P. Aczel. Non-well-founded sets., volume 14 of Lecture Notes, Center for the Study of Language and Information. Stanford, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[8, 9, 11, 23, 24, 31] In the past, the development of a mathematical situation theory has been held back by a lack of availability of appropriate technical tools. But by now, the theory has assembled its mathematical foundations based on intuitions basically coming from set theory and logic [1, 8, 23, 25]. With a remarkably original view of information (which is fully adapted by situation theory) 27, 28] a logic, based not on truth but on information, is being developed [24] This logic will probably be an extension of first order logic [5] rather than being an alternative to it. ....

....various types of objects. Compared to the existing approaches [16, 40, 43] BABY SIT enhances the features listed above in the following ways: ffl Situations are viewed at an abstract level. This means that situations are sets of parametric infons, but they may be non well founded (circularity) [1, 9]. ffl Parameters are place holders and can be anchored to unique individuals in the anchoring situation. The anchoring situation is required to cohere. ffl A situation can be realized if its parameters are anchored, either partially or fully, by the anchoring situation. That is, only anchoring ....

P. Aczel. Non-Well-Founded Sets, CSLI Lecture Notes Number 14, Center for the Study of Language and Information, Stanford, Calif., 1988.

....possible worlds. Peregrin s [24] analysis concludes that a possible world in the intuitive sense can be explicated as a maximal consistent class of statements . This implies that to give the semantics of possible worlds we require techniques like coinduction [23, 17, 6] or non well founded sets [2], each of which is in some sense syntax dependent. In light of that, and Peregrin s conclusion that possible worlds are language dependent , our embedding of syntactic types (that express intension) in the semantics seems unavoidable. But the latter should not be interpreted as: the use of Godel ....

P. Aczel. Non-Well-Founded Sets. Center for the Study of Language and Information, Stanford University, 1988.

....to this remark. In this paper, we work within the interleaving paradigm, and ignore true concurrency issues [58] In fact, instead of working explicitly with labelled transition systems quotiented by strong bisimulation, we will take advantage of Peter Aczel s work on non well founded sets [11], and work with synchronisation trees as canonical representations of strong bisimulation equivalence classes. We define the synchronisation trees over L as the largest solution of the set equation STL = L Theta STL ) More precisely , we take STL as the final coalgebra of the functor X ....

....take STL as the final coalgebra of the functor X 7 (L Theta X) The existence of this final coalgebra is guaranteed by Aczel s theory. Moreover, the final coalgebra property supports a method of definition by (non well founded, but guarded ) recursion, and a principle of coinduction. See [11] for further details. Note that a synchronisation tree p yields a labelled transition system (TC(p) Gamma ; p) TC(p) fpg [ fTC(q) j 9a: a; q) 2 pg ( a; q) 2 p: Henceforth, we shall use the term process interchangeably with synchronisation tree . 12 We will also make a ....

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P. Aczel. Non-well-founded sets. CSLI Lecture Notes 14. Center for the Study of Language and Information, 1988.

....c safe V al a [13, 43, 58] The definitions for traceC (p0 ; c) and traceA(p0 ; a) are in Section 2.3. 2. 2 Inductively and Coinductively Defined Sets The flowchart traces in the previous section can be infinite, and proofs on infinite traces are best worked with coinductive techniques [2, 54, 40], which we now review. The following presentation is summarized from Cousot and Cousot [14] We begin with the classical inductive definition. Let U be a universe of terms, and let F : P(U) P(U) be continuous with respect to the powerset lattice hP(U) i. The set defined inductively by F is ....

P. Aczel. Non-Well-Founded Sets. Lecture Notes 14, Center for Study of Language and Information, Stanford, CA, 1988.

.... it is used for testing process equivalence [18] in Model Checking as a notion of equivalence between Kripke Structures [20] in Web like databases for providing operational semantics to query languages [17] in Set Theory, for replacing extensionality in the context of non well founded sets [13]. With our approach, the structure of an HMM is reduced by computing bisimulation equivalence relation between states of the model, so that equivalent states can be collapsed. We employed both the notions of probabilistic and standard bisimulation. We will prove that bisimulation reduces the ....

....1: if hu i ; v i i 2 E, then there exists hu 1 i ; v 1 i i 2 E s.t. v 0 b v 1 . In order to minimize the number of nodes of a graph, we look for the maximal bisimulation on G. Such a maximal bisimulation always exists, it is unique, and it is an equivalence relation over the set of nodes of G [13]. The minimal representation of G = hN; Ei is therefore the graph: hN= fh[m] n] i : hm; ni 2 Egi which is usually called the bisimulation contraction of G. Using the algorithm in [19] the problem can be solved in time O(jEj log jN j) for acyclic graphs and for some classes of cyclic ....

Aczel, P.: Non-well-founded sets. Lecture Notes, Center for the Study of Language and Information 14 (1988).

....theories non assuming this strong constraint (re )emerged in many communities. Various proposals for (axiomatic) non well founded set theories (and universes) were developed and probably the first one was [7] by Forti and Honsell. Following Barwise and Moss (cf. 2] we can say that the book [1] by Aczel can be considered as the definitive reference on non well founded sets. A more recent and important book on the topic is [3] Sets can be seen as nothing but directed graphs. Edges represent membership: m n means that m has n as an element. The resulting set theoretic semantics for ....

....definitive reference on non well founded sets. A more recent and important book on the topic is [3] Sets can be seen as nothing but directed graphs. Edges represent membership: m n means that m has n as an element. The resulting set theoretic semantics for graphs, introduced and developed in [1], is based on the natural notion of picture of a graph. When the graph is acyclic it goes as follows: the leaves are nodes representing the empty set ; going from leaves to internal nodes any other node can be depicted by the set whose elements are the pictures of the nodes reached by their ....

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P. Aczel. Non-well-founded sets. Vol. 14 of Lecture Notes, Center for the Study of Language and Information. Stanford, 1988.

....3, 18, 35, 2] however, consider well founded sets only. An interesting extension of the concept of set is that of hyperset [9] that is sets in which, roughly speaking, membership can form cycles. Hypersets we are considering here are very closely related to Aczel s non well founded sets (cf. [1]) but they all have nite cardinality and are hybrid , in that their construction involves free Herbrand functors. Usefulness of hypersets as a convenient data model has been investigated in [10] and [8] Examples of hypersets arise when modeling a circular process or structure by means of sets. ....

....as a convenient data model has been investigated in [10] and [8] Examples of hypersets arise when modeling a circular process or structure by means of sets. Moreover, hyperset theory has been applied in a number of areas of logic, linguistics, and computer science. As Barwise says in (Foreword to [1]) It seemed that in order to understand common knowledge (a crucial feature of communication) circular propositions, various aspects of perceptual knowledge and self awareness [ we either had to give up the tools of set theory which are so well loved in mathematical logic, or we ....

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P. Aczel. Non-well-founded sets., volume 14 of Lecture Notes, Center for the Study of Language and Information. Stanford, 1988. 43

....is a canonical way of finding internally fully abstract domains of bisimulation, where two elements are equal if and only if they are bisimilar. It follows from a simple but very general argument that final coalgebras are fully abstract (see Aczel s final coalgebra model for nonwellfounded sets [Acz88], and also [RT93] We shall show that 2 it follows from general coalgebraic considerations [AR89,Bar93,RT93] that both our functors D and M 1 have a final coalgebra, which consequently are internally fully abstract with respect to (discrete and continuous) probabilistic bisimulation. Therefore ....

P. Aczel. Non-Well-Founded Sets. CSLI Lecture Notes 14. Center for the Study of Languages and Information, Stanford, 1988.

....: 23 1. Introduction 3 1. Introduction A transition system is usually defined as a set together with a relation on that set. It is a simple observation, possibly first made in [Ken87] and [Acz88], that equivalently a transition system can be represented as a coalgebra by viewing its relation as a (nondeterministic) function. This representation gives rise to a natural (and standard) notion of homomorphism of transition systems. Moreover, a bisimulation relation simply turns out to be ....

P. Aczel. Non-Well-Founded Sets. Number 14 in CSLI Lecture Notes. Center for the Study of Languages and Information, Stanford, 1988.

....is a canonical way of finding internally fully abstract domains of bisimulation, where two elements are equal if and only if they are bisimilar. It follows from a simple but very general argument that final coalgebras are fully abstract (see Aczel s final coalgebra model for nonwellfounded sets [Acz88], and also [RT93] Here, final means that there exists a unique homomorphism from any coalgebra to the final one. Finality is to the world of coalgebras what initiality is to the world of algebras, cf. MG85] We shall show that it follows from general coalgebraic considerations [AR89, Bar93, ....

P. Aczel. Non-Well-Founded Sets. CSLI Lecture Notes 14. Center for the Study of Languages and Information, Stanford, 1988.

....is a canonical way of finding internally fully abstract domains of bisimulation, where two elements are equal if and only if they are bisimilar. It follows from a simple but very general argument that final coalgebras are fully abstract (see Aczel s final coalgebra model for nonwellfounded sets [Acz88], and also [RT93] Here, final means that there exists a unique homomorphism from any coalgebra to the final one. One can argue that finality is to the world of coalgebras what initiality is to the world of algebras, cf. MG85] We shall show that it follows from general coalgebraic ....

P. Aczel. Non-Well-Founded Sets. CSLI Lecture Notes 14. Center for the Study of Languages and Information, Stanford, 1988.

....The scope of this new definition of bisimulation is far more general than the original one, and applies to many types of systems, including not only Park and Milner s concurrent processes, but also various kinds of data types and automata. Using the (generalized) notion of bisimulation, Aczel [Acz88] formulated a principle of coinduction, in very much the same way as Milner had introduced his bisimulation proof method : In order to prove that two processes are behaviourally equivalent (bisimilar) it is su#cient to establish the existence of a bisimulation relation between them. This ....

P. Aczel. Non-well-founded sets. Number 14 in CSLI Lecture Notes. Center for the Study of Languages and Information, Stanford, 1988.

....with the corresponding being is doing characterization, applies more generally to any final coalgebra. Coinduction as a proof principle for greatest fixed points of monotone operators is already around for some time. For final coalgebras of the powerset functor, it has been introduced in [Acz88]. In [RT93] the principle is stated in its generality for arbitrary functors. The word coinduction suggests a duality between induction and coinduction. This is explained by the observation that induction principles apply to initial algebras . Somewhat more concretely, the duality can be ....

P. Aczel. Non-well-founded sets. Number 14 in CSLI Lecture Notes. Center for the Study of Languages and Information, Stanford, 1988.

....BB88, DOPR96, Sto96, AD97] however, consider well founded sets only. An interesting extension of the set concept is the hyperset of [BM91] roughly speaking, sets in which membership can form cycles. The hypersets we consider here are very closely related to Aczel s non well founded sets (cf. [Acz88]) but they all have finite cardinality and are hybrid, in that their construction involves free Herbrand functors. Usefulness of hypersets as a convenient data model is investigated in [BM93] and [BM86] Examples of hyperset arise when modeling a circular process or structure by means of sets. ....

....a convenient data model is investigated in [BM93] and [BM86] Examples of hyperset arise when modeling a circular process or structure by means of sets. Moreover, hyperset theory has been applied in a number of areas of logic, linguistics, and computer science. As Barwise says in the foreword to [Acz88], It seemed that in order to understand common knowledge (a crucial feature of communication) circular propositions, various aspects of perceptual knowledge and self awareness . we either had to give up the tools of set theory which are so well loved in mathematical logic, or we had to ....

[Article contains additional citation context not shown here]

P. Aczel. Non-Well-Founded Sets, volume 14 of Lecture Notes, Center for the Study of Language and Information. Stanford University, Stanford, CA, 1988. 44

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P. Aczel. Non-well-founded sets., volume 14 of Lecture Notes, Center for the Study of Language and Information. Stanford, 1988.

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Aczel P: Non-well-founded Sets. No. 14 in CSLI Lecture Notes. Center for the Study of Language and Information, Stanford 1988

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P. Aczel. Non-well-founded sets. Number 14 in CSLI Lecture Notes. Center for the Study of Languages and Information, Stanford, 1988.

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P. Aczel. Non-Well-Founded Sets. Number 14 in CSLI Lecture Notes. Center for the Study of Language and Information, Stanford, CA, 1988.

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P. Aczel. Non-well-founded sets, volume 14 of Lecture Notes, Center for the Study of Language and Information. Stanford, 1988.

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Aczel., P., \Non-well-founded sets." Lecture Notes, Center for the Study of Language and Information 14, Stanford, 1988.

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P. Aczel. Non-Well-Founded Sets. Number 14 in CSLI Lecture Notes. Center for the Study of Language and Information, Stanford, CA, 1988.

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P. Aczel. Non-well-founded sets. Vol. 14 of Lecture Notes, Center for the Study of Language and Information. Stanford, 1988.

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P. Aczel. Non-well-founded sets. CSLI Lecture Notes 14. Center for the Study of Language and Information, 1988.

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P. Aczel. Non-well-founded sets. CSLI Lecture Notes 14. Center for the Study of Language and Information, 1988.

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Peter Aczel. Non-well-founded Sets. Number 14 in Lecture Notes. Center for the Study of Language and Information, Stanford University, 1988.

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