S. Bloom and Z. Esik. Iteration Theories. EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... on control categories comes from our previous work about fixed point operators on the # calculi in [4] and [6] The equational theories of fixed point operators in call by name # calculi have been studied extensively, and now there are some canonical axiomatizations including iteration theories [1] and Conway theories, equivalently traced cartesian categories [3] see [12] for recent results) Because the # calculus is an extension of the simply typed call by name # calculus, it looks straightforward to consider fixed point operators in the # calculus and indeed we have considered an ....

S. Bloom and Z. Esik. Iteration Theories. EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1993.

....monotone operators seems possible. We might pursue this option in the future. # Corresponding author : Equipe PPS, Universite Paris 7 Denis Diderot, Case 7014, 2 Place Jussieu 75251 PARIS Cedex 05, Vincent.Danos pps.jussieu.fr 1 Preliminaries Definition 1 (LMP) S, #, h : L [0, 1]) is a Labelled Markov Process (LMP) if (S, #) is a measurable space, for all a L, A #, h(a, s, A) is # measurable as a function of s and for all s S, h(a, s, A) is a subprobability as a function of A. Some particular cases: 1) when S is finite and # = 2 we have the familiar probabilistic ....

....for all s S, h(a, s, A) is a subprobability as a function of A. Some particular cases: 1) when S is finite and # = 2 we have the familiar probabilistic automaton, 2) when h(a, s, A) doesn t depend on s or on a we have the familiar (sub)probability triple. An example of the latter situation is ([0, 1], h) with h(a, s) B) #(B) with # the Lebesgue measure on Borelians. Definition 2 (shifts) For a L, r [0, 1] one defines endomaps of #, shifts and strict shifts as: h(a, s) A) r h(a, s) A) r Shifts are cocontinuous and strict shifts are continuous, but they have the empty set ....

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S. Bloom and Z. Esik. Iteration theories. EATCS Monographs on Theoretical Computer Science, 1993.

....the duality is a kind of De Morgan duality [2] iv Yoshihiko Kakutani Recursion and iteration Recursion is indispensable for programming languages and has been studied very extensively. However, most of such widely known studies, which includes uniformity, dinaturality, and diagonal property [1], is for call by name languages rather than for call by value ones. Therefore, we are to add a recursion operator to the n calculus and to consider the duality. By the duality between values and continuations, we can get recursion on continuations in the call by value calculus from ....

S. Bloom and Z.  Esik. Iteration Theories. EATC Monographs on Theoretical Computer Science. Springer-Verlarg, 1993.

....of regular events. Besides formal languages and automata, Kleene algebras also arise, for instance, in the context of relation algebra (c.f. 20] and logics, analysis and construction of programs (c.f. 11] We follow Kozen s de nition [13] These structures are called Kozen semirings in [3]. For concise proofs of all statements in this text see [24] Some formal proofs with the Isabelle proof checker can be found in [25] A semiring is an algebra (A; 0; 1) of a set A, two binary operations of multiplication and addition and two constants 0 and 1. Thereby (A; 0) is a ....

....There is a deductive system for the universal Horn theory of Kleene algebra such that an equation between Kleene algebra terms follows from the axioms of Kleene algebra i the two terms denote the same regular set. Completeness results for related structures have been given, for instance, in [5,3]. Remind that there is no nite equational axiomatization of the algebra of regular events [19] By theorem 2, every identity in Kleene algebra can be decided by automata. In contrast, Kleene algebra is incomplete for the Horn theory of regular events [13] decidability of the Horn theory of ....

S. L. Bloom and Z.  Esik. Iteration Theories. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1991.

....from g , 3 which is not related at all to x :x. Using the infinite parallel rewriting of [7] instead, we have that g rewrites to , the bottom element of the cpo of terms, and is indeed the canonical interpretation of the term x :x, according to the Iteration Algebras framework [3]. An infinite term made of infinitely many nested redexes of collapsing rules (as g in this example) will be called a hypercollapsing 3 Actually such a derivation is not strongly convergent, and thus it is not considered admissible in [22] tower , using the terminology of [22] This ....

....of cpo s and continuous functions. ut We denote with Sigma CAlg the category of continuous algebras and strict continuous homomorphisms. We recall now the basic definitions and the main results on initial algebras and rational terms that will be used along the paper; these are borrowed from [3, 19, 17], to which we refer the interested reader. It is well known that, for each signature Sigma , the category Sigma Alg has an initial object, often called the word algebra and denoted by T Sigma . Its elements are all the terms freely generated from the constants and the operators of Sigma , ....

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S. Bloom and Z. ' Esik. Iteration Theories. EATCS Monographs on Theoretical Computer Science. Springer Verlag, 1993.

....relating e.g. term graph rewriting to rational term rewriting (as introduced in [14] by exploiting the categorical framework. Lastly, in Section 5. 5 we sketch an historical overview of the various algebraic characterisation of the notion of fixed point, mostly related to iteration theories [7]. 1. Cyclic) Term Graphs This section introduces (ranked, possibly cyclic) term graphs as isomorphism classes of (ranked) labelled graphs. Since our main concern is to stress the underlying algebraic structure, the following presentation of term graphs slightly departs from the standard ....

....theories was further motivated by the correspondence between rational trees and certain varieties of iterative algebras [25, 31] thus providing a characterisation for behaviourally equivalent flowchart schemes. This is the line that brought the development of Iteration Theories, as recollected in [7, 8]. What we may consider the first results on the completeness of a linear syntax for the presentation of flowchart schemes and its relationship with their behavioural semantics are due to Bloom and Esik [6] This is the line bringing to the in depth analysis of such axiomatics represented ....

S. Bloom and Z. ' Esik. Iteration Theories. EATCS Monographs on Theoretical Computer Science. Springer Verlag, 1993.

....studied per se as a categorical tool [20] However, they already surfaced in the literature related to algebraic theories. In fact, there is a strong connection between traced categories and iteration theories (that is, algebraic theories with an equational characterization of (least) fix point [2, 3]) as it is pointed out in the works on flownomial calculus [32, 33] Definition 9 (traced monoidal categories) A traced monoidal category C is a five tuple hC 0 ; Omega ; e; ae; tri, where hC 0 ; Omega ; e; aei is a symmetric strict monoidal category, 3 which is equipped with a family of ....

S. Bloom and Z. ' Esik. Iteration Theories. EATCS Monographs on Theoretical Computer Science. Springer Verlag, 1993.

....infinite terms having a finite number of distinct subterms. Such a topic is already considered for example in [17] but it would be interesting to see if it could be addressed using categorical techniques along the line of this paper. Some preliminary considerations suggest that Iteration Theories [4] could fruitfully be used to this aim, in place of algebraic theories. ....

S. Bloom and Z. ' Esik. Iteration Theories. EATCS Monographs on Theoretical Computer Science. Springer Verlag, 1991.

.... that least simultaneous fixed points can be found in sequence by a form of Gaussian elimination, see e.g. 33] More generally, the equational theory between fixed point terms ( terms) induced by the leastfixed point operator, has been axiomatized as the free iteration theory of Bloom and Esik [3]. This theory is known to be decidable. Also, Eilenberg [6] and Plotkin [25] gave an order free characterisation of the least fixed point operator as the unique fixed point operator satisfying a condition known as uniformity, expressed with respect to the subcategory Cppo of strict maps in ....

....properties of fixed point operators in arbitrary categories of domainlike structures. In Section 2, we consider the basic notions of (parameterized) fixed point operator, Conway operator and iteration operator, developed from analogous notions in Bloom and Esik s study of iteration theories [3]. Our definitions are straightforward adaptations of Bloom and Esik s to the general setting of a category with finite products. In particular, the notion of iteration operator is intended to capture all desirable equational properties of a fixed point operator, as exemplified by the many ....

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S. Bloom and Z. Esik. Iteration Theories. EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1993.

....N and whose morphisms N m Gamma N n are n tuples of m ary primitive recursive functions, is a category with finite products and hence an algebraic theory. This category is a paradigmatic example of the notion of iteration theory introduced by Elgot: Categorical Logic 37 see the book by Bloom and Esik [ 1993 ] A reader who takes this advice should be warned that [ Bloom and Esik, 1993 ] adopts a not uncommon viewpoint that algebraic theories can be identified with categories with finite coproducts. Since the 2 category of categories with finite coproducts and functors preserving such is ....

....functions, is a category with finite products and hence an algebraic theory. This category is a paradigmatic example of the notion of iteration theory introduced by Elgot: Categorical Logic 37 see the book by Bloom and Esik [ 1993 ] A reader who takes this advice should be warned that [ Bloom and Esik, 1993 ] adopts a not uncommon viewpoint that algebraic theories can be identified with categories with finite coproducts. Since the 2 category of categories with finite coproducts and functors preserving such is equivalent (under the 2 functor taking a category to its opposite category) to the ....

S. L. Bloom and Z. ' Esik. Iteration Theories. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, New York, 1993.

.... domain theory [10, 4] We show that, under mild conditions, the induced fixed point operator endows the appropriate category with a unique well behaved parameterized fixed point operator (Theorem 3) By a general completeness result (Theorem 2) it follows that the axioms of iteration theories [1] are complete for deriving all valid fixed point equations. The general completeness result follows from an, apparently new, syntactic characterisation of the iteration theory equations as inducing a maximally consistent typically ambiguous theory (Theorem 1) 1 Syntax and Axioms We assume given ....

....0 [s=x oe ] We say that T is consistent if there are two terms t; t 0 of the same type such that T 6 t = t 0 . We say that T is closed consistent if there are two such terms that are closed. Our interest is in theories in which is a fixed point operator. The notion of a Conway Theory [1] provides a straightforward axiomatization of some fundamental properties of such theories. A theory T is said to be a Conway theory if it satisfies two axioms: 1. For any t( z; y ) oe and t 0 ( z; x oe ) T x: t[t 0 =y] t[ y: t 0 [t=x] y] oe. Dinaturality. 2. For any ....

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S. Bloom and Z. ' Esik. Iteration Theories. EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1993.

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S. Bloom and Z. Esik. Iteration Theories. EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1993.

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S. Bloom and Z. Esik. Iteration Theories. EATCS Monographs on Theoretical Computer Science. Springer, 1993.

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S. L. Bloom and Z. Esik. Iteration Theories. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1993.

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S. Bloom and Z. Esik. Iteration Theories. EATC Monographs on Theoretical Computer Science. Springer-Verlarg, 1993.

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B ' E93. S. L. Bloom and Z. ' Esik, Iteration Theories. EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1993.

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