D. Turi. Functorial Operational Semantics and its Denotational Dual. PhD thesis, Free University, Amsterdam, June 1996. Accessible from !http://www.dcs.ed.ac.uk/home/dt/?.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....as a technical tool, needed to establish a general completeness result (Theorem 2 of the present paper) Nevertheless, it seemed a natural notion. This was later substantiated by Turi and Plotkin in their category theoretic account of the operational and denotational semantics of GSOS systems [26, 27]. In their work, a GSOS system gives rise to a monad on the category of transition systems with functional bisimulations as morphisms. The algebras of this monad turn out to be exactly the GSOS models in the sense of De nition 3.1. The idea of using sequent calculus for process veri cation was ....

D. Turi. Functorial Operational Semantics and its Denotational Dual. PhD thesis, Free University, Amsterdam, 1996.

....isomorphic. In particular in the case of topological spaces, we shall only have a mono 7c from ( c) to [ c) for each object c. We then insist on the condition that 7 is a distributivity law for comonads (the name is by analogy with distributivity laws for monads as in [24] they originated in [49]) and that [ preserves pullbacks along the distributivity law. The role of such distributivity laws will be apparent from the proof of Proposition 69. This appeal to distributivity law subsumes the above, simple case where preserves the comonad [ just take 7c to be the identity. This is ....

Daniele Turi, Functorial operational semantics and its denotational dual, Ph.D. thesis, Free University, Amsterdam, June 1996.

....nding appropriate liftings of monads on categories C C C to the subscone category Subscone C C C (see Section 3) Outline. We de ne liftings of monads to scones in Section 2; this is simpler than for subscones, and of independent interest. This requires distributivity laws, slightly extending [25]. We then lift monads to subscones in Section 3. The important case where the target category C C C is a product of two categories is investigated in Section 4: this is where binary logical relations arise, allowing us to compare two models. We terminate our lifting construction by lifting monad ....

....T T T (hA 1 ; A 2 i) hT T T 1 (A 1 ) T T T 2 (A 2 )i. Related work. We have already said that there is no unique notion of monad lifting. One of the simplest is the lifting, proposed in [3] T of T T T , which maps the object hS; f; Ai of the scone to hS; f ; j A j; T T T (A)i. Turi [25] considers lifting monads to the category of coalgebras of a given endofunctor. This is a special case of our framework, when C C C = C (and T T T = T ) and moreover only objects of the form S jSj are taken into consideration, and only morphisms of the form hg; gi. This de nes the category ....

D. Turi. Functorial Operational Semantics and its Denotational Dual. PhD thesis, Free University, Amsterdam, June 1996.

....incorporating fairness constraints in the models by moving from presheaves to sheaves. Finally, it is not clear to us how our approach relates to the abstract understanding of bisimulation provided by coalgebras [4, 114] The hope is that the recent work and ongoing research of Turi and Plotkin [131, 132] will help provide the missing links. Recall that PomL is the category of (finite) pomsets over L. Appendix A Basic Definitions of Enriched Category Theory A.1 Enriched categories In this appendix we review some concepts from enriched category theory that we needed from Chapter 4. In ....

Daniele Turi. Functorial Operational Semantics and its denotational dual. PhD thesis, The CWI, Amsterdam, 1996.

....functor is the class WF of well founded sets, with identity as the structure map. The final coalgebra for this functor is NWF, the category of sets with the anti foundation axiom, again with identity as the structure map. For additional reading on fixed points for P, see [BM96] Acz88] and [Tur96]. For existence theorems for both initial algebras and final coalgebras, see [Bar92] James Worrell extends this discussion in [Wor00] 1.5.2. Induction and recursion. See also [JR97] for a nice exposition of this material. The principle of definition by recursion is an explicit application of ....

....# T#(S) In other words, the principle of induction on WF as an initial algebra is the usual foundation axiom. Put another way, the foundation axiom is equivalent to the assumption that the class of all sets is an initial algebra for P (although here, we ve only shown one implication see [Tur96] for the other) It is worth mentioning that the property of minimality isn t unique to initial algebras. On the contrary, any algebra which is a quotient of the initial algebra is also minimal, and so satisfies an inductive proof principle. Conceptually, if #A, ## is a quotient of the initial ....

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Daniele Turi, Functorial operational semantics and its denotational dual, Ph.D. thesis, Free University, Amsterdam, June 1996.

....object category 1. Thus (0) 1, 1) i.e. the category with two objects and one arrow) while (c) 0 for all other nitely presentable categories c. We now turn to the construction of an endofunctor over a signature . 4 CMCS 01 Preliminary Version Working over Set, e.g. see [23], one de nes a functor F (X) a f2 (n) X n which calculates the terms of depth 1 whose variables come from the set X. For example, given the signature of example 3.2, the associated endofunctor is FM (X) 1 X 2 . Such polynomial endofunctors (i.e. built from products and coproducts) are ....

D. Turi. Functorial Operational Semantics and its Denotational Dual. PhD thesis, Free University, Amsterdam, 1996.

....D to W Q;q f : W Q;q (C) W Q;q (D) where W Q;q f = inl W Q;q (D) ffi (f id Q ) inr W Q;q (D) # : 20 Here f idQ : C Q D Q is the natural map. Furthermore, we have ffl Q;q;C : W Q;q (C) V given by ffl Q;q;C = incl C ; q] # : Definition [Turi and Rutten [TR] Turi [T]] F is uniform if there are a set Q, a map q : Q V and a natural transformation ae : F PW Q;q such that P ffl Q;q;V ffi ae FV = incl FV , where incl F(V ) is the inclusion of F(V ) in V . Our formulation of this concept is slightly more general than the original formulation because we allow ....

Turi, D. 1996. Functorial Operational Semantics and its Denotational Dual. Ph.D. thesis, CWI, Amsterdam. 30

....5 shows how a good part of the development of Vicious Circles [3] for example, can be unified and simplified by using the much more general work that we do in this paper. One particular focus of our study is the notion of a uniform functor on sets, introduced by Turi and Rutten [11] and Turi [9]. We generalize their notion slightly, prove the main result about such functors (that their greatest fixed points are final coalgebras) and we also use our earlier work to check that many functors of interest are uniform. Taken together, the work of this paper shows that much of the extant ....

....is the final F coalgebra morphism for e. The uniqueness is as above. a Definition The Anti Foundation Axiom (AFA) is the assertion that for every set b and every e : b Pb, there exists a unique s : b V such that s = Ps ffi e. The map s is called the solution to the system e. Lemma 5. 3 (Turi [9]) AFA is equivalent to the assertion that hV; i V;PV i = hV; id V i is a final P coalgebra. Proof This is not just a trivial application of Lemma 5.2.3, since the latter statement reads for all classes b while present statement only uses sets. Nevertheless, one can show that set form implies ....

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Daniele Turi, Functorial Operational Semantics and its Denotational Dual, (Ph.D. thesis, CWI, Amsterdam, 1996).

....## E preserve regular monos. Further, suppose that U : E ## E has a right adjoint, H : E ## E (so that U is comonadic) Then the category E of coalgebras is also co Birkho . Proof. It is well known that the coalgebraic forgetful functor has a right adjoint i it is comonadic. See, for instance, [Tur96] for details. One shows directly that U re ects regular monos and so inherits the epi regular mono factorizations from E . This implies that E is regularly well powered. Because HA is injective if A is injective, and U re ects regular monos, E has enough injectives. Hereafter, we assume the ....

Daniele Turi. Functorial Operational Semantics and its Denotational Dual. PhD thesis, Free University, Amsterdam, June 1996.

.... crucial observations: i) Since the state transition function vA : FPA P(P(A) Theta [A p FPA ] in the final A f coalgebra FMA is bijective we have as well a A f coalgebra FMA = FPA ; vA ) as a A f algebra FM Gamma1 A = FPA ; v Gamma1 A ) both with the same carrier FPA [6,12]. ii) The definition of hide : FPA FPA obs and thus the concept of nondeterministic processes in [4] is based on a complex mixture of coalgebraic and algebraic techniques (see remark 5.2) This difference between the fully coalgebraic concept of deterministic processes in [4] and the mixed ....

D. Turi. Functorial operational semantics and its denotational dual. PhD Thesis, Free Univ. Amsterdam, 1996.

....with automata as objects. This suits our purposes. The feedback operation that we describe for automata may also be found in [12, 18] and is standard in dataflow networks and control theory) It is becoming increasingly clear that coalgebras are fundamental in the study of processes, see e.g. [1, 17, 19] (but also in areas like object oriented programming [8] or hybrid systems [9] Coalgebras describe abstract dynamical systems via a state transition function, and determine the notion bisimulation (observational indistinguishability of states) associated with such systems. Terminal coalgebras ....

D. Turi. Functorial operational semantics and its denotational dual. PhD thesis, Free Univ. Amsterdam, 1996.

....approach, however, often appears quite ad hoc, just think, for example, of how one would prove set theoretically the existence of a coiterative function into streams. In recent years, a categorical explanation of coinduction has appeared, based on the notion of coalgebra, see e.g. [Acz88, AM89, Acz93, RT93, RT94, Tur96, TP97, Len98]) Work supported by Esprit Working Group Types , MURST 97 Cofin. Sistemi Formali. grant, TMR Linear FMRX CT98 0170. 1 we shall refer to it as coalgebraic coinduction. Coalgebraic coinduction has proved to be extremely fertile ( HL95, Jac96, Len96, Rut96, Jac97, RV97, HLMP98, Mos, Len99] ....

....Coinduction up to [ see Section 4. 2 for more details) 7 2 Coalgebraic Description of Coinduction and Coiterative Morphisms In this section, we present the categorical description of coinduction based on the notion of coalgebra, for capturing equivalences induced by coiterative morphisms ([Acz88, AM89, RT93, RT94, Rut96, Tur96, TP97, Len98]) In this setting, the categorical counterparts of set theoretic bisimulations are F bisimulations, i.e. spans of coalgebra morphisms ( TP97] One of the advantages of a categorical description is that we can deal uniformly with coinductively defined objects and coiterative morphisms. In fact, ....

D.Turi. Functorial Operational Semantics and its Denotational Dual, PhD thesis, 1996.

....data structures from programs. Functional programs can also be structured according to the effects they produce. This can be possibe by using monads [33] as structuring device. Monads are well known mathematical structures with wide application in programming [30, 20] and formal semantics [26, 31, 32]. Previous works [12, 17, 25, 28] have studied how to combine both structuring mechanisms, giving rise to recursive operators that deal with effects modeled by monads. A result that can be concluded from those works is that encapsulating effects with monads leads to a smooth framework for ....

....with effects. In fact, monads permit to focus on the relevant structure of programs disregarding details about the specific effect that a program produces. Comonads are mathematical structures, dual to monads in a categorical sense, that have been used almost entirely in semantics (see e.g. [6, 7, 31, 32]) In recent years, however, there has been a growing interest in investigating the usefulness that comonads may have in programming. Some results in this concern are given by Kieburtz [21] who argues that comonads account naturally for effects that derive from program context. This paper ....

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D. Turi. Functorial Operational Semantics and its Denotational Dual. PhD thesis, Free University, Amsterdam, June 1996.

.... are the categories of classes of hypersets, i.e. non wellfounded sets satisfying the Antifoundation Axiom X 1 of [FH83] called AFA in [Acz88] However, despite the vast amount of published literature on final semantics in categories of sets and hypersets, e.g. Acz88,AM89,Bar93,Bar94,RT93,RT94] [BM96,MD96,Tur96,TR98], a definitive, final, picture is just only beginning to emerge [Moss00] Each paper utilizes its peculiar definition, and little attention has been given to comparing the various results or to achieving generality. In this paper, we try to give a general form of Aczel s Special Final Coalgebra ....

D.Turi. Functorial Operational Semantics and its Denotational Dual, PhD thesis, 1996.

.... is taken from [KM96] and is ultimately due to [Bar93] Sections 7 and 8 generalize similar results from [Rut95] The results on final systems in Section 9 are from [RT93] The results presented in Section 10 are from [Bar93, Bar94] which build on [AM89] Their presentation has been influenced by [Tur96] and [KM96] The example of the extended natural numbers in Sections 11 and 12 was developed jointly with Bart Jacobs and Bill Rounds. The comparison of induction and coinduction in Section 13 extends the characterization in [RT94] which was given in terms of congruences and bisimulations (see ....

....dynamical systems, which occurs in [Dev86] This section does not present any new results but for the observation that one of the essential ingredients in symbolic dynamics is the notion of cofree system. In addition to the references mentioned above, work on (final) coalgebraic semantics includes [Tur96], which gives a systematic comparison of final coalgebra and initial algebra semantics for concurrent languages; Rei95] on object oriented programming; HL95] on a model for the lambda calculus; Len96] on a higher order concurrent language; Jac96a] on behaviour refinement in object oriented ....

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D. Turi. Functorial operational semantics and its denotational dual. PhD thesis, Vrije Universiteit, Amsterdam, September 1996. 54

....Furthermore, it is not difficult to see that the foundation axiom is equivalent to the statement that the universe V , together with id : V V , forms an initial algebra. The anti foundation axiom, on the other hand, is equivalent to the statement that hV; idi forms a final coalgebra. See [Tur96] for a nice presentation of these claims. That the anti foundation universe is a final coalgebra for P was originally presented in [Acz88] 9 According to [JR97] the notion of a bisimulation relation first arose in studies of concurrency. Set theory with the anti foundation axiom is the first ....

....of the final coalgebra can be characterized by an infinitary modal logic. The first step in any generalization of this result is to abstract the properties of functors on SETS to properties that apply to functors on a broader class of categories. Some of this abstraction has already been given in [Tur96]. However, much work remains in applying the notion of characterization of objects to arbitrary categories. ....

Daniele Turi. Functorial Operational Semantics and its Denotational Dual. PhD thesis, Free University, Amsterdam, June 1996. 17

....labelled transition systems, corresponding to the endofunctor P (L 20) where L is a set of labels, the corresponding comonad can again be described in terms of labelled trees, where we now have to label the edges of the tree by elements of L as well as labelling the nodes by elements of A. See [20]. Yet another possibility is to consider the endofunctor P on Set that takes a set to the set of its nonempty finite subsets. The category of coalgebras now amounts to the category of finitely branching transition systems such that a transition is always possible. In this case, since P ....

D. Turi, Functorial operational semantics and its denotational dual, Ph.D. thesis, Vrije Universiteit te Amsterdam (1995).

....to investigate the general status and theory of the operator introduced in [HL95] which has remained largely unexplored since then. In particular, we will show that there are rich classes of cartesian applicative structures, and plenty of cartesian theories. Using concepts from final semantics [Acz88, RT93, Rut96, Tur96, Len98], one can easily express the difference between the traditional applicative coinduction and cartesian coinduction. Both kinds of coinduction correspond to the categorical coinduction principles which arise when the set of closed terms, 0 , or more in general an applicative structure) is ....

....(pEq D caas) is an aas D = D; vD ; ffl D ; fpngn2 ) such that the ordered applicative structure (D; vD ; ffl D ) is a Phi pEq D coinductive applicative structure. 3 The final perspective Following the Final Semantics Paradigm, introduced by Aczel [Acz88] and further developed in [AM89, RT93, Rut96, Tur96, Rut96, Len98], coinduction principles can receive a very neat categorical explanation. It is interesting to point out that many of the monotone operators described in Section 2 were actually suggested by this categorical analysis. We work in Set , the category whose objects are sets of a universe of the ....

D.Turi. Functorial Operational Semantics and its Denotational Dual, PhD thesis, CWI, 1996.

....And the logical reasoning principle that is needed for such final coalgebras is not induction but coinduction. There is no single reference in which this link between state based dynamical systems and coalgebra is made explicitly (with all its ramifications) but important insights can be found in [21, 69, 5, 25, 40, 1, 2, 51, 15, 58, 63, 64, 61, 20, 37, 35, 34, 67, 65, 38]. This list is incomplete and does not do justice to the various contributors to this area, but it hopefully gives the reader an impression of some of the developments. These notions of coalgebra and coinduction are still relatively unfamiliar, and it is our aim in this tutorial to explain them in ....

....may be described as construction versus observation. It may be found in process theory [50] data type theory [21, 25, 5, 40] including the theory of classes and objects in objectoriented programming [61, 30, 37, 35] semantics of programming languages [46] denotational versus operational [64, 67, 6]) and of lambda calculi [58, 59, 20, 31] automata theory [53] system theory [65, 34] natural language theory [10, 62] and many other fields. We assume that the reader is familiar with definitions and proofs by (ordinary) induction. As a typical example, consider for a fixed data set A, the set ....

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D. Turi. Functorial operational semantics and its denotational dual. PhD thesis, Free Univ. Amsterdam, 1996.

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D. Turi. Functorial Operational Semantics and its Denotational Dual. PhD thesis, Free University, Amsterdam, June 1996. Accessible from !http://www.dcs.ed.ac.uk/home/dt/?.

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SIAM J. Comput., 11:761--783. Turi, D. (1996). Functorial Operational Semantics and its Denotational Dual. PhD thesis, Free University, Amsterdam. Available from !http://www.dcs.ed.ac.uk/home/dt?.

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D. Turi. Functorial Operational Semantics and its Denotational Dual. PhD thesis, Free University, Amsterdam, June 1996. Accessible from !http://www.dcs.ed.ac.uk/home/dt/?.

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D. Turi. Functorial Operational Semantics and its Denotational Dual. PhD thesis, Free University, Amsterdam, June 1996. Accessible from !http://www.dcs.ed.ac.uk/home/dt/?.

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Daniele Turi. Functorial Operational Semantics and Its Denotational Dual. PhD thesis, Free University Amsterdam, 1996.

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D. Turi. Functorial Operational Semantics and its Denotational Dual. PhD thesis, Free University, Amsterdam, 1996.

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Daniele Turi. Functorial Operational Semantics and Its Denotational Dual. PhD thesis, Free University Amsterdam, 1996.

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D. Turi. Functorial operational semantics and its denotational dual. PhD thesis, Free Univ. Amsterdam, 1996.

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Daniele Turi. Functorial Operational Semantics and its Denotational Dual. PhD thesis, Free University of Amsterdam, June 1996.

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Tur96. D.Turi. Functorial Operational Semantics and its Denotational Dual, PhD thesis, 1996.

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Tur96. D.Turi. Functorial Operational Semantics and its Denotational Dual, PhD thesis, CWI, 1996.

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