. Such formal approximation relations are fairly standard (see for example [7, 22, 24]) so we simply give these conditions at lifted and recursive types: e oe P iff 9d 2 [ oe] e = lift (d) implies 9P 1 :P Lift(P 1 ) and d oe P 1 , r U(oe) P iff r = or 9P j :P c j (P j ) and 9d j :r = in j (d j ) and d j oe j [ oe =X0 ] P j . We shall need the following ....

[Article contains additional citation context not shown here]

Gordon D. Plotkin. Denotational semantics with partial functions. Stanford CSLI 1985.

....of M. M is based on the equational fragment of the FIX logic of [CP92] but contains a single parameterised recursive datatype which is used to model computations engaged in I O, and does not (explicitly) contain a fixpoint type. Following Plotkin s use of a metalanguage to study object languages [Plo85] we equip the programs (closed expressions) of M with an operational semantics. Theorem 2 shows the good fit between the domain theoretic semantics of M and its operational semantics: we prove that the denotational semantics is sound and adequate with respect to the operational semantics. To ....

....that will play a key role in the proof of Theorem 3.1. We shall show that there is a T indexed family of relations ( oe [ oe] Theta f P j 9oe(P :oe) g j oe 2 T ) satisfying certain conditions. Such formal approximation relations are fairly standard (see for example [CG93] Pit94b] and [Plo85] so we simply give these conditions at function, lifted and recursive types: ffl f oe)oe 0 P iff f = or 9E: P x: E and 8d oe P 0 : f(d) oe 0 E[ P 0 =x] ffl e oe P iff 9d 2 [ oe] e = d] implies 9P 0 : P Lift(P 0 ) and d oe P 0 , ffl r oe U(oe) P iff ....

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G.D. Plotkin. Denotational semantics with partial functions. Unpublished lecture notes from CSLI summer school, 1985.

....(or titfor tat ) behaviour results in quite redundant interactions. This dissociates the resulting code from any feasible implementation. Can one obtain better code from a game theoretic model of a programming language Results. This work addresses the above mentioned issues taking FPC [30] as an example programming language. FPC is essentially the functional core of ML; we have chosen it for its rich type structure and call by value operational semantics. Moreover its primitive character is useful to examine the basic features of our approach in a simple, though non trivial, ....

....(A; T ) to the arena 1 T Omega A Omega T , which by the previous considerations does not extend to a functor G 2 G. Thus one cannot interpret type judgements T 1 ; Tn ff as functors G n G, and consequently one cannot interpret recursive types as parameterised free algebras [11, 9, 30, 32]. Our approach to solving this problem is to restrict attention to a subcategory of G in which type judgements can be interpreted as functors, and find the recursive types there. This idea is traditional in domain theory. For example, it occurs when solving domain equations of functors of mixed ....

G. Plotkin. Denotational semantics with partial functions. Lecture at C.S.L.I. Summer School, 1985.

....Computing (Springer Verlag, Berlin, 1991) pp 162 189. y Research supported by the CLICS project (ESPRIT BR Action nr 3003) 1 Introduction Higher order metalogics based on typed lambda calculi (such as Scott s LCF [20] and Plotkin s formalizations of domain theoretic denotational semantics [19]) have been used to give semantics to programming languages via formal translations of programming language syntax into the types and terms of the metalogic. The basic features of such translations are their compositionality (i.e. the translation of a compound program expression depends only on ....

G. D. Plotkin, Denotational semantics with partial functions, unpublished lecture notes from CSLI Summer School, 1985.

....definition. The standard approach to introducing recursive types in a language involves modifying the semantics of some of the type constructors so that all recursive domain equations will have solutions; typically this means that the sum or product types will not be extensional (see for example [GS90, Plo85, SP82]) Our work developed from a desire to have recursive types in a language whose equational semantics includes extensionality for both products and sums (i.e. both products and sums are categorical ) It is well known that such a semantics in which all recursive domain equations have solutions ....

G.D. Plotkin. Denotational semantics with partial functions. Lecture notes, C.S.L.I. Summer School, Stanford, 1985.

....of [ R(f:x:m; n) v , the FIX logic rules and the induction hypothesis, we have x i : i ; f : 0 ) x: 0 m: and x i : i n: 0 ; from which x i : i R(f:x:m; n) is immediate. 2 Dynamic Adequacy of QL We shall prove a theorem based on Plotkin s methods given in [13]. We write D( for the composition [ CPO u: v ( x) QL FIX CPO where [ CPO is the standard domain theoretic semantics of FIX. We de ne a relation C between elements d 2 D( and canonical forms c: by induction on the structure of ; more precisely, we shall de ne a ....

G.D. Plotkin. Denotational semantics with partial functions. Unpublished lecture notes from CSLI summer school, 1985.

....it is in spirit of axiomatic domain theory to make the weakest assumptions possible for the desired results to follow. Second, we want as good a correspondence as possible between the category theory and the typed calculi modelled. Standard recursively typed calculi (such as Plotkin s metalanguage [18]) have partial function spaces (corresponding to the Kleisli exponentials of a suitable strong monad) but not total function spaces. However, it must be admitted that most real world models are cartesian closed (with the possible exception of Rosolini s oe domains [20] Fixed point objects A ....

....partial orders, possibly without least element) and continuous functions. For the monad T , we take the lift functor on PreDom (which adds a new least element to an CPO) this has an associated commutative strong monad. C T is now pPreDom, the category of CPOs and partial continuous functions [18]. C T is the category Dom , of those CPOs with a least element and strict (i.e. least element preserving) continuous functions. Incidentally, pPreDom and Dom are equivalent categories, this is not true in general for C T and C T . D is the category, Dom, of CPOs and all continuous ....

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G. D. Plotkin. Denotational semantics with partial functions. Lecture notes, C.S.L.I. Summer School, 1985.

....all computable partial functions are definable. By separating out the recursive types which only allow bounded computations, we obtain a finer resolution of where nontermination is possible than that provided by other type systems. In particular, we compare our work to Plotkin s versions of PCF [43, 44] and Hagino s Categorical Data Types [21, 22] Indeed, our work may be viewed as an extension of Hagino s language, which is roughly equivalent in expressive power to , into a system equivalent in power to PCF. The existence of a clean category theoretic semantics for the language has been ....

G.D. Plotkin. Denotational semantics with partial functions. Lecture notes, C.S.L.I. Summer School, Stanford, 1985.

....functions between domains, with respect to which the fixed point is characterised by the property of uniformity. Further, the monad determines a category of partial functions, which is arguably the most suitable category for program semantics. The development of this viewpoint can be found in [23, 2, 18, 29, 3]. In recent years it has become apparent that many natural categories of predomains arise as full subcategories of elementary toposes. For example, the category of complete partial orders and continuous functions is a full reflective subcategory of the Grothendieck topos, H, considered in [6, ....

G.D. Plotkin. Denotational semantics with partial functions. Lecture notes, CSLI Summer School, 1985.

....categorical domain theory as needed for the denotational semantics of deterministic programming languages. We particularly consider a metalanguage FPC, a typed functional language with sums, products, exponentials and recursive types equipped with a call byvalue operational semantics (see [Plo85, Gun92]) We wish to axiomatise domain theoretic models of FPC and prove that they provide a computationally sound and adequate denotational semantics. Such a theorem holds for pCpo [Plo85] the category of small cpos (posets, possibly without bottom, closed under lubs of chains) and partial ....

....sums, products, exponentials and recursive types equipped with a call byvalue operational semantics (see [Plo85, Gun92] We wish to axiomatise domain theoretic models of FPC and prove that they provide a computationally sound and adequate denotational semantics. Such a theorem holds for pCpo [Plo85], the category of small cpos (posets, possibly without bottom, closed under lubs of chains) and partial continuous functions. The aim of this paper is to generalise to a wide class of order enriched categories (Section 2) one can compare this endeavour to [SP82, Fre90, Fre92] where a similar ....

G.D. Plotkin. Denotational semantics with partial functions. Lecture at C.S.L.I. Summer School, 1985.

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