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Example text

In logic programming, where the programmer acts as speaker, the processor as auditor, and the user as questioner, soundness of the programming system guarantees that all outputs constitute correct answers. Various forms of completeness guarantee that answers will always be produced when they exist. 2. There is a close formal correspondence between programming systems and pairs of proof and query systems: inputs correspond to questions, programs correspond to sets of hypotheses, computations to proofs, and outputs to theorems (for a di erent correspondence, in which programs in the form of lambda terms correspond to natural deduction proofs, see How80, Tai67, CAB+ 86]|compare this to the interpretation of formulae as queries and proofs as answers Mes89]).

If P is complete, then semantic correctness implies provable correctness. Going back to the communication analysis of previous sections, let Ks be the speaker's implicit knowledge, let K0a be the auditor's initial implicit knowledge, and let T0a be the auditor's initial explicit knowledge. When the speaker utters a set of formulae Tu , consistent with her implicit knowledge, the auditor's implicit knowledge improves as before to K1a =1 K0a \0 Models(Tu), and the auditor's explicit knowledge improves to Ta = Ta Tu .

For example, the primitives cons (construct ordered pair), car ( rst component of pair), and cdr (second : component of pair) in Lisp are de ned by the two equations (car (cons (x y)) = x) and (cdr (cons (x y)) =: y) McC60]. Primitive operators that manipulate term structure, or provide program control structures, are usually dened by explicitly given small nite sets of equations. Primitive operators for basic mathematical operations are de ned by large or in nite sets of equations that must be described rather than listed.

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