D. Niwi'nski. Fixed points vs. infinite generation. In Proc. 3rd IEEE Symp. on Logic in Comput. Sci., pages 402--409, 1988. 20  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... For Emerson and Jutla [2] O is the class of all closed terms of the calculus on trees (cf. 9, 4, 7, 1] for which the Complementation property is easy: every term can be syntactically transformed into its dual where L( is the complement of L( On the other hand, Niwi nski has proved [8] the Equivalence property for the calculus without intersection, which applies also to non closed terms. But even if does not contain intersection, its dual may contain intersection. Thus, Emerson and Jutla, still using a game theoretical argument, prove that for every closed term there is an ....

....1 If F is recognizable, then F are recognizable. This lemma is nothing but Lemmas 3.2 and 3.4 of [10] However, for completeness of this paper, we prove it in the last section of this paper. 6.3 The complementation theorem The complementation theorem is now easy. By Niwi nski s theorem [8], for any recognizable set K of trees there is a closed term , not using the intersection operator, such that K = By Proposition 8, K = by Proposition 9, K = and by Theorem 1, K is recognizable. For completeness we give a sketch of the proof of the Niwi nski s result, in a ....

D. Niwi'nski. Fixed points vs. infinite generation. In Proc. 3rd IEEE Symp. on Logic in Comput. Sci., pages 402--409, 1988. 20

....tree, CTL is exactly as expressive as S2S with quantification restricted to paths, and that ECTL is exactly as expressive as S2S with quantification restricted to chains. For n 2, however, SnS with quantification restricted to chains is strictly more expressive than ECTL . ffl Niwinski [44] shows that a calculus based on powerset al..gebras of trees is exactly as expressive as a type of infinite tree grammar which in turn is exactly as expressive as SnS. ffl Huttel [26] shows that a calculus generalising F(fl; by having n successor operations is exactly as expressive as SnS up to ....

D. Niwi'nski. Fixed points vs. infinite generation. In Proc. 3rd Ann. Symp. Logic in Computer Science, pages 402--409, 1988.

....complementation construction for Rabin automata. The natural translation from modal mu calculus to SnS has been formulated at least in [4, 6, 56, 59] and translations between Rabin automata and what is essentially the strongly aconjunctive fragment of modal mu calculus were given by Niwinski in [70, 71]. Emerson and Jutla [33] described a translation from the full mu calculus to Rabin automata by using the correspondence between mu calculus and alternating tree automata, and reducing these to ordinary automata by a construction related to Safra s [80] Furthermore, Chapter 3 Automata 63 ....

....as in Lemma 3.2.18. 2 Related translations between Rabin automata and parity automata can be found in [65] Translations between ordinary Rabin automata and what is essentially the restricted fragment of the modal mu calculus Kn, i.e. ordinary FRautomata, have also been described by Niwinski in [70, 71]. The relation between the index of Rabin automata and the alternation class of the corresponding formula was first stated there. We can now summarise the equiexpressiveness results obtained so far concerning the modal mu calculus Kn, first recurrence automata and Rabin automata. Theorem 3.2.24 ....

Niwinski, D.: Fixed points vs. infinite generation, in Proceedings of the 3rd IEEE Symposium on Logic in Computer Science, 1988, pp. 402-409

....a modal mu calculus with label set f1; ng can define the Rabin recognizable tree languages up to an equivalence similar to the observational equivalence of Milner. 1 Introduction In [11] it was shown that the temporal logic ETL [10] can define exactly the class of regular languages. In [7] it was shown that a fixed point calculus whose signature apart from maximal and minimal fixed points and disjunction includes the usual operators on trees can define exactly the sets of infinite trees recognized by Rabin tree automata [8] this class of sets corresponds to the class of structures ....

Niwinski, D., Fixed Points vs. Infinite Generation, Proc. 3rd Symp. on Logic in Comp. Sci., Edinburgh 1988.

....; Bkig of pairs of sets of states. A run is accepting if there exists an index i for which the set G i is visited infinitely often and the set B i is visited only finitely often. In [Rab69] Rabin describes 1 In fact, alternating parity tree automata are exactly as expressive as the calculus [Niw88, EJ91]. On the other hand, weak alternating tree automata are exactly as expressive as the alternation free fragment of calculus [KV98] a translation of formulas of monadic second order logic to Rabin tree automata. Today, Rabin automata are used in order to reason about specifications of the full ....

D. Niwinski. Fixed-points vs. infinite generation. In Proc. 3rd Symposium on Logic in Computer Science, 1988.

....recognizable tree sets is not closed under complement. More sets of trees become definable if we allow disjunctions as in ( with n members (for n = 1; 2 : The minimal n that suffices for the definition of a given set T of trees is called the (nondeterministic) Rabin index of T . Niwinski [Niw88] showed that the Rabin index induces indeed a strict hierarchy of tree language classes (called RABIN 1 , RABIN 2 , etc. below) This is in contrast to the case of words where this hierarchy collapses (because S1S is equivalent already to nondeterministic sequential Buchi automata) Another ....

D. Niwinski. Fixed points vs. infinite generation. In Proc. 3rd Ann. IEEE Symp. on Logic in Computer Science, pages 402 -- 409, 1988.

.... [B 62] Mul63] Rab69] Rab70] Str82] It has been shown also that RTA are more expressive than all the commonly used specification formalisms for finite state parallel programs [VW86] Wol89] and that they are expressively equivalent to the branching time propositional calculus [Koz83] Niw88] We show that in the class of RTA definable sets we have also the nice fact that any property is the intersection of a safety and a liveness property. In practice, it is often sufficient to consider only safety properties. For example, most of service properties of communication protocols are ....

D. Niwinski. Fixed points vs. infinite generation. In Proc. of Third. Symp. on Logic in Computer Science. Computer Society Press, 1988.

....infinite path in that tree is accepting, i.e. the largest priority occurring infinitely often on that path is even. It is possible to show that Phi 2 F if and only if there exists an accepting k P A derivation of Phi. The claim is proved by induction on k. See the proof of Theorem 6. 1 in [20] (or Theorem 3.2 in [21] for a similar argument. Theorem 1 follows by taking k = n. The case for k negative is trivial for k negative we have Phi 2 F P (A) if and only if Phi 2 A and there is a k P A derivation of Phi if and only if Phi 2 A. 3 Uniform Programs Let [ P; p] denote ....

D. Niwinski. Fixed points vs. infinite generation. In LICS '88, pages 402--409, 1988.

....path in that tree is accepting, i.e. the largest priority occurring infinitely often on that path is even. It is possible to show that Phi 2 F k P (A) if and only if there exists an accepting k P A derivation of Phi. The claim is proved by induction on k. See the proof of Theorem 6. 1 in [20] (or Theorem 3.2 in [21] for a similar argument. Theorem 1 follows by taking k = n. The case for k negative is trivial for k negative we have Phi 2 F k P (A) if and only if Phi 2 A and there is a k P A derivation of Phi if and only if Phi 2 A. 3 Uniform Programs Let [ P; p] denote ....

D. Niwinski. Fixed points vs. infinite generation. In LICS '88, pages 402--409, 1988.

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