By Hartmut Ehrig
The goal of this e-book is to give basics of algebraic requisites with admire to the next 3 points: basics within the feel of a delicately encouraged advent to algebraic standards, that is effortless to appreciate for laptop scientists and mathematicians; basics within the experience of mathematical theories that are the root for exact definitions, structures, effects, and correctness proofs; and basics within the experience of options, that are brought on a conceptual point and formalized in mathematical phrases. The e-book is both suitableas a textual content e-book for graduate classes and as a reference for researchers and method developers.
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Extra resources for Fundamentals of Algebraic Specification 1: Equations and Initial Semantics
Example text
N) f([ci]) = for ci ~ am' ... a1' Calph We will show the following further properties: (6) For each t (7) For cl,c2 ~ ~ TOp there is c Cop cl = c2 ~ COP = Calph u Cstring s. t. c _ t implies = c2 cl Both properties together mean that each congruence class [t] for t exactly represented by one canonical term c and (5) COp. ~ ~ TOp is This together with (4) shows that ;he component functions f alph : QalPh '" A and f string''Qstring '" A of f are bijective. morphism as required in (0 Hence f'T .
Then the following is true: X = 0, then e is valid in A if and only if eval(L) = eval(R). the empty mapping If This is because 0:0 - A is the only assignment and thus eval is the unique evaluation. If X ~ 0 and especially Xs ~ 0, then there is no assignment function, and thus e = (X,L,R) is true in A no matter, what Land R look like. If A has no empty base sets, then validity of equations e = (X,L,R) and e' (X' ,L,R) is equivalent over A. In this case we can leave X unspecified and choose the one which contains exactly those variables occuring in Land R.
Stacks are slightly more difficult than strings because we also have to consider error and exception cases. We are not going to present a theory of error and exception handling here. but the ideas given in this example can be extended to several other abstract data types of interest in computer science. 13 EXAMPLE (ADT{STACK» 1. Concept A stack in computer science is a storage device where data items from a given are stored in a sequence. domain We only have access to the topmost item. Access to inner items is only possible by first removing from the top one by one all items above the items to be accessed.