Bolotov, A.

Hermo, M.

Lucio, P.

English

WestminsterResearch

WestminsterResearchhttp://www.westminster.ac.uk/westminsterresearchA tree-style one-pass tableau for an extension of ECTL+Bolotov, A., Hermo, M. and Lucio, P.A paper presented at the 25th Workshop on Automated Reasoning: Bridging the Gap between Theory and Practice, University of Cambridge, Apr 2018, University of Cambridge.The WestminsterResearch online digital archive at the University of Westminster aims to make the research output of the University available to a wider audience. Copyright and Moral Rights remain with the authors and/or copyright owners.A tree-style one-pass tableau for a extension of ECTL+.Alexander Bolotov1 Montserrat Hermo2 Paqui Lucio21 University of Westminster, W1W 6UW, London, UK.2 University of the Basque Country, 20018-San Sebastián, Spain.Abstract: Only restricted versions of fairness are expressible in the well-known branching temporal logicsECTL and ECTL+, while the full expressiveness of branching-time logic in CTL? makes this logic extremelychallenging for the application of the tableau technique. Tree-shaped one-pass tableaux are well suited for theautomation and are amenable for the implementation. We present here a sound and complete method of tree-style one-pass tableau for a sub-logic of CTL? which is more expressive than the logic ECTL+allowing theformulation of some fairness constraints with ‘until’ operator. The provided example follows by an algorithmfor constructing a systematic tableau that enables to prove completeness.1 IntroductionFor the specification of the reactive and distributed sys-tems, or, most recently, autonomous systems, where themodelling of the possibilities ‘branching’ into the future isessential, the branching-time logics (BTL) give us an ap-propriate framework. The most used class of formalismsare ‘CTL’ (Computation Tree Logic) type logics: CTL it-self, ECTL (Extended CTL) [2] that was defined to enablesimple fairness constraints but not their Boolean combi-nations and ECTL+ ([3]) which further extends ECTL al-lowing Boolean combinations of ECTL fairness constraints(but not permitting their nesting). The literature on fairnessconstraints, even in linear-time setting, lacks the analysis oftheir formulation with a ‘stronger’ temporal operator - U(‘until’) such as (AU B) or AU B. Here we bridgeLamport Notation / Formulae expressibleCTL-type name here but not aboveB(U ,◦) / CTLB(U ,◦,♦)/ ECTL E(♦q)B+(U ,◦,♦) / ECTL+ E(♦q ∧ ♦r)B+(U ,◦, U ) A((pU q) ∧ (sU ¬q))B?(U ,◦) / CTL? A♦(◦p ∧ E◦¬p)Figure 1: BTL classification.this gap, providing an analysis of such complex fairnessconstraints with U (also allowing the nesting of temporaloperators) in the branching-time setting weaker than CTL?.Thus, we consider the logic that extends ECTL+ with themodalities U and U . While the addition of the formerdoes not increase the ECTL+ expressiveness1, AU (B)cannot be expressed in the ECTL+ language. The fairnessconstraint A(pU q) can be read as ‘q is true along all pathsof the computation except possibly their finite initial inter-val, where p is true’.In Figure 1 we fit our logic into the hierarchy ofbranching-time logics: ‘B’ is used for ‘Branching’, fol-lowed by the set of only allowed modalities as parame-1(AU B) can be expressed in ECTL+ by (A ∨B) ∧ ♦B.ters; B+ indicates admissible Boolean combinations of themodalities and B? reflects ‘no restrictions’ in either con-catenations of the modalities or Boolean combinations be-tween them. We present a tree-style one-pass tableau forthe logic B+(U ,◦, U ) continuing the analogous devel-opments in linear-time case [4].B+(U ,◦, U ) language is defined with linear-timetemporal operators  (always), ◦ (next time), and U (un-til), and path quantifiers - A (on all future paths) and E (onsome future path). The state (σ) and path (pi) formulae aredefined below (state formulae are wff).σ ::= L | σ1 ∧ σ2 | σ1 ∨ σ2 | Api | Epipi ::= pi1 ∧ pi2 | pi1 ∨ pi2 | ◦σ | σ U σ | σ U (σ) | σ |(σ U σ)2 The tableau methodWhile our tableaux are AND-OR trees with nodes labelledby sets of state formulae, the only rule which introducesAND-nodes is the next-state rule:Σ,A◦Φ1, . . . ,A◦Φn,E◦Ψ1, . . . ,E◦ΨmAΦ1, . . . ,AΦn,EΨ1 & . . . & AΦ1, . . . ,AΦn,EΨmFigure 2: NEXT-STATE RULE. (Σ is a (possibly empty) setof literals; Φi,Ψi are non-empty sets of formulae.)The next-state rule labels a branch that splits into mbranches, at the node labelled by the premise of this rule.The conclusion of this rule uses the & symbol to reflect togeneration of m AND-successor nodes. We also apply α-and β-rules: an α-rule expands a branch at the node labelledby its premise, with a node labelled by the conclusion; a β-rule splits a branch by two or three OR-nodes labelled bythe formulae in its conclusion (separated by |).We handle inputs in a new, branching-time, setting in‘analytic" way, extending similar construction for linear-time logic [4]. This extension is possible due to the defini-tion of the ‘context’ in which eventualities are to be fulfilledin this new setting. Our β+-rules are characteristic (andcrucial!) for our construction. They tackle difficult casesof formulae in B+(U ,◦, U ). The β+-rules, similarly toΣ,E((σ1 U σ2) ∧Π)Σ,E((σ1 U σ2) ∧ ◦(σ1 U σ2) ∧Π)Figure 3: α-RULE (EU ). ( Σ is a (possibly empty) set ofstate-formulae and Π is a (possibly empty) conjunction ofpath-formulae.)Σ,A((σ) ∨Π)Σ, σ,A((◦σ) ∨Π) | Σ,AΠFigure 4: β-RULE (Aσ). ( Σ, σ is a set of state-formulaeand Π is a (possibly empty) disjunction of path-formulae.)β-rules, split a branch into two or three branches; these arethe only rules that utilise ‘context’ to force the soonest sat-isfaction of the eventualities. The context is given by thesets of state (Σ) and path (Π) formulae. While, the outer-context was already used in the linear-time tableaux [4], forbranching-time, the new concept of the ‘inner-context’ isintroduced. Figure 5 shows a β+ rule that handles a dis-junction of formulae, including U in the scope of the Aquantifier.Σ,A((σ U σ′) ∨Π)Σ, σ′ | Σ, σ′,A(◦((σ ∧ (¬Σ ∨ ϕΠ))U σ′) ∨Π) | Σ,AΠFigure 5: β+-RULE (AU σ)+. ( Σ, σ, σ′ is a set of state-formulae and Π is a (possibly empty) disjunction of path-formulae, ϕΠ is the state-formula introduced by Def. 1.)Definition 1 Let Π be a disjunction of path-formulae of thethree forms σ,σ U σ′ and (σ U σ′) where σ and σ′ arestate-formulae. The formula ϕΠ to be the following dis-junction of state-formulae:∨σ∈Πσ ∨∨σ U σ′∈Πσ′ ∨∨(σ U σ′)∈ΠE(TU σ′).3 ExamplesOur example of an open tableau illustrates the use of theinner context (see Fig. 6). This tableaux is constructed bya systematic algorithm that keeps (along a branch) exactlyone –if there is some– marked eventuality forcing its ful-filment. In Fig. 6, the semicolon inside the A-quantifierstands for disjunction, the marked eventuality is in black-boxes and (Q◦) is the next-state rule. Note that Π consistsof a path-formula p, so ϕΠ is just p. Since the markedeventuality is TU ¬p and the outer-context Σ is empty, thesubformula (σ ∧ (¬Σ ∨ ϕΠ) in the conclusion of the rulein Fig. 5 is just p. It is notable that this inner-context p en-ables the central branch to loop, given a model of the initialformula that –in this branch– does not force the eventuality,but satisfies the other disjunct in the A-quantifier: p.Figure 6: Open Tableau4 ConclusionWe presented a tree-style one pass tableaux method fora new logic in the family of BTL – B+(U ,◦, U ) –which extends the expressiveness of ECTL+ fairness bya new class of fairness constraints utilising the U opera-tor. The full details and the correctness proof are given in[1]. The tableaux rules that invoke the inner-context, en-abled us to handle a particularly difficult class of formu-lae: A-disjunctive formulae with eventualities. The proofof correctness of β+-rules is based on identifying relevantstate-formulae inside specific path-modalities. This opensthe prospect to study more expressive logics (eg CTL?)by identifying subformulae that do not affect the ‘context’which allows to simplify given structures. The presentedtechnique, being the extension of a similar one for thelinear-time setting, is amenable for implementation. In therefinement and implementation of our new algorithm wewill rely on similar techniques used in the implementationof its linear-time analogue.References[1] A. Bolotov, M. Hermo, and P. Lucio. Ex-tending fairness expressibility of ECTL+:a tree-style one-pass tableau approach,http://www.sc.ehu.es/jiwlucap/techreport18.pdf.Technical report, February 2018.[2] E. A. Emerson and J. Y. Halpern. Decision proceduresand expressiveness in the temporal logic of branch-ing time. Journal of Computer and System Sciences,30(1):1 – 24, 1985.[3] E. A. Emerson and J. Y. Halpern. Sometimes and notnever revisited: On branching versus linear time tem-poral logic. J. ACM, 33(1):151–178, January 1986.[4] J. Gaintzarain, M. Hermo, P. Lucio, M. Navarro, andF. Orejas. Dual systems of tableaux and sequents forPLTL. Journal of Logic and Algebraic Programming,78(8):701–722, 2009.

A tree-style one-pass tableau for an extension of ECTL+

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A tree-style one-pass tableau for an extension of ECTL+

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