A. W. Mostowski. Hierarchies of weak automata and weak monadic formulas. Theoretical Computer Science, 83:323--335, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is accepted by A. It turns out that the same diagonal argument can be applied to weak alternating automata introduced by Muller, Saoudi, and Schupp to characterize the weakly definable sets of trees [11] providing a direct proof of the Mostowski s result on the hierarchy of alternating automata [10], instead of relying on a result of Thomas [15] 1 Alternating parity automata An alternating parity automaton is an alternating automaton (see [12] where the acceptance criterion is given by a parity condition. Namely, it is a tuple hA; Q; ffi; n; ri where ffl the alphabet A is a finite set ....

A. W. Mostowski. Hierarchies of weak automata and weak monadic formulas. Theoretical Computer Science, 83:323--335, 1991.

....deterministic and The authors were supported by Polish KBN grant No. 7 T11C 027 20 Preprint submitted to Elsevier Science 16 July 2001 nondeterministic automata on in nite trees was proved [9] in 1986, but an analogous problem for alternating automata had remained open for a while. Mostowski [8] investigated the hierarchy of so called weak alternating tree automata and showed its strictness using a reduction to a hierarchy of weak monadic second order quanti ers formerly examined by Thomas [17] Skurczy nski [16] further showed that even nondeterministic weak tree automata can recognize ....

....indices induce a hierarchy of weak automata in natural way. We will call it weak hierarchy, as opposed to strong hierarchy, i.e. the hierarchy of Mostowski indices of alternating tree automata. 3 Strictness of the weak hierarchy In this section we present a new proof of the result of Mostowski [8] stating the strictness of the weak hierarchy. Let = fa; bg. 6 Let L 0;0 be the language of trees having only b on the leftmost path. Let L 1;1 be the language of trees such that a occurs on the leftmost path. Let L 1;2 be the language of tress such that for every k the subtree rooted in the ....

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A. W. Mostowski. Hierarchies of weak automata and weak monadic formulas. Theoretical Comput. Sci., 83:323-335, 1991.

....is accepted by A. It turns out that the same diagonal argument can be applied to weak alternating automata introduced by Muller, Saoudi, and Schupp to characterize the weakly definable sets of trees [11] providing a direct proof of the Mostowski s result on the hierarchy of alternating automata [10], instead of relying on a result of Thomas [15] 1 Alternating parity automata An alternating parity automaton is an alternating automaton (see [12] where the acceptance criterion is given by a parity condition. Namely, it is a tuple hA; Q; ffi; n; ri where ffl the alphabet A is a finite set ....

A. W. Mostowski. Hierarchies of weak automata and weak monadic formulas. Theoretical Computer Science, 83:323--335, 1991.

....ae of A on s is accepting. Notice the difference between FR and Buchi automata: in former we require that the first infinitely often occurring state is accepting, in latter that some such state is. There is a natural way of mapping Rabin automata to FR automata, related to the constructions of [15] and [8] on automata with parity acceptance conditions. Definition20. Let A = Q; q 0 ; ffi; G 1 ; R 1 ) Gm ; Rm ) be a det. Rabinautomaton, and let C be the set of all permutations of the string 12 : m. Define q(s) 2 Q for every s 2 Sigma by ffl q(ffl) q 0 and ffl q(s ....

Mostowski, A. W.: Hierarchies of weak automata and weak monadic formulas, in Theoretical Computer Science, vol. 83, 1991, pp. 232-335

....here represents a decidable part of equality. It is known that the equality of terms can not be directly included in the language of set constraints. Our approach is based on a reduction of set constraints to the monadic class given in a recent paper by L. Bachmair, H. Ganzinger, and U. Waldmann [2]. References [1] A. Aiken, D. Kozen, and E. L. Wimmers, Decidability of systems of set constraints with negative constraints. Technical Report 93 1362, Computer Science Department, Cornell University, June 1993. 2] L. Bachmair, H. Ganzinger, and U. Waldmann, Set constrains are the monadic ....

....class given in a recent paper by L. Bachmair, H. Ganzinger, and U. Waldmann [2] References [1] A. Aiken, D. Kozen, and E. L. Wimmers, Decidability of systems of set constraints with negative constraints. Technical Report 93 1362, Computer Science Department, Cornell University, June 1993. [2] L. Bachmair, H. Ganzinger, and U. Waldmann, Set constrains are the monadic class. In Eight Annual Symposium on Logic in Computer Science, pages 75 83, 1993. 3] R. Gilleron, S. Tison and M. Tommasi, Solving systems of set constraints with negated subset relationships. In Proceedings of the 34 ....

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Mostowski A. W., Hierarchies of weak automata and weak monadic formulas, Theoretical Computer Science 83 (1991) 323--335.

.... independently for obtaining no memory strategies in [Mst91a] and [EJ91] where it is called parity condition ) The simulation of Muller acceptance using Buchi s version of the LAR, as presented here, yields smaller transition graphs with Rabin chain condition than previous constructions in [Mst91b] and [Ca94] Gurevich and Harrington showed the much more general result that strategies with an LAR memory suffice for B( Sigma 0 2 ) games even over infinite graphs ( Forgetful Determinacy Theorem [GH82] YY90] Ze94] By lack of space we cannot enter this interesting subject here. For ....

A.W. Mostowski, Hierarchies of weak automata and weak monadic formulas, Theor. Comput. Sci. 83 (1991), 323-335.

.... automata are easily converted into Muller tree automata: Given a Rabin chain tree automaton with acceptance component Omega Gamma fix a system F of final state sets by including all sets F which satisfy the Rabin chain condition ( applied to F in place of In(aej ) The converse is also true ([Mst91b], Car94] We give a simple proof, using a data structure of Buchi [Bu83] Theorem 6.5 For any Muller tree automaton one can construct an equivalent Rabin chain tree automaton. Proof. Let A = Q; A; q 0 ; Delta; F) be a Muller tree automaton, assuming without loss of generality that Q = f1; ....

A.W. Mostowski, Hierarchies of weak automata and weak monadic formulas, Theor. Comput. Sci. 83 (1991), 323-335.

....is successful. An language L A is said to be Rabin recognizable if it consists of all words accepted by a Rabin automaton. A special form of Rabin automaton will be useful in Section 3, the Rabin chain automaton (or: parity automaton) introduced in [Mos84] and applied independently in [Mos91] and [EJ91] for the complementation of tree automata. A Rabin chain automaton is a Rabin automaton whose list of designated pairs (E k ; F k ) of state sets form an increasing chain: E 1 F 1 E 2 F 2 : Em Fm : It is well known that any Rabin automaton is equivalent to a Rabin ....

....for the complementation of tree automata. A Rabin chain automaton is a Rabin automaton whose list of designated pairs (E k ; F k ) of state sets form an increasing chain: E 1 F 1 E 2 F 2 : Em Fm : It is well known that any Rabin automaton is equivalent to a Rabin chain automaton ([Mos91], Car94] A simpler simulation can be based on the idea of latest appearance record ( GH82] or order vector ( Buc83] The desired automaton has as states all permutations of states of the given automaton (order vectors) where each vector is extended by a pointer ( hit in Buchi s ....

A.W. Mostowski. Hierarchies of weak automata and weak monadic formulas. TCS, 83(2):323 -- 35, 1991.

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