Street, RS., Propositional dynamic logic of looping and converse in elementary decidable, Information and Control, 54:121-141,1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that, in order to study (a variant of) PDL, we only need to consider the equivalence classes of in the class of all Kripke structures. 4.3 On decidability of PDL with repeat and other philosophical topics An important question to ask is whether the logic of RPDL remains decidable. Streett [34] has argued that this is the case. His argument appears to be, however, incorrect. It seems to go awry in an inductive proof of the central lemma 4.14 in his paper, where an illegal application of the induction hypothesis is used. This error calls for some attention. Remember the Fisher Ladner ....

....The relation 6 is, however, not well founded. Consider the following example of an infinitely decreasing chain, with a primitive, a ]OE [a] a ]OE [a ]OE Delta Delta Delta Hence, we may not perform induction over this relation. This is exactly what has happened in [34]. Inspection of 6 shows that this happens only in this case: the operator causes the trouble. This implies that, when we want to perform induction on the structure of OE, we have to prove the case OE = hff i manually; we have no induction hypothesis to invoke. 43 Kozen and Tiuryn [17] ....

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Street, R.S., "Propositional Dynamic Logic of Looping and Converse Is Elementary Decidable", Infor. and Control 54 (1982), 121--141. 68

..... is the set of all the states that appears infinitely often in the sequences. 3. CLASSIFICATION OF w w AUTOMATA There are six types of w automata depending of the acceptance condition F : B automaton (B chi) 3] M automaton (Muller) 4] R automaton (Rabin) 5] S automaton (Street) [6], L automaton (Kurshan) 7] and automaton (Manna y Pnueli) 8] If M is a B automaton the F S is a set of states. Therefore, a run r is accepted by a B automaton M if inf(r) F f (1) Thus, some states s f S are specified as accepting states. In order for a word to be accepted, these ....

Street, RS., Propositional dynamic logic of looping and converse in elementary decidable, Information and Control, 54:121-141,1982.

....(s; t) 2 R M Psi n. According to these definitions, any model M of OE rel is also a model of K if all states of M numerically satisfy all atomic formulae corresponding to number restrictions. We show that if OE rel is satisfiable then we can construct a model in which this is indeed the case. [Str82] shows that the tree model property holds for CPDL (see [Str82] for a formal definition of tree model) This result carries over immediately to CPDL Gamma , and therefore every satisfiable CPDL Gamma formula admits a model which is a tree, if we view each state as a node and each ....

....M of OE rel is also a model of K if all states of M numerically satisfy all atomic formulae corresponding to number restrictions. We show that if OE rel is satisfiable then we can construct a model in which this is indeed the case. Str82] shows that the tree model property holds for CPDL (see [Str82] for a formal definition of tree model) This result carries over immediately to CPDL Gamma , and therefore every satisfiable CPDL Gamma formula admits a model which is a tree, if we view each state as a node and each transition between states as an arc labeled with the corresponding ....

R.S. Street. Propositional dynamic logic of looping and converse is elementarily decidable. Information and Control, 54:121--141, 1982.

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Street, RS., Propositional dynamic logic of looping and converse in elementary decidable, Information and Control, 54:121-141,1982.

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