Computing in Horn Clause Theories by Peter Padawitz

By Peter Padawitz

At least 4 examine fields detennine the theoretical heritage of specification and deduction in desktop technology: recursion idea, automatic theorem proving, summary information kinds and tenn rewriting platforms. As those parts procedure one another a growing number of, the robust differences among practical and relational perspectives, deductive and denotational ways in addition to among specification and programming are relieved in favour in their integration. The publication won't reveal the traces 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 impact a couple of of the fields pointed out above. bankruptcy 1 touches on historic problems with specification and prototyping and delimits the subjects dealt with during this booklet from others that are on the center 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 requisites starting from mathematics via string manipulation to raised information buildings and interpreters of programming languages. a few of these examples function a connection with illustrate definitions and effects, others may well throw a gentle at the powerful hyperlink among requisites and courses, that are achieved by way of utilizing deduction principles. hence now we have incorporated examples of ways to take advantage of software trans/ormation tools in specification design.

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Xn=Yn P. Y1 ..... Yn E X. AX>. AX). consists of all atoms p such that AXuEAX I- p. t. AX) (cf. 2). 1 Let = BINARV (cf. 5). A derivation of the atom 11<100 from AXuEAX via the Horn clause calculus reads as follows: a 1 < bO += a < b 01<10+=0<1 0<1 01 < 10 01al )Ca)C 10al0 Y1 < yz += Xi < xz, x 1aY1, Xzayz 1 < 10 += 01 < 10, 0lal, 10al0 1 < 10 += 01al,10al0 1 < 10 a 1 < bO += a < b 11 < 100 += 1 < 10 11 < 100 € AX (subst i tut ion) € AX (cut) AX € EAX (substitution) € EAX (substitution) (cut) (cut, cut) € € AX (substitution) (cut).

For instance, if dom(f) = (s,O,1 ,2, 10, 11), f(s) =0, f(O) = 1, «1) =2, f(2) =5, f(10) =3 and f(l1) = 4, then f stands for the tree 48 In general, the empty sequence £ denotes the root, while a sequence of the form wn with w E IN * and n E IN represents the (n+ I)th successor of the node given by w. This interpretation leads to the following a)(ioms that characterize a set D £; IN * as a tree domain: • • wO E D implies wED, w(n+ 1) E D implies wn E D. A first approach to a corresponding Horn clause specification follows the lines of STORE (cf.

SEQ-OF-NAT factseq _: nat---+seq n : nat factseq 0 == 1&E vars: axms: fact seq nl == (fact nl)&(factseq n) Burstall and Darlington. p. FACT-SEQ where fact O. fact each other. Instead. e. fact t. is 50. propose the following optimization of 1. fact 2 •... are not computed independently of element of the sequence (denoted by the ground used to compute fact t/ and thus factseq t/. The transformat ion is factseq2 n == factseq2 0 == factseq2 nl == == «fact n)*nl.

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