D. Pavlovi'c, "Guarded Induction on Final Coalgebras," in B. Jacobs et al (eds) Proc. Coalgebraic Methods in Computer Science, ENTCS vol.18 (1998), 143--160.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....C is Set, the usual category of sets. What we shall use is that C has a final object, and all limts, and also that for each natural number n, the set a n (an object of C) corresponds to Hom(n; a) the set a n of functions from n to a. Fix a functor F and a final F coalgebra hc; i. Pavlovi c [7] is interested in operations ffi : c n c n which have unique fixed points. His paper discusses two conditions on operations which guarantee the existence and uniqueness of fixed points. The simplest condition is that of being prefixing. To simplify things a bit, we work with n = 1. An ....

..... We define j by: for all sets A, j A is b 7 F b(w) That is, for b 2 A, consider b : 1 A, then F b : F1 FA, and finally F b ffi w : 1 FA. This gives an element of FA. The naturality of j is easily verified. And then the diagram above shows that ffi = f kffiw = ffl ffi j c . a Pavlovi c [7] is mainly concerned with the guarded operations on final coalgebras. We shall not repeat the definition here, but we note that in this example, the guarded operations properly contain the substitutive ones. For example, the operation ff 7 (0; ff 0 ; 0; ff 1 ; 0; ff 2 ; 0; Delta Delta Delta ....

D. Pavlovi'c, "Guarded Induction on Final Coalgebras," in B. Jacobs et al (eds) Proc. Coalgebraic Methods in Computer Science, ENTCS vol.18 (1998), 143--160.

.... again: exp exp = 1 : exp) 1 : exp) 1 1) exp exp) 2 : exp exp) But how do we know that there is a fixpoint How do we know whether an equation like y = 0 : 3 : y Gamma 5 sin x x 2 ) has a unique solution A general answer to such questions is provided in [11]. 2.2 Using Taylor series To see things in a familiar setting, note that the Taylor representation again induces an isomorphism: A T = F NaN F NaN hO;Di F NaN F NaN R Theta A R ThetaT = F NaN F NaN R e T F NaN F NaN SS hhead;taili F NaN F NaN R Theta R ....

....order equations [3, ch. 10] In fact, the general Cauchy Kowalevskaya method for solving initial value problems [6, ch. 3] boils down to a recursive procedure for computing the Taylor coefficients, and can be presented in a coinductive setting as above, suitably extended for several variables [11]. Of course, many important problems are not in a form suitable for this simple approach; and those that are often lead to intractable recurrence relations on the coefficients. But the methods devised for dealing with such cases, also tend to lead to stream algebras and their homomorphisms, albeit ....

D. Pavlovi'c, Guarded induction on final coalgebras. University of Sussex Report CS 98/1, pp. 18

....and process calculus, this issue has mostly been addressed at the level of syntax [2, 7, 13] The extant characterizations of the guardedness are syntactic conditions that apply to specific languages, and ensure unique fixpoints (e.g. 13, sec. 3. 2] We initiated an abstract semantic analysis in [16], but only covered a special class of stream flavored coalgebras. In the meantime, Moss [14] has proposed a different approach, that avoids this restriction on coalgebras, but only captures what seems to be a fairly restricted subclass of the guarded opera2 tions that come about in practice. The ....

....In the meantime, Moss [14] has proposed a different approach, that avoids this restriction on coalgebras, but only captures what seems to be a fairly restricted subclass of the guarded opera2 tions that come about in practice. The present paper outlines a new approach, different from both [16] and [14] and provides a full, unrestricted account of all guarded operations on arbitrary final coalgebras. Even for streams, the existence and the uniqueness of the fixpoints of guarded equations is not as trivial as it may appear from the above examples. For instance, if interpreted along the ....

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D. Pavlovi'c, Guarded induction on final coalgebras. Electr. Notes in Theor. Comput. Sci. 11(1998) 1--18

....principle [2] Before that, it was introduced and used mostly in the semantics of concurrency [13] It has by now been presented from many different angles: 1,8,12,16 18] to name just a few contributors. Why would so foundational a principle wait for the late 20th century to be discovered In [14,16] the idea was put forward that coinduction is new only by name, while it had actually been around for a long time, concealed within the infinitistic methods of mathematical analysis. Roughly, induction arithmetic coinduction analysis The infinitary constructions in elementary calculus are ....

....x 0 . Real numbers are thus presented as streams of pairs of integers. The mentioned coalgebraic structure, although never spelled out, is then employed in the subsequent constructions, as well as in the discussion touching upon some of the still very active themes, such as guarded induction [14,15], or the role of redundancy in representation. It may seem ironic that what we now consider to be a very general computational method had an early brief appearance, even on the background, among the primordial hacks . But this is probably just an instance of the irony of language. In any case, ....

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D. Pavlovi'c, Guarded induction on final coalgebras. Electr. Notes in Theor. Comput. Sci. 11(1998) 1--18 Pratt

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