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

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

Program building is ready turning requirements of software program into implementations. contemporary learn geared toward bettering the method of application building exploits insights from summary algebraic instruments corresponding to lattice conception, fixpoint calculus, common algebra, classification conception, and allegory theory.
This textbook-like instructional provides, along with an advent, 8 coherently written chapters by way of top professionals on ordered units and entire lattices, algebras and coalgebras, Galois connections and glued 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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Read or Download Algebraic and Coalgebraic Methods in the Mathematics of Program Construction: International Summer School and Workshop Oxford, UK, April 10–14, 2000 Revised Lectures PDF

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Additional resources for Algebraic and Coalgebraic Methods in the Mathematics of Program Construction: International Summer School and Workshop Oxford, UK, April 10–14, 2000 Revised Lectures

Example text

Much of the motivation comes from the need to have semantic models supporting recursion. 1 Lattices Consider the posets depicted in Figure 9. In (a) we have ↑a ∩ ↑b = ∅. In (b) we find that ↑a ∩ ↑b = {c, d}. Similar considerations apply to the down-set operator. For points x, y in a poset P there may be a point z ∈ P such that ↓x ∩ ↓y = ↓z, or this may fail either because the intersection is empty or because it is not of the form ↓z. By contrast, if we look at a powerset ℘(X) we find easily that, for any subsets A, B of X, there exists C ∈ ℘(X), namely C = A ∪ B, such that ↑A ∩ ↑B = ↑C; and similarly for ↓.

In such a setting we may wish to model negation in a less restrictive way, retaining the property that P ∧ ¬P = F (‘not both of P and ¬P are true’), but discarding the Law of the Excluded Middle, P ∨ ¬P = T, and substituting something weaker. 5. We may also wish to allow truth values other than the booleans F and T: for example, we might want to accommodate a third value, P, representing ‘possible’ or ‘not yet determined’. Or we might, as in probabilistic models and in fuzzy logic, wish to allow truth values lying in the interval [0, 1].

Then, by (i), F −1 : U(Q) → U(P ) is a well defined map. (a) Prove that F is an order-embedding if and only if F −1 maps U(Q) onto U(P ). (b) Prove that F maps P onto Q if and only if F −1 : U(Q) → U(P ) is one-to-one. 11). 13 Down Is Nice Too We can define down-sets of a poset P in just the same manner as we defined up-sets and form the poset down-sets, O(P ), carrying the inclusion ordering. The symbol O is traditional here—O stands for ‘order ideal’, a synonym for ‘down-set’. 10. Mini-exercise Draw a labelled diagram of O(P ) for the poset P in Figure 6.

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