Computing in Horn Clause Theories by Peter Padawitz

By Peter Padawitz

At least 4 examine fields detennine the theoretical history of specification and deduction in computing device technology: recursion thought, automatic theorem proving, summary info kinds and tenn rewriting platforms. As those parts strategy one another progressively more, the powerful differences among useful and relational perspectives, deductive and denotational methods in addition to among specification and programming are relieved in favour in their integration. The booklet won't reveal the strains of this improvement; conversely, it starts off out from the nucleus of Hom clause common sense and brings forth either recognized and unknown effects, such a lot of which have an effect on a couple of of the fields pointed out above. bankruptcy 1 touches on old problems with specification and prototyping and delimits the themes dealt with during this e-book from others that are on the middle of comparable paintings. bankruptcy 2 offers the elemental notions and notations wanted for the presentation and interpretation of many-sorted Horn clause theories with equality. bankruptcy three offers a couple of pattern Hom clause standards starting from mathematics via string manipulation to better information constructions and interpreters of programming languages. a few of these examples function a connection with illustrate definitions and effects, others may possibly throw a gentle at the robust hyperlink among standards and courses, that are done through using deduction ideas. therefore we now have integrated examples of the way to exploit application trans/ormation tools in specification design.

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T. all A E A. t. t. 2 The Horn Clause Calculus Let F be a set of formulas. g. all clauses over SIG. and let R be a set of inference rules. Given C S; F and c E F, a derivation of c from C via R is a sequence . we simply write Co I- c. To be more precise. we should call (*) a rule instance rather than a rule. An inference rule only bears the schema of formulas before and after a transformati on step.

Some of them correspond to the theory of a single SIG-structure, say B. t. A if A coincides with B. It is not only the sort of axioms that determines on which semantical concept should be adopted. Different kinds of questions also suggest different theories. For instance, if we expect a positive answer, free or initial semantics is mostly the right choice, while final semantics is more suitable for finding a refutation. We begin with the Horn clause calculus consisting of the Substitution Rule and the Cut Rule.

The ba1anced predicate of BALANCED2 can be adapted immediately to 2-3trees: BALANCED3 base: preds : vars: TWO-THREE-TREE _ is balanced: tree height _ is _ : tree,nat m,n,k : nat 45 axms: x,y,z : tree x is balanced += height x is n height U) is 0 height [x]m[y] is kl += height x is k, height y is k height [x]m[y]n[z] is kl += height x is k, height y is k, height z is k When specifying the insertion operator with subsequent rebalancing we have to take into account that rebalancing must be finished before further insertions are started.

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