Download PDF by Roy Crole (auth.), Roland Backhouse, Roy Crole, Jeremy: Algebraic and Coalgebraic Methods in the Mathematics of

By Roy Crole (auth.), Roland Backhouse, Roy Crole, Jeremy Gibbons (eds.)

ISBN-10: 3540436138

ISBN-13: 9783540436133

Program development is ready turning requisites of software program into implementations. fresh study geared toward bettering the method of application building exploits insights from summary algebraic instruments resembling lattice idea, fixpoint calculus, common algebra, classification conception, and allegory theory.
This textbook-like instructional offers, in addition to an creation, 8 coherently written chapters via major professionals on ordered units and entire lattices, algebras and coalgebras, Galois connections and stuck aspect calculus, calculating sensible courses, algebra of software termination, routines in coalgebraic specification, algebraic equipment for optimization difficulties, and temporal algebra.

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It is common to have such a specification language, and a separate programming language. One might then take a specification, and construct a program from it. The central idea of this chapter is to reason about programs using a language of relations. A function is indeed a relation, and moreover relations satisfy “logical” rules such as conjunction and disjunction. In this chapter, a language and theory of relations is developed which is particularly suitable for specifying programs. The specifications can then be refined, using either calculational techniques or formal theorems, to yield functional programs.

Frequently used properties of maps are contained in the next Mini-exercise. 34 Hilary A. Priestley Mini-exercise (i) Any order-embedding is clearly monotone. Show that it is also one-toone (you will need (po2), antisymmetry of , in P ). Show that not every one-to-one monotone map is an order-embedding (get an example using 2-element posets). (ii) Let F : P → Q and G : Q → R be maps between posets P, Q, R. Show that if F and G are monotone (order-embeddings, order-isomorphisms) then so is the composite G ◦ F : P → R.

Given any poset P (with /P or without ⊥), we form P⊥ (called P ‘lifted’) as follows. Take an element ⊥ ∈ and define on P⊥ := P ∪ {⊥} by x y if and only if x = ⊥ or x y in P. For example, take the natural numbers IN with the antichain order, =. Then IN⊥ is as shown in Figure 7. P⊥ is just {⊥} ⊕ P . A poset of the form S⊥ , where S is an antichain, is called flat. 7 New Posets from Old: Sums and Products Antichains and chains, and the lifting construction, are examples of constructing new posets from existing ones by forming suitable order-theoretic sums.

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Algebraic and Coalgebraic Methods in the Mathematics of Program Construction: International Summer School and Workshop Oxford, UK, April 10–14, 2000 Revised Lectures by Roy Crole (auth.), Roland Backhouse, Roy Crole, Jeremy Gibbons (eds.)


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