Abstract
A new typed, higher-order logic is described which appears particularly well fitted to reasoning about forms of computation whose operational behaviour can be specified using the Natural Semantics style of structural operational semantics [5]. The logic’s underlying type system is Moggi’s computational metalanguage [11], which enforces a distinction between computations and values via the categorical structure of a strong monad. This is extended to a (constructive) predicate logic with modal formulas about evaluation of computations to values, called evaluation modalities. The categorical structure corresponding to this kind of logic is explained and a couple of examples of categorical models given.
As a first example of the naturalness and applicability of this new logic to program semantics, we investigate the translation of a (tiny) fragment of Standard ML into a theory over the logic, which is proved computationally adequate for ML’s Natural Semantics [10]. Whilst it is tiny, the ML fragment does however contain both higher-order functional and imperative features, about which the logic allows us to reason without having to mention global states explicitly.
Research supported by the CLICS project (ESPRIT BR Action nr 3003).
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References
R. L. Croie and A. M. Pitts, New Foundations for Fixpoint Computations, Proc. 5th Annual Symposium on Logic in Computer Science, Philadelphia (IEEE Computer Society Press, Washington, 1990) 489–497.
R. L. Croie and A. M. Pitts, New Foundations for Fixpoint Computations: FIX-Hyperdoctrines and the FIX-Logic, University of Cambridge Computer Laboratory Technical Report No. 204, August 1990.
M. Dummett, Elements of Intuitionism (Oxford University Press, 1977).
C. Gunter and D. S. Scott, Semantic Domains. Chapter in Handbook of Theoretical Computer Science (North-Holland, Amsterdam, 1990).
G. Kahn, Natural Semantics. In K. Fuchi and M. Nivat (eds), Programming of Future Generation Computers (Elsevier Science Publishers B. V. ( North-Holland ), Amsterdam, 1988 ) 237–258.
J. W. Klop, Combinatory Reduction Systems, Amsterdam Mathematical Center Tracts 129 (1980).
D. Kozen and J. Tiuryn, Logics of Programs. Chapter in Handbook of Theoretical Computer Science (North-Holland, Amsterdam, 1990).
J. Lambek and P. J. Scott, Introduction to Higher Order Categorical Logic, Cambridge Studies in Advanced Mathematics 7 (Cambridge University Press, 1986).
F. W. Lawvere, Equality in Hyperdoctrines and the Comprehension Schema as an Adjoint Functor. In A. Heller (ed.), Applications of Categorical Algebra (Amer. Math. Soc., Providence RI, 1970 ) 1–14.
R. Milner, M. Tofte and R. Harper, The Definition of Standard ML (The MIT Press, Cambridge Massachussetts, 1990 ).
E. Moggi, Computational lambda-calculus and monads, Proc. 4th Annual Symposium on Logic in Computer Science, Asilomar CA (IEEE Computer Society Press, Washington, 1989) 14–23.
E. Moggi, Notions of Computations and Monads, preprint, 1989.
E. Moggi, Lecture notes on An Abstract View of Programming Languages, July 1989.
P. D. Mosses, Denotational Semantics. Chapter in Handbook of Theoretical Computer Science ( North-Holland, Amsterdam, 1990 ).
B. Nordström, K. Petersson and J. M. Smith, Programming in Martin-Löf’s Type Theory, An Introduction (Oxford University Press, 1990 ).
G. D. Plotkin, Call-by-Name, Call-by-Value and the.1-Calculus, Theoretical Computer Science 1 (1977) 125–159.
G. D. Plotkin, LCF considered as a programming language, Theoretical Computer Science 5 (1977) 223–255.
G. D. Plotkin, A Structural Approach to Operational Semantics, Aarhus University Computer Science Department Report DAIMI FN-19, 1981.
G. D. Plotkin, Denotational semantics with partial functions, unpublished lecture notes from CSLI Summer School, 1985.
D. S. Scott, A type-theoretic alternative to CUCH, ISWIM, OWHY, unpublished manuscript, University of Oxford, 1969.
R. A. G. Seely, Hyperdoctrines, Natural Deduction and the Beck Condition, Zeitschr. f. math. Logik und Grundlagen d. Math. 29 (1983) 505–542.
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© 1991 British Computer Society
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Pitts, A.M. (1991). Evaluation Logic. In: Birtwistle, G. (eds) IV Higher Order Workshop, Banff 1990. Workshops in Computing. Springer, London. https://doi.org/10.1007/978-1-4471-3182-3_11
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DOI: https://doi.org/10.1007/978-1-4471-3182-3_11
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