Logic and programming languages

ABSTRACT

Logic has been long interested in whether answers to certain questions are computable in principle, since the outcome puts bounds on the possibilities of formalization. More recently, precise comparisons in the efficiency of decision methods have become available through the developments in complexity theory. These, however, are applications to logic, and a big question is whether methods of logic have significance in the other direction for the more applied parts of computability theory. Programming !anguages offer an obvious opportunity as their syntactic formalization is well advanced; however, the semantical theory can hardly be said to be complete. Though we have many examples, we have still to give wide-ranging mathematical answers to these queries: What is a machine? What is a computable process? How (or how well) does a machine simulate a process? Programs naturally enter in giving descriptions of processes. The definition of the precise meaning of a program then requires us to explain what are the objects of computation (in a way, the statics of the problem) and how they are to be transformed (the dynamics).

So far the theories of automata and of nets, though most interesting for dynamics, have formalized only a portion of the field, and there has been perhaps too much concentration of the finite-state and algebraic aspects. It would seem that the understanding of higher-level program features involves us with infinite objects and forces us to pass through several levels of explanation to go from the conceptual ideas to the final simulation on a real machine. These levels can be made mathematically exact if we can find the right abstractions to represent the necessary structures.

The experience of many independent workers with the method of data types as lattices (or partial orderings) under an information content ordering, and with their continuous mappings, has demonstrated the flexibility of this approach in providing definitions and proofs, which are clean and without undue dependence on implementations. Nevertheless much remains to be done in showing how abstract conceptualizations can (or cannot) be actualized before we can say we have a unified theory.

References

1. Bohm, C., Ed. λ-Calculus and Computer Science Theory. Lecture Notes in Computer Science, Vol. 37. Springer-Verlag, New York, 1975.
2. Clark, K. L., and Cowell, D. F. Programs, Machines, and Computation. McGraw-Hill, New York, 1976.
3. Crossley, J. N., Ed. Algebra and Logic Papers from the 1974 Summer Res. Inst. Australian Math. Soc., Monash U. Clayton, Victoria, Australia. Lecture Notes in Mathematics, Vol. 450, Springer-Verlag, 1976.
4. Donahue, J. E. Complementary Definitions of Programming Language Semantics. Lecture Notes in Computer Science, Vol. 42, Springer-Verlag, 1976.
5. Eilenberg, S. Automata, Languages, and Machines. Academic Press, New York, 1974.
6. van Emden, M. H., and Kowalski, R. A. The semantics of predicate logic as a programming language. J. ACM 23, 4 (Oct. 1976), 733--742.
7. Manes, E. G., Ed. Category Theory Applied to Computation and Control. First Int. Symp. Lecture Notes in Computer Science, Vol. 25, Springer-Verlag, New York, 1976.
8. Manna, Z. Mathematical Theory of Computation. McGraw-Hill, New York, 1974.
9. Milne, R., and Strachey, C. A. Theory of Programming Language Semantics. Chapman and Hall, London, and Wiley, New York, 2 Vols., 1976.
10. Plotkin, G. D. A powerdomain construction. SIAM J. Comptng. 5 (1976), 452--487.
11. Rabin, M. O., and Scott, D. S. Finite automata and their decision problems. IBM J. Res. and Develop. 3 (1959), 114--125.
12. Scott, D. S. Data types as lattices. SIAMJ. Comptng. 5 (1976), 522--587.
13. Stoy, J. E. Denotational Semantics--The Scott-Strachey Approach to Programming Language Theory. M.I.T. Press, Cambridge, Mass.
14. Tennent, R. D. The denotational semantics of programming languages. Comm. ACM 19, 8 (Aug. 1976), 437--453.

Index Terms

• Mathematics of computing

  • Continuous mathematics

    • Calculus

      • Lambda calculus

  • Mathematical analysis

    • Calculus

      • Lambda calculus

• Theory of computation

  • Formal languages and automata theory

    • Formalisms

      • Algebraic language theory

  • Logic

    • Constraint and logic programming

  • Semantics and reasoning

    • Program semantics

      • Algebraic semantics

      • Denotational semantics