By J. Avenhaus, K. Madlener (auth.), Ding-Zhu Du, Ker-I Ko (eds.)

This publication incorporates a number of survey papers within the parts of algorithms, lan guages and complexity, the 3 parts during which Professor Ronald V. booklet has made major contributions. As a fonner pupil and a co-author who've been encouraged through him without delay, we want to devote this ebook to Professor Ronald V. publication to honor and have fun his 60th birthday. Professor booklet initiated his outstanding educational profession in 1958, graduating from Grinnell university with a Bachelor of Arts measure. He acquired a grasp of Arts in educating measure in 1960 and a grasp of Arts measure in 1964 either from Wesleyan collage, and a physician of Philosophy measure from Harvard collage in 1969, less than the counsel of Professor Sheila A. Greibach. Professor Book's learn in discrete arithmetic and theoretical com puter technology is mirrored in additional than a hundred and fifty clinical courses. those works have made a robust effect at the improvement of a number of components of theoretical laptop technological know-how. A extra distinctive precis of his clinical learn looks during this quantity separately.

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**Example text**

For denoting specifications we identify a conditional rewrite system with a set of conditional equations. This is done in the next definition. 6. 2 Let spec = (sig, R) be a hiemrchical specification over speCo = (sigo,Ro). +l t 19 if s ---7? t or for some rule f ===? 1 -t r in R o, p E O(s) and substitution a we have sJp == a(l), t == s[a(r)]p and ---7? satisfies a(r). Then ---7Ro= UiEN ---7? b) We define an approximation (---7i)iEN of ---7 R on Term(F) by ---70=---7 Ro and s ---7i+l t if s ---7i t or for some rule fj I).

Second, tl = t2 may be valid without E-support of this fact. For example [19], let A be given by Fa = {true, false} and Eo = 0 and let not be partially defined by El = {not(true) = false}. Then not(not(false)) = false is valid, since this equation holds in all total Fo-generated models B of spec: We either have not(false) = true or not (false) = false in B, in both cases not(not(false)) = false holds in B. But there is no Eo U El-support to prove not(not(false)) = false to be valid. Now we comment on inductive theorem proving.

We associate to spec the quotient algebra Aspec = Term(F)/ = E of the free term algebra Term(F) according to a suitably defined E-equality =E. Note also that we have to define carefully how to apply a positive/negative conditional THEOREM PROVING IN HIERARCHICAL CLAUSAL SPECIFICATIONS 3 equation, in particular how to evaluate the negative conditions. Since we do not restrict to positive conditional equations, an initial model of spec in the sence of first order logic may not exist. Nevertheless, Aspec is defined such that it contains (an isomorphic copy of) A and is initial in a natural class of models of spec.