Tài liệu CONCUR 2004 – Concurrency Theory- P19 docx

526 M. Viswanathan and R. Viswanathan with the HFL formula flc-diag expressing a property of the form prescribed by Theorem 2. The properties of the terms decode and init defined in Table 4 are given by the following theorem: Theorem 3. Let be a closed well-named FLC formula over the action set A. 1. 2. Consider any subformula and FLC environment For any function such that for every free in The heart of the construction is decode that shows how to decode (in HFL) the transition system representing an FLC formula. Its definition given in Table 4 is easiest understood on the basis of its property given in Theorem 3, with the variable read as standing for the function representing an environment and the variable in each of the cases read as standing for the singleton set On an argument the formula is decoded in cases according to its outermost form which in turn is inferred based on which of the propositional constants holds in (standing for For all constructs other than variables and fixed points, their corresponding cases can be understood by close analogy with the HFL-translation of these constructs given in Table 3 together with the understanding that and yield singleton sets including the corresponding subformulas of and that for constant literals term yields the set of states in which the literal holds. If is a variable, we evaluate the environment on the set (as given by the property of which is yielded by the term If is a fixed point formula, we correspondingly bind (using or v ) a new variable and decode the subformula of (given by but in an environment that is obtained by modifying the current environment to map (given by to (the case-term used for the environment argument to in the fixed point cases yields this updated environment). This ensures that when decoding the subformulas of any use of a variable corresponding to this recursive definition will be decoded as The decoding of the fixed-point cases explains the presence of the environment argument in defining decode. Finally, it is worth noting that: (1) decode is a recursive definition of a higher-order function, and (2) because decode is defined by case-analysis, it is not monotonic in the argument (standing for the formula being decoded). These features of HFL are therefore crucial to its definition. As an easy corollary of Theorem 3, we get the relevant properties of the terms flc-sem and flc-diag. Corollary 3. For any closed well-named FLC formula over the action set A, we have that TEAM LinG Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. A Higher Order Modal Fixed Point Logic 527 1. 2. iff Combined with Theorem 2 this gives us that the HFL formula flc-diag is a characteristic formula for a property of finite transition systems that is inex- pressible in FLC, and thus HFL is strictly more expressive than FLC even over finite transition systems. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. Müller-Olm, M.: A Modal Fixpoint Logic with Chop. In: Proceedings of the Sym- posium on the Theoretical Aspects of Computer Science Volume 1563 of Lecture Notes in Computer Science., Springer (1999) 510–520 Pnueli. A.: In transition from global to modular temporal reasoning about pro- grams. In: Logics and Models of Concurrent Systems. NATO ASI Series. Springer- Verlag (1984) 123–144 Misra, J., Chandy, K.M.: Proofs of network processes. IEEE Transactions on Software Engineering SE-7 (1981) 417–426 Abadi, M., Lamport, L.: Composing specifications. ACM Transactions on Pro- gramming Languages and Systems 15 (1993) 73–132 Abadi, M., Lamport, L.: Conjoining specifications. ACM Transactions on Pro- gramming Languages and Systems 17 (1995) 507–534 McMillan, K.: Circular compositional reasoning about liveness. In Pierre, L., Kropf, T., eds.: CHARME 99: Correct Hardware Design and Verification. Volume 1703 of Lecture Notes in Computer Science., Springer-Verlag (1999) 342-345 Henzinger, T.A., Qadeer, S., Rajamani, S.K., Tasiran, S.: An assume-guarantee rule for checking simulation. In: Gopalakrishnan, G., and Windley, P., eds: FMCAD 98: Formal Methods in Computer-aided Design, Volume 1522 of Lecture Notes in Computer Science., Springer-Verlag (1998) 421–432 Viswanathan, M., Viswanathan, R.: Foundations of Circular Compositional Rea- soning. In: Proceedings of the International Colloquim on Automata, Languages and Programming. Lecture Notes in Computer Science, Springer (2001) Kozen, D.: Results on the propositional Theoretical Computer Science, 27 (1983) 333–354 Stirling, C.: Modal and Temporal Logics. In Handbook of Logic in Computer Science. Volume 2. Claredon Press, Oxford, UK (1992) 477–563 Emerson, E.A., Jutla, C.S., Sistla, A.P.: On model checking for the and its fragments. Theoretical Computer Science 258 (2001) 491–522 Lange, M., Stirling, C.: Model Checking Fixed Point Logic with Chop. In: Proceed- ings of the Foundations of Software Science and Computation Structures. Volume 2303 of Lecture Notes in Computer Science., Springer (2002) 250–263 Lange, M.: Local model checking games for fixed point logic with chop. In: Pro- ceedings of the Conference on Concurrency Theory, CONCUR’02. Volume 2421 of Lecture Notes in Computer Science., Springer (2002) 240–254 Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley (1979) Burkart, O. Caucal, D., Moller, F., Steffen, B.: Verification on Infinite Structures. In: Handbook of Process Algebra. Elsevier Science Publishers (2001) 545–623 TEAM LinG Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. 528 M. Viswanathan and R. Viswanathan 16. 17. 18. Hirshfeld, Y., Jerrum, M., Moller, F.: A polynomial algorithm for deciding bisim- ilarity of normed context-free processes. Theoretical Computer Science 15 (1996) 143–159 Sénizergues, G.: Decidability of bisimulation equivalence for equations graphs of finite out-degree. In: Proceedings of the IEEE Sysmposium on the Foundations of Computer Science. (1998) 120–129 Tarski, A.: A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics 5 (1955) 285–309 TEAM LinG Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. Author Index Abdulla, Parosh Aziz 35 Amadio, Roberto M. 68 Andrews, Tony 1 Baldan, Paolo 83 Baudru, Nicolas 99 Berger, Martin 115 131 Bollig, Benedikt 146 Borgström, Johannes 161 Bozga, Liana 177 Brázdil, Tomáš 193 Briais, Sébastien 161 Brookes, Stephen 16 Bruns, Glenn 209 Bugliesi, Michele 225 Caires, Luís 240 Cîrstea, Corina 258 Clarke, Edmund 276 Colazzo, Dario 225 Corradini, Andrea 83 Crafa, Silvia 225 Dal Zilio, Silvano 68 Danos, Vincent 292 Ene, Cristian 177 Gay, Simon 497 Groote, Jan Friso 308 Hirschkoff, Daniel 325 Jagadeesan, Radha 209 Jeffrey, Alan 209 Jonsson, Bengt 35 König, Barbara 83 340 355 Krivine, Jean 292 193, 371 Lakhnech, Yassine 177 Laroussinie, F. 387 Leroux, Jérôme 402 Leucker, Martin 146 Lozes, Étienne 240 Ma, Qin 417 Maranget, Luc 417 Markey, Nicolas 387, 432 Melliès, Paul-André 448 Mokrushin, Leonid 340 Morin, Rémi 99 Nestmann, Uwe 161 Nilsson, Marcus 35 O’Hearn, Peter W. 49 Pattinson, Dirk 258 Qadeer, Shaz 1 Rajamani, Sriram K. 1 Raskin, Jean-François 432 Ravara, António 497 355 Rehof, Jakob 1 Riely, James 209 Saksena, Mayank 35 Schnoebelen, Philippe 371, 387 193 355 Sutre, Grégoire 402 Tabuada, Paulo 466 Talupur, Muralidhar 276 Thiagarajan, P.S. 340 Touili, Tayssir 276 Varacca, Daniele 481 Vasconcelos, Vasco 497 Veith, Helmut 276 Viswanathan, Mahesh 512 Viswanathan, Ramesh 512 Völzer, Hagen 481 Walukiewicz, Igor 131 Willemse, Tim 308 Winskel, Glynn 481 Xie, Yichen 1 Yi, Wang 340 TEAM LinG Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. This page intentionally left blank TEAM LinG Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. This page intentionally left blank TEAM LinG Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. This page intentionally left blank TEAM LinG Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark. . (2002) 25 0–2 63 Lange, M.: Local model checking games for fixed point logic with chop. In: Pro- ceedings of the Conference on Concurrency Theory, CONCUR 02 Verlag (1984) 12 3–1 44 Misra, J., Chandy, K.M.: Proofs of network processes. IEEE Transactions on Software Engineering SE-7 (1981) 41 7–4 26 Abadi, M., Lamport,