By T. H. Tse
Established methodologies are a well-liked and strong instrument in details structures improvement. many alternative ones exist, every one making use of a few versions and so a specification needs to be switched over from one shape to a different throughout the improvement strategy. to unravel this challenge, Dr. Tse proposes a unifying framework at the back of well known established types. He techniques the matter from the viewpoints of algebra and class thought. He not just develops the frameworks but in addition illustrates their functional and theoretical usefulness. therefore, this e-book will offer perception for software program engineers into how methodologies may be formalized, and should open up various purposes and difficulties for theoretical machine scientists.
Read or Download A Unifying Framework for Structured Analysis and Design Models: An Approach Using Initial Algebra Semantics and Category Theory PDF
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Additional resources for A Unifying Framework for Structured Analysis and Design Models: An Approach Using Initial Algebra Semantics and Category Theory
A homomorphism h: A -> B is defined as an isomorphism if and only if there exists a homomorphism /:: B -> A such that k o h = /\ and h o k = iB. In this case we write £ = A"1. Two algebras A and B over the same signature are said to be isomorphic to each other if and only if there exists an isomorphism h: A —> B. 1 Let A be an initial algebra over the signature £. If B is another algebra over the same signature, then B is initial if and only if A is isomorphic to B. Proof: Suppose A is isomorphic to B.
Let k be any homomorphism from X to A. By definition, it has the following two properties: (kl) For any symbol a in E A , k(ax) = a , . , k(tn)) We note that (kl) is equivalent to (/zl) because cx in (/:1) is in fact the term " a " in (hi). , t ) in (k2) is in fact the term t in (/z2). Hence any homomorphism k: X —> A satisfies (hi) and (A2). In other words, there exists a unique homomorphism h mapping X to A. O We can therefore arrive at the main theorem on the proposed term algebra. 4 The term algebra T^ is an initial algebra over Z.
By definition, it has the following two properties: (kl) For any symbol a in E A , k(ax) = a , . , k(tn)) We note that (kl) is equivalent to (/zl) because cx in (/:1) is in fact the term " a " in (hi). , t ) in (k2) is in fact the term t in (/z2). Hence any homomorphism k: X —> A satisfies (hi) and (A2). In other words, there exists a unique homomorphism h mapping X to A. O We can therefore arrive at the main theorem on the proposed term algebra. 4 The term algebra T^ is an initial algebra over Z.
A Unifying Framework for Structured Analysis and Design Models: An Approach Using Initial Algebra Semantics and Category Theory by T. H. Tse