F. Laroussinie and Ph. Schnoebelen. A Hierarchy of Temporal Logics with Past. Theoretical Computer Science, 148:303--324, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....CTL , where truth values of atoms may depend on the branch of evaluation, and then reduce the weak one variable fragment to this logic. The main technical instrument in both proofs is the method of quasimodels [4] 2 Decidability of non local PCTL The propositional language PCT L [3, 7] extends propositional logic with temporal connectives U;S ( until, since ) and a path quantifier E ( there exists a branch (or history) The dual path quantifier A ( for all branches (or histories) is defined as an abbreviation: Aj = E:j. Other standard abbreviations we need are: 3 F j = ....

F. Laroussinie and Ph. Schnoebelen. A hierarchy of temporal logics with past. In number 775 of LNCS, pages 47--58, Springer-Verlag, 1994.

....fragment to this logic. The main technical instrument in both proofs is the method of quasimodels [11, 13] For possible applications of decidable fragments of first order temporal logics, the reader may consult, e.g. 6] 2 Decidability of non local PCT L The propositional language PCT L [5, 14] extends propositional logic with temporal connectives U, S ( until, since ) and a path quantifier E ( there exists a branch (or history) The dual path quantifier A ( for all branches (or histories) is defined as an abbreviation: Aj= E:j. Other standard abbreviations we need are: 3 F j = Uj ....

F. Laroussinie and Ph. Schnoebelen. A hierarchy of temporal logics with past. In number 775 of LNCS, pages 47--58, Springer-Verlag, 1994.

....of ltl Past with only U and S, and provides an optimal (pspace) automata theoretic algorithm. 4] studies the complexity of several fragments of ltl obtained by limiting the temporal height and the number of atomic propositions. Branching time temporal logics with past have been investigated in [13,15]. Outline of the paper. In the sequel, we rst formally de ne the structures, logics and problems under study, and sum up our results. In section 2 we prove the np completeness results, and in section 3, the pspace completeness results. We summarize our study and conclude in section 4. 1 pltl: ....

F. Laroussinie and Ph. Schnoebelen. A hierarchy of temporal logics with past. Theoretical Computer Science, 148(2):303324, 1995.

....or observers have been used. We show that all these properties can be trivially captured (and model checked) using CTL[DC] With this usefulness of past time properties, there have been several formulations extending LTL and CTL with past. Their model checking problem has also been investigated [11, 12]. However, to the best of our knowledge, there have been no tools supporting model checking of CTL with past. Our tool CTL marks perhaps a rst such avaiable tool. Moreover, the fact that we can integrate our approach with a wide variety of design notations, i.e. SMV, Verilog, Esterel, and tools ....

....model checked) using CTL[DC] The properties in previous section are also illustrative. CTl[DC] seems well suited to capture response and (discrete) real time properties. There have been several formulations extending LTL and CTL with past. Their model checking problem has also been investigated [11, 12]. However, to the best of our knowledge, there have been no tools supporting model checking of CTL with past. Our tool CTLDC marks perhaps a rst such available tool. Moreover, the fact that we can integrate our approach with a wide variety of design notations such as SMV, Verilog, Esterel, and ....

F. Laroussinie and P. Schnoebelen, A hierarchy of temporal logics with past, Theoretical Computer Science, 140(1), 1995.

....closer to the logical equivalence and thus leading to a smaller index will be crucial. We believe that the potential for such improvements is high at the price of much less understandable definitions. For the treatment of past an alternative and potentially more efficient approach in the line of [LS95] elimination of past modalities in CTL might come to mind, but the techniques used there can at least not directly be transferred to LCSA because of the intricate interaction between past and gossip modalities. ....

F. Laroussinie and P. Schnoebelen, A hierarchy of temporal logics with past, Theoretical Computer Science 148 (1995), 303--324. 15

....closer to the logical equivalence and thus leading to a smaller index will be crucial. We believe that the potential for such improvements is high at the price of much less understandable definitions. For the treatment of past an alternative and potentially more efficient approach in the line of [LS95] elimination of past modalities in CTL might come to mind, but the techniques used there can at least not directly be transferred to LCSA because of the intricate interaction between past and gossip modalities. ....

F. Laroussinie and P. Schnoebelen, A hierarchy of temporal logics with past, Theoretical Computer Science 148 (1995), 303--324.

....closer to the logical equivalence and thus leading to a smaller index will be crucial. We believe that the potential for such improvements is high at the price of much less understandable definitions. For the treatment of past an alternative and potentially more efficient approach in the line of [LS95] elimination of past modalities in CTL might come to mind, but the techniques used there can at least not directly be transferred to LCSA because of the intricate interaction between past and gossip modalities. ....

F. Laroussinie and P. Schnoebelen, A hierarchy of temporal logics with past, Theoretical Computer Science 148 (1995), 303--324.

....of such secrecy definitions using view authorizations. The backward modalities Since and Last are introduced and interpreted in a linear sense mainly to express knowledge and causality respectively. A result of this is that efficient model checking is impossible. Indeed, it is shown in [23] that CTL with a linearly interpreted past modality Since has the same expressivity as CTL [14] and thus we cannot expect an efficient model checking tool for CSL since it will cost at least a (single) exponential upper bound, as does CTL . However, we are interested in investigating symbolic ....

F. Laroussinie and P. Schnoebelen. A hierarchy of temporal logics with past. In STACS, 1994.

....Ptl. Our logic is influenced by Osl [SSC95] and the logic of [Aba89] that goes back to [GPSS80] where also a sound and complete proof system is given. In most approaches, a future oriented temporal logic is adopted. We include past oriented operators as well. This does not add expressive power [LS95] but it adds convenience for specification, and it does not complicate the logic too much. 4.1 Temporal logic Let # be an object system signature, and let # be a class signature. Definition 30. The syntax of Ptl is given by Ptl : Atom (Ptl # Ptl) Ptl U Ptl) Ptl S Ptl) ....

F. Laroussinie and Ph. Schnoebelen. A Hierarchy of Temporal Logics with Past. Theoretical Computer Science 148 (1995), 303-324.

....may have happened in the next state, formally . a , X M a, where X is the temporal next operator. For the sake of conciseness, we concentrate on future directed temporal operators. The corresponding past directed ones offer more specification convenience but do not increase specification power [LS95] Definition 3 The syntax of L is given by L : P j false j (L ) L) j (L U L) j ( M L) The predicates in P are flexible, i.e. we intend to give them time dependent meanings. The other symbols are rigid , i.e. we intend to give them time independent meanings. false is the usual logical ....

F. Laroussinie and Ph. Schnoebelen. A hierarchy of temporal logics with past. Theoretical Computer Science, 148:303--324, 1995.

....is the fragment of PCTL # where every future time modality (U or X) is immediately under the scope of path quanti er (A or E) As far as expressive power is concerned, and comparing logics with the same initial equivalence criterion we used in Section 5. 2, the following results are proved in [LS95] PCTL # can be translated to CTL # , PCTL is strictly more expressive than CTL, the CTL F 1 fragment of PCTL can be translated to CTL. The translations are not succinct: CTL F 1 can be exponentially more succinct than CTL (a consequence of [Wil99a] and PCTL # can be ....

....that PLTL model checking can be done in time O( S ) as for LTL and CTL # . The interested, or unconvinced, reader will nd a longer argumentation in [LS00a] Proposals with branching past can be found e.g. in [Rei89, Wol89, Sti92, Kam94, KP95] These de nitions for PCTL and PCTL # are from [LS95] These logics are equivalent to the CTL lp and CTL # lp (lp for linear past ) from [KP95] Our PCTL # further coincides with the OCL logic from [ZC93] The PCTL # from [HT87] di ers from our PCTL # since its path quanti ers always forget the past (see [LS95, LS00a] for a formalism allowing both ....

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F. Laroussinie and Ph. Schnoebelen. A hierarchy of temporal logics with past. Theoretical Computer Science, 148(2):303324, 1995.

.... easier to write and more natural [17] However, allowing past time makes verification algorithms harder to implement (though not necessarily from a complexity theoretic viewpoint) Additionally, all the main temporal logics with past time admit translations to their pure future fragment [11, 7, 6, 8, 24, 15, 26]. Forgettable past and the N modality. Being able to refer to past moments is often useful, but there also exist situations where it is convenient to forget the past. Consider for example, the following temporal formula G(alarm ) F problem) Spec1) where F means at some past time . ....

....that (Spec1) holds after any reset, the formula G reset ) G(alarm ) F is not exactly what we aim at. With (Spec2) a problem that occurred before the reset may account for the alarm ringing, which is probably not what we had in mind. For this kind of situations, Laroussinie and Schnoebelen [15, 16] proposed to use a new modality, N (read Now , or from now on ) that forgets all the past moments (see below for the formal semantics) With N, one can state the intended property of the alarm example via e.g. G reset ) NG(alarm ) F Not much is known about N, except that the ....

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F. Laroussinie and Ph. Schnoebelen. A hierarchy of temporal logics with past. Theoretical Computer Science, 148(2):303--324, 1995.

....propose model checking algorithms for a TL with past. None of the widely available model checking tools supports past time constructs. Translation between logics. Instead of building new model checking tools for TL with past, we suggest an alternative, so called translation based, approach [LS95, LS96] larger logics are translated into CTL (or related logics) so that the existing model checkers, e.g. SMV [McM92] can be used with no adaptation at all. Contrasting its many advantages, the main drawback of this approach is that the diagnostic a model checker sometimes provides refers to ....

....logics have been known since [GPSS80] They were used to argue that past time does not add theoretical expressivity. They were not suggested as an actual practical approach to the model checking problem for extended logics. Our contribution. In this paper, we extend our previous results [LS95] in several directions : we prove a translation theorem for NCTL, a fragment of PCTL (i.e. CTL Past) that extends the CTL F Gamma1 solved in [LS95] and we show that the translation is correct even in a framework with fairness. By necessity, NCTL only permits a restricted use of the Since ....

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F. Laroussinie and Ph. Schnoebelen. A hierarchy of temporal logics with past. Theoretical Computer Science, 148(2):303--324, 1995.

....are equivalent. Now a possible strategy to prove Theorem 6.1 is to rewrite any state formula 2 L P into an equivalent L br P formula. 6. 2 Translating L P state formulae into L br P formulae Here we use separation methods inspired from [Gab87, GPSS80] and adapted to CTL like logics as in [LPS93, LS94]. L P state formulae, which can be thought as formulae of the form 8 , may contain subformulae where some occurrences of X are not immediately preceded by a 8 operator. In order to translate them into equivalent branching time formulae, we have to propagate the 8 operator in front of each ....

F. Laroussinie and Ph. Schnoebelen. A hierarchy of temporal logics with past. In Proc. STACS'94, Caen, France, LNCS 775, pages 47--58. Springer-Verlag, February 1994.

....propose model checking algorithms for a TL with past. None of the widely available model checking tools supports past time constructs. Translation between logics. Instead of building new model checking tools for TL with past, we suggest an alternative, so called translation based, approach [16,17]: larger logics are translated into CTL (or related logics) so that the existing model checkers, e.g. SMV [22] can be used with no adaptation at all. Contrasting its many advantages, the main drawback of this approach is that the diagnostic a model checker sometimes provides refers to its input ....

....pure future logics have been known since [8] They were used to argue that past time does not add theoretical expressivity. They were not suggested as an actual practical approach to the model checking problem for extended logics. Our contribution. In this paper, we extend our previous results [16] in several directions : we prove a translation theorem for NCTL, a fragment of PCTL (i.e. CTL P ast) that extends the CTL F Gamma1 solved in [16] and we show that the translation is correct even in a framework with fairness. By necessity, NCTL only permits a restricted use of the Since ....

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F. Laroussinie and Ph. Schnoebelen. A hierarchy of temporal logics with past. Theoretical Computer Science, 148(2):303--324, 1995.

....can be defined as 9 Uf and 8Ff as 8 Uf . 2 A folk result on CTL In this setting it is well known that the 8 U operator can be expressed in terms of 9 U and 8F: 8[f U g] j 8Fg :9[ g) U ( f :g) 1) This fact may be used to simplify proofs by induction over the structure of CTL formulas (see e.g. [1, 8]) or to ease model checking (see e.g. 3, 4] In such cases 8F is much simpler than 8 U. 9 U also is often simpler to deal with than 8 U. Indeed, looking at the quantifier alternation in the semantic clause for 8 U, we see that it has the general form 8 9 k8 i : and then is ....

F. Laroussinie and Ph. Schnoebelen. A hierarchy of temporal logics with past. In Proc. 11th Ann. Symp. Theoretical Aspects of Computer Science (STACS'94), Caen, France, Feb. 1994, volume 775 of Lecture Notes in Computer Science, pages 47--58. Springer-Verlag, 1994.

....and extends [KP95, DS98] We advocate a translation based approach for the verification of CTL Past and present a translation algorithm from NCTL (a carefully delineated fragment of PCTL) into CTL. This result extends expressivity results about branching time logics with past presented in [LS95] By necessity, NCTL only permits a restricted use of the past time modalities. We show, through an extensive example (the well known Lift example [Bar87, Hal89] that these restrictions are not too drastic in practice. Indeed, we only isolated the NCTL fragment as a by product of writing our ....

....show, through an extensive example (the well known Lift example [Bar87, Hal89] that these restrictions are not too drastic in practice. Indeed, we only isolated the NCTL fragment as a by product of writing our Lift specification in PCTL. Related work. ffl Our PCTL logic was already proposed in [LS95] where a first translation algorithm is proposed. KP95] proposes extensions of CTL and CTL with two different kinds of Past (see section 3) Their CTL lp actually is our PCTL without its Now combinator. They are concerned with verification and expressivity but not with translation issues. ....

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F. Laroussinie and Ph. Schnoebelen. A hierarchy of temporal logics with past. Theoretical Computer Science, 148(2):303--324, 1995.

....for reactive systems. Regarding LB , L U and L sb , this article shows, through three translation theorems of the general form L L 0 , that they can all be translated into any other. Our translation theorems use specific techniques we developed for branching time temporal logics with past [16]. Usually, the main technical difficulty is to establish a so called separation theorem. Our motivations are not only theoretical. The translations we describe are constructive, easy to implement, and potentially useful in the automated analysis of reactive systems. For example, by showing how to ....

....be separated. Remark 4.3 In a linear time framework, 8, 9] use a different, less general, definition of separated formulae: a formula is separated (in Gabbay s sense) if it is a boolean combination of pure past and pure future formulae. Our definition is required in branching time frameworks (see [16]) For example (2) does not hold for Gabbay s definition of separated formulae: haihbi has no equivalent as a boolean combination of pure past and pure future HML bf formulae. Now we conclude the proof of Theorem 4.1 with Proposition 4.4 HML sep bf i HML. Proof Use haif j i to eliminate ....

[Article contains additional citation context not shown here]

F. Laroussinie and Ph. Schnoebelen. A hierarchy of temporal logics with past. In Proc. STACS'94, Caen, France, LNCS 775, pages 47--58. Springer-Verlag, February 1994.

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F. Laroussinie and Ph. Schnoebelen. A Hierarchy of Temporal Logics with Past. Theoretical Computer Science, 148:303--324, 1995.

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F. Laroussinie and Ph. Schnoebelen. A hierarchy of temporal logics with past. Theoretical Computer Science, 148(2):303-324, 1995.

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F. Laroussinie and P. Schnoebelen. A hierarchy of temporal logics with past. In volume 775 of Lecture Notes in Computer Science, pp. 47--58, Springer-Verlag, 1994.

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F. Laroussinie and Ph. Schnoebelen. A hierarchy of temporal logics with past. Theoretical Computer Science, 148(2):303324, 1995.

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