R. Goldblatt, Logics of Time and Computation, CSLI Lecture Notes, vol. 7, Center for the Study of Language and Information, Ventura Hall, Stanford, CA, 1987. Distributed by Chicago University Press.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....It includes boolean constructs (union, intersection, and complement) existential qualification and universal qualification for building complex concept expressions, while role can only be atomic. ALC corresponds to the well known modal logic K i [108] which is the basic normal multimodal logic [60, 63, 22, 66]. Satisfiability of an ALC concept (satisfiability of a K i formula) is known to be PSPACE complete while logical implication for ALC (for K i ) is EXPTIME complete. C is the description logic obtained from ALC by adding the following role constructs: union, chaining, reflexive transitive ....

.... Gamma fA (1a) H (1a) j A (1a) H (1a) 2 Pi (x)g for all x 2 S The following three lemmas state the basic properties of M , and M . Lemma 7 Let M be a model of fl( Phi) Then, for every formula OE 2 CL(fl( Phi) and every x 2 S ; x j= OE iff M;m(x) j= OE: See for example [60] for the definition of filtration in Modal Logic. 38 Proof We prove the lemma by induction on the formation of OE (called formula induction in the following) We assume, without loss of generality, Delta] to be expressed by means of : Delta , and that the converse operator is applied ....

R. Goldblatt. Logics of time and computation, volume 7 of Lecture Notes. Center for the Study of Language and Information, second edition, 1992.

....Mat97] Monniaux [Mon01] shows that BAN and GNY logics are decidable, while Massacci [Mas97] gives a tableaux calculus for the (undecidable) logic of access control of [ABLP93] We refer the reader to [Mon01] for more information on such logics. Multi modal logics like Propositional Dynamic Logic [Gol87] have also been used to model the changing states of a program. Finally, propositional bi modal tense logics give a very simple and elegant model of the flow of time [HC96] Checking that a down loaded applet meets the security criteria is now reduced to proving, on board, that an appropriate ....

R. I. Goldblatt. Logics of Time and Computation. CSLI Lecture Notes Number 7, Center for the Study of Language and Information, Stanford, 1987.

....this section is to define the notion of a rooted feature algebra, and to derive some results for rooted feature algebras which will be useful in the following two sections. This notion is very similar to the notion of a generated submodel as introduced for modal and multimodal logic (see e.g. [5]) However, because of the presence of atoms in our formalism, the definition of rooted feature algebras is more complex. Let S be a satisfiable feature term, and let the feature algebra I together with the element d 2 D be a witness for the satisfiability of S, that is, let . Then D may ....

R. Goldblatt. Logics of Time and Computation. CSLI Lecture Notes 7. Center for the Study of Language and Information, Stanford University, Cal., 1987.

.... nucleus in the theory of topoi and sheafification [Joh82] From this topological perspective, Goldblatt studied a system identical to PLL accommodating Lawvere s suggestion that the # modality means it is locally the case that by interpreting this in various ways to mean at all nearby points [Gol81,Gol93]. The algebraic properties of such operators (on complete Heyting algebras) have been explored by Macnab [Mac81] who calls them modal operators . In this paper we show how PLL can be naturally seen as a special CS4 theory or CS4 algebra in the sense that it can be obtained from CS4 by adding ....

R. Goldblatt. Mathematics of Modality. CSLI Lecture Notes No. 43. Center for the Study of Language and Information, Stanford University, 1993.

....C of a modal formula # is then inductively de ned by . ff ] p] # 2 #(c) # #] C [ #] #] #(c) The meticulous reader is invited to check that this de nition coincides with the usual semantics of propositional modal logic (cf eg [7]) Given a formula # ML, the rank of #, which represents the nesting depth of # operators, is then given inductively by rank(ff) 0, rank(# #) max rank(#) rank(#) rank(p) 1 for p Prop, rank(##) rank(#) 1. Semantically, the rank can be thought of the number of transition ....

....Proof. Let (C ) be given as in the above proposition. If x 1, we have x = # n (x) # n # (x) where : C (K, #) is the map given by nality. 4.2 The Canonical Model Let M be the functor P PProp, Prop a countably in nite set. The canonical model (see for example [4, 7]) for the modal logic is the M coalgebra (L, # # ML : # is maximally consistent #R : L # PL ## # # V : L # PProp Prop The canonical model is nal in the category ThML which has M coalgebras as objects and morphisms f : C, #) D, #) are functions f : C for ....

Robert Goldblatt. Logics of Time and Computation, volume 7 of CSLI Lecture Notes. Center for the Study of Language and Information, Stanford University, 1992. Second Edition.

.... as it pertains to DL can be found in the authors text [Harel et al. 2000] There are by now a number of books and survey papers treating logics of programs, program verification, and Dynamic Logic [Apt and Olderog, 1991; Backhouse, 1986; Harel, 1979; Harel, 1984; Parikh, 1981; Goldblatt, 1982; Goldblatt, 1987; Knijnenburg, 1988; Cousot, 1990; Emerson, 1990; Kozen and Tiuryn, 1990] In particular, much of this chapter is an abbreviated summary of material from the authors text [Harel et al. 2000] to which we refer the reader for a more complete treatment. Full proofs of many of the theorems cited ....

R. Goldblatt. Logics of time and computation. Technical Report Lect. Notes 7, Center for the Study of Language and Information, Stanford Univ., 1987.

....of atomic propositions. Given a T coalgebra (C, the semantics I Il(c,v) C C of a modal formula E AA is then inductively defined by 2 INTRODUCTORY EXAMPLES 5 The meticulous reader is invited to check that this deflation coincides with the usual semantics of propositional modal logic (cf eg [7]) Given a formula 6 , the rank of , which represents the nesting depth of operators, is then given inductively by rank(if) 0, rank( max rank( rank( rank(p) 1 for p G Prop, rank( rank( 1. Semantically, the ra can be thought of the number of transition steps, a formula contains ....

....is a surjection. Proof. Let (C , be given as in the above proposition. If x C TI, we have x (x) nne (x) where : C, K,n) is the map given by finality. 4.2 The Canonical Model Let M be the functor 7 9 x 79Prop, Prop a countably infinite set. The canonical model (see for example [4, 7]) for the modal logic AA is the M coalgebra (L, Aa, L A: L 79L Av: L ] Prop q) C AA: q) is maximally consistent q q rq Prop The canonical model is final in the category l h which has M coalgebras as objects and morphisms f: C, D, 5) are functions f: C D such that for all c ....

Robert Goldblatt. Logics of Time and Computation, volume 7 of CSLI Lecture Notes. Center for the Study of Language and Information, Stanford University, 1992. Second Edition.

....prerequisites of action; it must include an account of ability as opposed to mere physical possibility . The most in uential work in this area is Moore s theory of knowledge and action [74, 76] His framework can be described as a combination of rst order dynamic logic (a modal logic of action) [27] with an S4 modal logic of knowledge [36, 41] 55, p. 439] However: Strictly speaking, the framework is not the modal logic just described, but the encoding in rst order logic of the semantics of this modal logic; thus the part of the logic dealing with action is closely related to the ....

Goldblatt, R.: Logics of Time and Computation. (CSLI Lecture Notes, 7.) Center for the Study of Language and Information, Stanford University, Stanford, CA, 1987.

....Its set of intended models can be constructed in the temporal sense described above. One specific behaviour of the system corresponds to one of these temporal models. We will work out this general idea for the case of a meta level architecture. More details on temporal logic can be found in [2] [20]. 7 3 Static and dynamic view on the object level reasoning In this section we use the notion of a partial model to formalize the information state of the object level reasoning component at a certain moment. A transition of one information state to another one can be formally described by a ....

....of partial models (or simulation structures) in meta level architectures ( 37] 19] see also [32] What is different in our case is that the partial models may be dynamic. Furthermore, similarities can be found to the approach called dynamic interpretation of natural language (e.g. see [17] [20], 23] In this approach the dynamic interpretation of a sentence in natural language is defined as an operator that transforms the current information state into a new one where the content of the sentence is included. With respect to dynamics the type of meta level architecture covered here is ....

R. Goldblatt, Logics of Time and Computation. CSLI Lecture Notes, Vol. 7. 1987, Center for the Study of Language and Information.

....us with a new state (the rst component of (c) i) and an output (the second component of (c) i) 3. Consider TX = P(X) P(A) where A is a set (of atomic propositions) and P denotes the covariant powerset functor. Every T coalgebra (C; C P(C) P(A) gives rise to a Kripke model (see [10, 7]) P maps a set to its powerset and a function f : X Y to the function Pf de ned by (Pf) x) ff(x) j x 2 xg. 5 by putting K (C; C; R; V ) where C is the carrier (set of worlds) of the model, R is the successor relation, given by (c; c ) 2 R ( c 2 1 (c) and V : A P(C) is ....

....in the related paper [19] predicate liftings are syntactically de ned entities, and naturality, which we take as our de ning property, is derived. 18 We illustrate the concept of predicate liftings by showing that they generalise the interpretation of the 2 operator from Kripke models (see e.g. [7, 10]) to coalgebras of arbitrary signature functors. Example 6.2. Suppose TX = P(X) P(A) as in Example 2.2. Consider the operation (C) P(C) P(TC) de ned by (C) c) f(a; c ) 2 TC j c cg: An easy calculation shows, that this de nes a predicate lifting . Now consider a T coalgebra ....

R. Goldblatt. Logics of Time and Computation, volume 7 of CSLI Lecture Notes. Center for the Study of Language and Information, Stanford University, 1992. Second Edition.

....given an expression A, 2A means A is true in all possible worlds . Chellas, in [9] part 1, gives details. Alternatively, a temporal logic may be de ned by ordering the set of possible worlds by time and considering 2 to be true in all future worlds . Goldblatt describes several such logics in [11], chapter 8. More generally, we may consider a partial ordering of possible worlds by some arbitrary accessibility relation, with 2A meaning true in all accessible worlds . For Intensional HTML, the set of possible worlds is the set of possible versions of pages or parts of a page, for some ....

Robert Goldblatt. Logics of Time and Computation. Number 7 in CSLI Lecture Notes. Center for the Study of Language and Information, 1987.

....theory are e.g. 29, 37, 62] The paper by Thomas in the handbook of Theoretical Computer Science [55] gives an excellent overview of automata on infinite objects. Eilenberg s two volumes [13] are older but more encyclopedic. In the area of (modal and) temporal logic good background texts are e.g. [30, 56, 21]. For surveys Stirling s paper [51] is highly recommendable. Recommendable too are [15, 39] Finally, on the topic of classical second order theories, a useful reference is Gurevich s chapter [22] Acknowledgements Colin Stirling helped initiate the writing of these notes. Thomas s handbook ....

R. Goldblatt. Logics of Time and Computation. CSLI Lecture Notes Number 7. Center for the Study of Language and Information, 1987.

....11 1 as above is iso. Indeed, if p is surjective then a coalgebra is nal in Log(T ) i it is isomorphic in Coalg(T ) to a coalgebra T TT 1 with = id . 4.2 The Canonical Model Let M be the functor P PProp, Prop a countably in nite set. The canonical model (see for example [3,6]) for the modal logic ML is the M coalgebra (L; h R ; V i) L f ML : is maximally consistentg R : L PL 7 f : 2 ) 3 2 g V : L PProp 7 Prop The canonical model is nal in the category Log ML that has M coalgebras as objects and morphisms f : A; B; are ....

Robert Goldblatt. Logics of Time and Computation, volume 7 of CSLI Lecture Notes. Center for the Study of Language and Information, Stanford University, 1992. Second Edition.

....It includes boolean constructs (union, intersection, and complement) existential qualification and universal qualification for building complex concept expressions, while roles can only be atomic. ALC corresponds to the well known modal logic K i [53] which is the basic normal multimodal logic [33, 35, 16, 39]. Satisfiability of an ALC concept (satisfiability of a K i formula) is known to be PSPACEcomplete while logical implication for ALC (for K i ) is EXPTIME complete. C is the Description Logic obtained from ALC by adding the following role constructs: union, chaining, reflexive transitive closure, ....

R. Goldblatt. Logics of time and computation, volume 7 of Lecture Notes. Center for the Study of Language and Information, second edition, 1992. 9

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R. Goldblatt. Mathematics of Modality. CSLI Lecture Notes No. 43. Center for the Study of Language and Information, Stanford University, 1993.

.... . Since is injective, is as well. Since, by Proposition 1.4(ii) p T ( is also surjective, hence iso. Now de ne = T ( 2 4.2 The Canonical Model Let M be the functor P PProp, Prop a countably in nite set. The canonical model (see for example [3,6]) for the modal logic ML is the M coalgebra (L; h R ; V i) L f ML : is maximally consistentg R : L PL 7 f : 2 ) 3 2 g V : L PProp 7 Prop Which is the case for all examples in this paper with the exception of T = P . A su cient condition for p to be ....

Robert Goldblatt. Logics of Time and Computation, volume 7 of CSLI Lecture Notes. Center for the Study of Language and Information, Stanford University, 1992. Second Edition.

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Robert Goldblatt. Mathematics of Modality. CSLI Lecture Notes No. 43. Center for the Study of Language and Information, Stanford University, 1993. Distributed by Cambridge University Press.

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R. Goldblatt, Logics of Time and Computation, CSLI Lecture Notes, vol. 7, Center for the Study of Language and Information, Ventura Hall, Stanford, CA, 1987. Distributed by Chicago University Press.

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R. Goldblatt. Logics of Time and Computation. Center for the Study of Language and Information, Stanford, California, 2 edition, 1992.

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R. Goldblatt. Logics of Time and Computation, volume 7 of CSLI Lecture Notes. Center for the Study of Language and Information, Stanford University, 2nd. ed. edition, 1992.

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R. Goldblatt. Logics of Time and Computation, Second Edition. Number 7 in CSLI Lecture Notes. Center for the Study of Language and Information, 1992.

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Robert Goldblatt. Logics of Time and Computation. Number 7 in CSLI Lecture Notes. CSLI, Center for the Study of Language and Information, second edition, 1992.

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R I Goldblatt, Logics of Time and Computation, CSLI Lecture Notes Number 7, Center for the Study of Language and Information, Stanford, 1987.

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R. Goldblatt, Logics of Time and Computation. CSLI Lecture Notes, vol. 7. 1987, Center for the Study of Language and Information.

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Robert Goldblatt. Mathematics of Modality. CSLI Lecture Notes No. 43. Center for the Study of Language and Information, Stanford, California, 1993. Distributed by Cambridge University Press.

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