M. Otto. Capturing bisimulation-invariant Ptime. In Proceedings of the 4th Symposium on Logical Foundations of Computer Science, volume 1234 of Lecture Notes in Computer Science, pages 294--305. Springer-Verlag, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(see Gurevich [9] In the following we always use the notions logic and capturing in the sense of Gurevich. Appeared in Computer Science Logic, 11th International Workshop (CSL 97) Lecture Notes in Computer Science 1414, pp. 220 238. c Springer Verlag 1998 For example, recently Otto [17] considered bisimulation on Kripke models with a distinguished world. He proved that a straightforward extension of the propositional calculus captures bisimulation invariant P. The approach to prove this and similar results is through canonization. Let be an equivalence relation on a class ....

....model C with I k (C) I k (A) Theorem 2 Let k 3. If for each vocabulary there is an algorithm that, given an invariant I k (A) of a structure A, computes a model B with I k (B) I k (A) in time polynomial in the size of the smallest model C with I k (C) I k (A) then P = NP. Otto [17] slightly modified the question by asking if the following problem is in P (for k 3 and vocabularies ) L k [ INVERSION Input: A finite k structure I and a positive integer n in unary encoding. Question: Is there a structure A of size n such that I k (A) I Theorem 3 Let k ....

M. Otto. Capturing bisimulation-invariant Ptime. In Proceedings of the 4th Symposium on Logical Foundations of Computer Science, volume 1234 of Lecture Notes in Computer Science, pages 294--305. Springer-Verlag, 1997.

....in P if, and only if, C is definable in L. Actually, this notion needs to be refined by some additional effectivity conditions to avoid trivialities (see Gurevich [9] In the following we always use the notions logic and capturing in the sense of Gurevich. For example, recently Otto [17] considered bisimulation on Kripke models with a distinguished world. He proved that a straightforward extension of the propositional calculus captures bisimulation invariant P. The approach to prove this and similar results is through canonization. Let j be an equivalence relation on a class ....

....model C with I k (C) I k (A) Theorem 2. Let k 3. If for each vocabulary there is an algorithm that, given an invariant I k (A) of a structure A, computes a model B with I k (B) I k (A) in time polynomial in the size of the smallest model C with I k (C) I k (A) then P = NP. Otto [17] slightly modified the question by asking if the following problem is in P (for k 3 and vocabularies ) L k [ INVERSION Input: A finite k structure I and a positive integer n in unary encoding. Question: Is there a structure A of size n such that I k (A) I Theorem 3. Let k 3. ....

M. Otto. Capturing bisimulation-invariant Ptime. In Proceedings of the 4th Symposium on Logical Foundations of Computer Science, volume 1234 of Lecture Notes in Computer Science, pages 294--305. Springer-Verlag, 1997.

....set theoretic operations. Section 7 is devoted to describing the type of the linear ordering defined on the universe HFA 1 and of its restrictions to HFA and HF. Finally, Section 8 concludes the paper by comparing our present work and [22,23] with the related and independent paper of M. Otto [27]. The latter is devoted to definability and capturing bisimulation invariant PTIME in finitely dimensional modal calculus which, as we show, corresponds to the language Delta L 0 . Agreement. We will use abbreviations slo and slp o for strict linear order and, respectively, strict ....

....in [3] Bisimulation and corresponding approach to querying for finite graphs representing data has been discussed also in [19] and [8] but not from the set theoretical viewpoint. There exists also the following interesting connection with modal logic. In a closely related paper of M. Otto [27] it was captured the class of bisimulation invariant PTIME computable global predicates over finite Kripke models (which are also graphs) in terms of finitely dimensional modal calculus. To this end, a linear preorder on Kripke models, essentially the same as our OE, was defined. Actually we ....

M. Otto, Capturing Bisimulation-Invariant PTIME. In: S. Adian, A. Nerode (Eds.): Logical Foundations of Computer Science, 4th International Symposium, LFCS'97 , Yaroslavl, Russia, July 1997, Proceedings. Lecture Notes in Computer Science, Vol. 1234, Springer, 1997, 294-305.

....application of Lemma 3.17 moreover shows that also the fragment of FP 2 that corresponds to the extension of L by universal FO 2 sentences without equality is undecidable [23] Another interesting family of extensions of L are the k dimensional calculi. They have been introduced by Otto [28] who shows that these languages can express precisely those properties of Kripke structures that are invariant under bisimulation and decidable in polynomial time. Unfortunately, these languages do not inherit the nice algorithmic properties of L : already the satisfiability of the ....

....express precisely those properties of Kripke structures that are invariant under bisimulation and decidable in polynomial time. Unfortunately, these languages do not inherit the nice algorithmic properties of L : already the satisfiability of the two dimensional calculus is highly undecidable [28]. 3.5 Well orderings and CL 2 The following problems are recursively equivalent (both in their general and in their finitistic versions) the FO 2 theory of several built in well founded relations; the FO 2 theory of several built in well orderings; the satisfiability problem for ....

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M. Otto, Capturing bisimulation-invariant Ptime, in Proc. 4th Symposium on Logical Foundations of Computer Science, 1997, LNCS 1234, 1997, pp. 294--305.

.... Pi 1 and C Pi 1 , as a sample application of Theorem 4.3. We show that the corresponding query classes Ptime L (those boolean Ptime queries on finite Kripke structures that are closed under bisimulation equivalence , respectively C ) can be captured. For Ptime L Pi 1 compare also [Ott97b]. Proposition 4.4 For the complete invariants I Pi (for bisimulation equivalence) and I Pi C (for bisimulation equivalence with counting) of Definition 3.1: Ptime L Pi 1 j Ptime ffi I Pi and Ptime C Pi 1 j Ptime ffi I Pi C : Sketch of proof. We show that both I Pi and ....

M. Otto. Capturing bisimulation-invariant Ptime. In Proc. of the 4th Symposium on Logical Foundations of Computer Science, 1997, 294--305, Lecture Notes in Computer Science 1234, Springer 1997.

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