Download Algebra and Coalgebra in Computer Science: 4th International by Giorgio Bacci, Vincent Danos, Ohad Kammar (auth.), Andrea PDF

By Giorgio Bacci, Vincent Danos, Ohad Kammar (auth.), Andrea Corradini, Bartek Klin, Corina Cîrstea (eds.)

This ebook constitutes the refereed complaints of the 4th foreign convention on Algebra and Coalgebra in computing device technological know-how, CALCO 2011, held in Winchester, united kingdom, in August/September 2011. The 21 complete papers awarded including four invited talks have been rigorously reviewed and chosen from forty-one submissions. The papers file result of theoretical paintings at the arithmetic of algebras and coalgebras, the best way those effects can aid equipment and strategies for software program improvement, in addition to adventure with the move of the ensuing applied sciences into commercial perform. They conceal themes within the fields of summary types and logics, really good types and calculi, algebraic and coalgebraic semantics, and approach specification and verification. The e-book additionally contains 6 papers from the CALCO-tools Workshop, colocated with CALCO 2011 and devoted to instruments in response to algebraic and/or coalgebraic principles.

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I∈F It then can be shown that is still associative and multiplication distributes over from both the left and the right [DKV09]. Note that in a ω-continuous semiring S we have 0 a for all a ∈ S. t. the canonical order ≤, but the nonnegative reals do. It is easy to see that Kleene’s fixed-point theorem applies to polynomial systems over ω-continuous semirings:3 3 The theorem is often also attributed to Tarski. In fact, it can be seen as a slight extension of Tarski’s fixed-point theorem for complete lattices [Tar55], or as a particular case of Kleene’s first recursion theorem [Kle38].

Stream(X). will systematically produce all finite streams one by one, starting from the [] stream. Suppose now we remove the base case and obtain the program P2: stream([H|T]) :- number(H), stream(T). - stream(X). fails, since the model of P2 does not contain any instances of stream/1. The problems are two-fold: (i) the Herbrand universe does not contain infinite terms; (ii) the least Herbrand model does not allow for infinite proofs, such as the proof of stream(X); yet these concepts are commonplace in computer science, and a sound mathematical foundation exists for them in the field of hyperset theory [2].

Fix a pumpable tree t with pumpable factorization uvxyz as schematically described in Figure 1(a) where the middle (grey) and the lower (dark grey) part are derived from the same nonterminal, and the top part (white) may be empty. If t has height at most n + 1, we set t1 := t and are done. Otherwise, one of the three parts of t (white, grey, or dark grey) contains a subtree of height at least n + 1. Since G only has n variables, this subtree is also pumpable. We only consider the case that the pumpable tree is on the left part of the white zone (other cases are similar).

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