D. Kozen and R. Parikh. A decision procedure for the propositional mu-calculus. In Proceedings of the 2nd Workshop on Logic of Programs, Lecture Notes in Computer Science 164, pages 313--325. Springer-Verlag, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....C v D, contradicting the hypotheses. 2 This result allows us to limit our attention to concept unsatisfiability only. In order to devise a method to check a ALC concept for unsatisfiability, we exhibit a correspondence between ALC and a well known logic of programs called modal mu calculus ([71, 73, 121, 122]) which has been recently investigated for expressing temporal properties of reactive and parallel processes ( 118, 75, 28, 132, 31] Formulas Phi; Psi; of modal mu calculus are formed inductively from atomic formulas A; and variables X; according to the following abstract ....

....function associated with I and ae, and the extension function associated with M and ae map, respectively, any concept C and the corresponding formula q(C) to the same subset of Delta = S. Hence the thesis follows. 2 It follows that we may transfer both decidability and complexity results ([73, 51, 106]) for the modal mu calculus to ALC. Thus, we can immediately state what is the complexity of reasoning with ALC concepts and ALC TBoxes. Theorem 56 Satisfiability of ALC concepts, satisfiability of ALC TBoxes, and logical implication in ALC TBoxes are EXPTIME complete problems. 112 The ....

D. Kozen and R. Parikh. A decision procedure for the propositional mu-calculus. In Proceedings of the 2nd Workshop on Logic of Programs, Lecture Notes in Computer Science 164, pages 313--325. Springer-Verlag, 1983.

....Centre of the Danish National Research Foundation. can be encoded into the calculus. On binary trees the logic is as expressive as monadic second order logic of two successors [18, 6] On the other hand the logic is manageable. Satisfiability problem for the logic was shown to be EXPTIME complete [15, 23, 4] which means that it is of the same complexity as for many much less expressive logics. The best known upper bound for the model checking problem is exponential but it is polynomial if nesting of fixpoints is bounded [3, 1] One of lacking elements in this picture was finitary complete ....

Dexter Kozen and R.J.Parikh. A decision procedure for the propositional mu-calculus. In Second Workshop on Logics of Programs, 1983.

....logic RTPL. That is: Given a logical property is it possible to automatically synthesize a satisfying finite TPG (provided any such exists) This problem is also known as the problem of model construction. The satisfiability problems for CTL and the modal calculus have been proven decidable [EC82, EH85, Wol85, KP83] whereas for TCTL and T the same problems are undecidable [ACD90] In [LLW95] a bounded satisfiability problem is proven decidable for the logic L , and as our logic RTPL has adopted the same concept of formula clocks and delay modalities as presented in L , we can also only prove the same kind ....

D. Kozen and R. Parikh. A decision procedure for the propositional mu-calculus. Lecture Notes in Computer Science, 1983.

....of the Danish National Research Foundation. 1 can be encoded into the calculus. On binary trees the logic is as expressive as monadic second order logic of two successors [18, 6] On the other hand the logic is manageable. Satisfiability problem for the logic was shown to be EXPTIME complete [15, 23, 4] which means that it is of the same complexity as for many much less expressive logics. The best known upper bound for the model checking problem is exponential but it is polynomial if nesting of fixpoints is bounded [3, 1] One of lacking elements in this picture was finitary complete ....

Dexter Kozen and R.J.Parikh. A decision procedure for the propositional mu-calculus. In Second Workshop on Logics of Programs, 1983.

....such as CTL and the Mu calculus. The techniques are general and uniform. For example, the techniques above can be combined to yield a single exponential decision procedure for the Mu calculus [EJ88] This was a problem which again was not obviously in elementary, much less exponential time (cf. [KP83], SE84] Secondly, automata can provide a general, uniform framework encompassing essentially all aspects temporal reasoning about reactive systems (cf. VW84] VW86] Va87] AKS83] Ku94] Automata themselves have been proposed as a potentially useful specification language. Automata, ....

....in expressive power to tree automata. This result was first given in [EJ91] We will discuss that argument and the information that can be extracted from it. But first we note that the result can be rather easily obtained by using translation through SnS concatenated with other known results. In [KP83] it was established that the Mu calculus can be translated into SnS: L SnS. Earlier, Ra69] established the basic result that tree automata are equivalent to SnS: ta j SnS. Later, Niw88] showed that for a restricted Mu calculus, call it R, we have ta j R. By definition, R L. Putting it all ....

Kozen, D., and Parikh, R. A Decision Procedure for the Propositional Mu-calculus, Proc. of the Workshop on Logics of Programs, CarnegieMellon University, Springer LNCS no. 164, pp. 176--192, June 6--8, 1983.

....without loss of generality. 5. Reasoning with Fixpoints In this section we concentrate on developing reasoning methods to check for satis#ability concepts involving #xpoints. In particular, we exhibit a correspondence between #ALCQ and a well known logic of programs, called modal mu calculus #Kozen, 1983; Kozen Parikh, 1983; Streett Emerson, 1984, 1989#, that has been recently investigated for expressing temporal properties of reactive and parallel processes #Stirling, 1992; Larsen, 1990; Cleaveland, 1990; Winsket, 1989; Dam, 1992#. To get a better insight on the correspondence between the ....

....of generality. 5. Reasoning with Fixpoints In this section we concentrate on developing reasoning methods to check for satis#ability concepts involving #xpoints. In particular, we exhibit a correspondence between #ALCQ and a well known logic of programs, called modal mu calculus #Kozen, 1983; Kozen Parikh, 1983; Streett Emerson, 1984, 1989#, that has been recently investigated for expressing temporal properties of reactive and parallel processes #Stirling, 1992; Larsen, 1990; Cleaveland, 1990; Winsket, 1989; Dam, 1992#. To get a better insight on the correspondence between the two logics, we #rst ....

[Article contains additional citation context not shown here]

Kozen, D., & Parikh, R. #1983#. A decision procedure for the propositional mu-calculus. In Proc. of the 2nd Work. on Logic of Programs, No. 164 in Lecture Notes in Computer Science, pp. 313#325. Springer-Verlag.

.... this yields a preorder checking method that outperforms other known algorithms [CS91] No such relationship has so far been established between timed automata and any of the proposed real timed logics; The satisfiability problems for CTL and the modal calculus have been proven decidable [EC82, EH85, Wol85, KP83]; thus given a logical property it is possible to automatically synthesize a satisfying finite automata (provided any such exists) In contrast, the satisfiability problems for both TCTL and T are undecidable [ACD90, HNSY92] In this paper we present results establishing both of the two above ....

D. Kozen and R. Parikh. A decision procedure for the propositional mu-- calculus. Lecture Notes in Computer Science, 1983.

....the invariance of certain statements, etc. Among the various temporal and modal logics that have been proposed in the process algebra literature for verifying properties of concurrent systems [17, 28, 43] we focus on one of the most powerful logics of programs which is called modal mu calculus ([32, 33, 56, 57, 19]) Modal mucalculus is a logic of programs, which is strictly more expressive than logics like PDL, DeltaP DL, CTL and CTL . It has been proposed as a logic for expressing temporal properties of reactive and parallel processes in [54, 36, 9, 62, 12, 55] We refer to the excellent tutorial ....

D. Kozen and R. Parikh. A decision procedure for the propositional mu-calculus. In Proceedings of the 2nd Workshop on Logic of Programs, number 164 in Lecture Notes in Computer Science, pages 313--325. Springer-Verlag, 1983.

....expressed in SnS and the validity problem for SnS is decidable, it follows, by Proposition 5.3, that the validity problem for CTL is decidable. It can be similarly shown that modal fixpoint formulas can be expressed in SnS, yielding the decidability of the validity problem for modal fixpoint logic [KP84]. Thus, SnS provides us with a general framework for proving decidability results for modal logics [Gab73, Gab75] Unfortunately, the reduction to SnS is not too useful, since the validity problem for SnS is nonelementary (that is, there is a lower bound on the time complexity of the form 2 : ....

D. Kozen and R. Parikh. A decision procedure for the propositional mu- calculus. In Logics of Programs, volume 164 of Lecture Notes in Computer Science, pages 313--325. Springer-Verlag, 1984.

....without loss of generality. 5. Reasoning with Fixpoints In this section we concentrate on developing reasoning methods to check for satisfiability concepts involving fixpoints. In particular, we exhibit a correspondence between ALCQ and a well known logic of programs, called modal mu calculus (Kozen, 1983; Kozen Parikh, 1983; Streett Emerson, 1984, 1989) that has been recently investigated for expressing temporal properties of reactive and parallel processes (Stirling, 1992; Larsen, 1990; Cleaveland, 1990; Winsket, 1989; Dam, 1992) To get a better insight on the correspondence between the ....

....of generality. 5. Reasoning with Fixpoints In this section we concentrate on developing reasoning methods to check for satisfiability concepts involving fixpoints. In particular, we exhibit a correspondence between ALCQ and a well known logic of programs, called modal mu calculus (Kozen, 1983; Kozen Parikh, 1983; Streett Emerson, 1984, 1989) that has been recently investigated for expressing temporal properties of reactive and parallel processes (Stirling, 1992; Larsen, 1990; Cleaveland, 1990; Winsket, 1989; Dam, 1992) To get a better insight on the correspondence between the two logics, we first ....

[Article contains additional citation context not shown here]

Kozen, D., & Parikh, R. (1983). A decision procedure for the propositional mu-calculus. In Proc. of the 2nd Work. on Logic of Programs, No. 164 in Lecture Notes in Computer Science, pp. 313--325. Springer-Verlag.

No context found.

D.Kozen and R.J.Parikh, A decision procedure for the propositional mu-calculus, in : Second Workshop on Logics of Programs, Lecture Notes in Computer Science, Vol. 164 (Springer, Berlin, 1983) 313-325. 66

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