D.S. Scott. A type theoretic alternative to CUCH, ISWIM, OWHY. Unpublished manuscript, University of Oxford, 1969.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Thus we can state Theorem 4.1. 1 by way of following induction principle: 8e : 8n : n] e oe n 2 S) oe oe(e) 2 S = S Just as least fixed points are definable using the universal property of the initial ( Gamma) algebra Omega Gamma so is Scott s induction principle for least fixed points [Sco69b] derivable from the above induction rule. Proposition 4.1.2 [ Scott Induction ] Let P ae D be an inclusive subset, and let f : D D be a continuous function. Then we have 2 P 8d 2 D :d 2 P oe f(d) 2 P fix(f) 2 P Proof In Theorem 4.1.1 take S = fn 2 Omega j h(n) 2 Pg where h is the ....

D.S. Scott. A type theoretic alternative to CUCH, ISWIM, OWHY. Unpublished manuscript, University of Oxford, 1969.

....and general way to give semantics to the first language in terms of a semantics for the second. In denotational semantics it is quite common to define the semantics of a programming language PL by translating it into a typed metalanguage ML. The idea is as old as denotational semantics (see [Sco93] so the main issue is whether it can be made into a viable technique capable of dealing with complex programming languages. Before being more specific about what metalanguages to use, let us discuss what are the main advantages of semantics via translation: to reuse the same ML for ....

D.S. Scott. A type-theoretic alternative to CUCH, ISWIM, OWHY. Theoretical Computer Science, 121, 1993.

....26 9 Conclusions 26 2 1 Introduction Recently, the concepts of game semantics have become of particular interest to researchers in programming language theory. The main reason why this is the case, is that they have been used to give fully abstract semantics to programming languages, such as PCF [Sco69]. In this report we aim to explain the basic concepts of game semantics using a functional language rather than the more standard mathematical notations. A game is played by two participants, the player and the opponent. Each of these participants can make moves in the game. The set of possible ....

D.S. Scott. A type--theoretic alternative to CUCH, ISWIM, OWHY. Manuscript, 1969.

....a program M has a value according to the operational semantics iff it is provably equivalent to a value (not necessarily the same) in the calculus, but they are too weak for proving equivalences of programs. The denotational approach may suggest important principles, e.g. fix point induction (see [Sco93, GMW79]) that can be found only after developing a semantics based on mathematical structures rather than term models, but it does not give clear criteria to single out the general principles among the properties satisfied by the model. The approach adopted in this paper generalises the one followed in ....

D.S. Scott. A type-theoretic alternative to CUCH, ISWIM, OWHY. Theoretical Computer Science, 121, 1993.

....property Ninetyone is needed to prove that f91 is total. A recent extension to the logic makes some progress towards these problems [4] 1.2. 2 LCF: Logic for Computable Functions LCF actually refers to a family of theories which grew out of Scott s investigations of models of the lambda calculus [17, 10, 14]. The primary logic motivating much of the development of LCF is called PPLAMBDA which formalizes typed, call by name functions. In PPLAMBDA, terms are written along with their types as s:oe. Types are omitted since they can be inferred from context. Each type oe contains an undefined element : ....

D. S. Scott. A type-theoretic alternative to CUCH, ISWIM, OWHY. Typed notes, Oxford, 1969.

....way to give semantics to the first language in terms of an available semantics for the second. In denotational semantics it is quite common to define the semantics of a programming language PL by translating it into a typed metalanguage ML. The idea is as old as denotational semantics (see [Sco93] so the main issue is whether it can be made into a viable technique capable of dealing with complex programming languages. Before being more specific about what metalanguages to use, let us discuss what are the main advantages of semantics via translation: ffl to reuse the same ML for ....

D.S. Scott. A type-theoretic alternative to CUCH, ISWIM, OWHY. Theoretical Computer Science, 121, 1993.

....this section we specialize the technique of giving semantics via translation to the case of programming languages. The general idea is to de ne the denotational semantics of a programming language PL by translating it into a typed metalanguage ML. The idea is as old as denotational semantics (see [Sco93]) so the main issue is whether it can be made into a viable technique capable of dealing with complex programming languages. Before being more speci c about metalanguages, let us discuss what are the main advantages in using them to give semantics via translation: to reuse the same ML for ....

....idea that predomains should be part of a set theoretic universe with an expressive logic. From ADT we take equational reasoning principles, which are valid in many categories used in Denotational Semantics, such as the category of cpos. We have not taken the more traditional approach of LCF (see [Sco93]) in which predomains come equipped with a partial order, because in some semantic categories (e.g. complete extensional PERs and e ective morphisms) the order structure is not the most important one, while in others (e.g. in dI domains and stable functions) some of the LCF axioms fail. The ....

D.S. Scott. A type-theoretic alternative to CUCH, ISWIM, OWHY. Theoretical Computer Science, 121, 1993.

....which free us from the irrelevant detail of syntax and focus our mind on the important structures underlying programming languages. We believe that there is a great potential to be exploited here. The c calculus open also the possibility to develop a new Logic of Computable Functions (see [12]) based on an abstract semantic of computations rather than domain theory, for studying axiomatically di erent notions of computation and their relations. Acknowledgements My thanks to M. Hyland, A. Kock (and other participants to the 1988 Category Theory Meeting in Sussex) for directing me ....

D.S. Scott. A type-theoretic alternative to CUCH, ISWIM, OWHY. Oxford notes, 1969.

....EL T is a (conservative) extension of typed predicate logic with equality. ML T is the term language of EL T , therefore ML T is a (conservative) extension of typed equational logic. As typed predicate logic one may take logics such as First Order Logic (FOL) Higher Order Logic (HOL) or LCF ([22, 18]) One may also consider a predicate logic with dependent types. In fact, we need to consider such a logic to be able to express the modalities of EL T . 2.1 Key features of ML T and EL T The syntactic categories of EL T are types, terms and formulas (as in rst order logic with dependent ....

D.S. Scott. A type-theoretic alternative to CUCH, ISWIM, OWHY. Oxford notes, 1969.

....a program M has a value according to the operational semantics i it is provably equivalent to a value (not necessarily the same) in the calculus, but they are too weak for proving equivalences of programs. The denotational approach may suggest important principles, e.g. x point induction (see [Sco93, GMW79]) that can be found only after developing a semantics based on mathematical structures rather than term models, but it does not give clear criteria to single out the general principles among the properties satis ed by the model. The approach adopted in this paper generalises the one followed in ....

D.S. Scott. A type-theoretic alternative to CUCH, ISWIM, OWHY. Theoretical Computer Science, 121, 1993.

....of programs with side e ects has shown the importance of the let constructor (see [Mas88, MT89a, MT89b] In the framework of the computational lambda calculus the importance of let becomes even more apparent. The denotational approach may suggest important principles, e.g. x point induction (see [Sco69, GMW79]) that can be found only after developing a semantics based on mathematical structures rather than term models, but it does not give clear criteria to single out the general principles among the properties satis ed by the model. Moreover, the theory at the heart of Denotational Semantics, i.e. ....

.... or ML (see [HMM90] while in [Mog89a] we propose a modular approach to Denotational Semantics based on the idea of monad constructor (i.e. an endofunctor on the category of monads over a category C) The metalanguage open also the possibility to develop a new Logic of Computable Functions (see [Sco69]) based on an abstract semantic of computations rather than domain theory, for studying axiomatically di erent notions of computation and their relations. Some recent work by Crole and Pitts (see [CP90] has consider an extension of the metalanguage equipped with a logic for inductive predicates, ....

D.S. Scott. A type-theoretic alternative to CUCH, ISWIM, OWHY. Oxford notes, 1969.

.... induction principle, with S Omega inclusive: 8e 2 Omega : 8d 2 Omega : d] e oe d 2 S) oe oe(e) 2 S 8d 2 Omega :d 2 S Just as least fixed points are definable using the universal property of the initial ( algebra Omega# so is Scott s induction principle for least fixed points [16] derivable from the above rule. In order to formulate this induction principle for a fixpoint object within the 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 ....

D.S.Scott, A type-theoretic alternative to CUCH, ISWIM, OWHY, unpublished manuscript, University of Oxford, 1969.

....G. Birtwistle (ed. IVth Higher Order Workshop, Banff 1990, Workshops in 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 ....

D. S. Scott, A type-theoretic alternative to CUCH, ISWIM, OWHY, unpublished manuscript, University of Oxford, 1969.

....operators and arbitrary algebraic data types. We will refer to this language as PCF, since it is based on the calculus considered in Plotkin s seminal paper [Plo77] with pairing and algebraic data types added. 1 If we include algebraic data types of natural numbers and booleans, as in [Plo77, Sco69], then it is easy to program any partial recursive function on the natural numbers. With algebraic data types of trees, lists, stacks, and so on, we may write common functional programs in the style of Miranda or Lazy ML, for example [Tur85] While most sequential implementations of lazy languages ....

....types, there is a traditional and accepted equational proof system. The axioms of this proof system include j equivalence (extensionality) for functions (j)eq x: oe: Mx = M; x not free in M; 1 The language itself seems attributable to Scott, since the basic ideas are presented in the manuscript [Sco69] and Plotkin s name for the calculus is clearly derived from Scott s LCF (Logic for Computable Functions) and the surjective pairing axiom (sp)eq h 1P; 2P i = P; where 1 and 2 are the first and second projection functions. These seem essential for proving common facts about functions and ....

D.S. Scott. A type--theoretic alternative to CUCH, ISWIM, OWHY. Manuscript, 1969.

....fixed point operators and arbitrary algebraic data types. We will refer to this language as PCF, since it is based on the calculus considered in Plotkin s seminal paper [43] with pairing and algebraic data types added. 1 If we include algebraic data types of natural numbers and booleans, as in [43, 50], then it is easy to program any partial recursive function on the natural numbers. With algebraic data types of trees, lists, stacks, and so on, we may write common functional programs in the style of Miranda or Lazy ML, for example [53] While most sequential implementations of lazy languages ....

....evaluation. We will do this in Section 2.8, where we show that a leftmost reduction strategy is complete for finding normal forms, provided the algebraic rules satisfy certain conditions. 1 The language itself seems attributable to Scott, since the basic ideas are presented in the manuscript [50] and Plotkin s name for the calculus is clearly derived from Scott s LCF (Logic for Computable Functions) CHAPTER 2. EXTENSIONALITY IN PCF WITH ALGEBRAIC TYPES 4 For any variant of PCF with equationally axiomatized algebraic data types, there is a traditional and accepted equational proof ....

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D.S. Scott. A type-theoretic alternative to CUCH, ISWIM, OWHY. Manuscript, 1969.

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