R. C. Backhouse et al. Fixed point calculus. Inform. Proc. Letters, 53:131-- 136, 1995. 20 Home/Search Document Details and Download
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A Calculational Approach to Program Inversion - Linacre (2003) (1 citation) (Correct)
....(2.7) The question of when this fixed point is unique has been answered by [31, 32, 30] We will talk more about that in Section 4.6. A calculus of fixed points becomes handy when the, usually simpler but less general, laws on folds do not apply. A summary of fixed point calculus can be found in [5]. Among the many rules, we will only cite the fixed point fusion theorem below and leave the others to be introduced when they are needed. The theorem says that, provided that h is a lower adjoint in a Galois connection, we have: h(f ) g The Galois connection is an important, re occurring ....
....that: #concat # 1 = X P(cpr X ) Refining sets to lists and putting the empty case back, we get the pointwise definition familiar to Haskell programmers, shown in Figure 3.2. This implementation, however, is very ine#cient because of overlapping recursive calls. For example, both ( 1] [2, 3, 4, 5]) and ( 1, 2] 3, 4, 5] are possible splits of [1, 2, 3, 4, 5] To compute its partitions, one will need to recurse on [2, 3, 4, 5] and [3, 4, 5] among others. To compute the partitions of [2, 3, 4, 5] however, another call to partitions [3, 4, 5] will be made. One therefore might wish to ....
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R. C. Backhouse. Fixed point calculus. In R. C. Backhouse, R. Crole, and J. Gibbons, editors, Algebraic and Coalgebraic Methods in the Mathematics of Program Construction, number 2297 in Lecture Notes in Computer Science. Springer-Verlag, 2002.
Least Reflexive Points of Relations - Desharnais, Möller (2002) (Correct)
....is often used to characterize relations that do not have infinite chains, we remark that R is progressively finite i# its converse is well founded. PROPOSITION 2. Let Q and R be relations and f(X) R;X Q. 1. If R is progressively finite, then f has a unique fixed point, viz. R # ;Q [1]. 2. If Q R and R is progressively finite, then Q is progressively finite. We now want to give a short proof of the result of [3] that, for an inflationary relation R on a progressively finite order, iteration from an arbitrary element necessarily leads to a reflexive point. To state this in ....
R. C. Backhouse et al. Fixed point calculus. Inform. Proc. Letters, 53:131--136, 1995.
Kleene Under a Demonic Star - Desharnais, Möller, Tchier (2000) (1 citation) (Correct)
....(0) Because we assume our KAs to be complete lattices (Definition 2.1) least and greatest fixed points of monotonic functions exist. Definition 3.3. Let (X, be an ordered set. The relation on the set of functions on X is defined by f # #(x X : f(x) g(x) Theorem 3.4. See, e.g. [3]) The following properties hold: a) f (g) monotonic) b) Let g be continuous and strict. Then f (f) g( h) fusion law) c) f g) f( g f) permutation law) Analogous laws hold for the greatest fixed point. This can be shown more easily than by direct proof using the ....
....lattice. The theorem entails the following corollary that highlights the importance of progressive finiteness in the simplification of fixed point related properties. Corollary 3.14. Let again f(x) x b. If a is progressively finite, then f has a unique fixed point, viz. a # b [3]. To conclude this section, we study analogous iterations at the level of types. Theorem 3.15. Let t be a type and set, for x TYP, h(x) #(a x) t and t x) a) x h(x) b) h) #(a # t) c) #(h) #(a # t) d) e) k) a # (f) #(k) a # . ....
R. C. Backhouse et al. Fixed point calculus. Inform. Proc. Letters, 53:131--136, 1995.
Computing Kernels in Directed Bichromatic Graphs - von Karger, Berghammer (1997) (1 citation) (Correct)
....For finitary graphs our result could be obtained as a corollary of Richardson s theorem, but like Schmidt and Strohlein, we prove our theorem without any finiteness assumption. Our proof is conducted entirely in Tarski s axiomatic calculus of binary relations [8] and the fixed point calculus [1]. This formal approach has two benefits. On the practical side, the proof is constructive and yields an algorith1 Since kernel theory is primarily concerned with directed graphs we adapted the notion of bichromaticity accordingly. In the literature, bichromaticity seems to be defined only for ....
Backhouse R. et al.: Fixed-point calculus, Inf. Proc. Letters 53, 131-136 (1995)
Lazy Kleene Algebra - Möller (2003) (Correct)
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R. C. Backhouse et al. Fixed point calculus. Inform. Proc. Letters, 53:131-- 136, 1995. 20
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