Satisfiability Games for Branching-Time Logics

The satisfiability problem for branching-time temporal logics like CTL*, CTL and CTL+ has important applications in program specification and verification. Their computational complexities are known: CTL* and CTL+ are complete for doubly exponential time, CTL is complete for single exponential time. Some decision procedures for these logics are known; they use tree automata, tableaux or axiom systems. In this paper we present a uniform game-theoretic framework for the satisfiability problem of these branching-time temporal logics. We define satisfiability games for the full branching-time temporal logic CTL* using a high-level definition of winning condition that captures the essence of well-foundedness of least fixpoint unfoldings. These winning conditions form formal languages of \omega-words. We analyse which kinds of deterministic {\omega}-automata are needed in which case in order to recognise these languages. We then obtain a reduction to the problem of solving parity or B\"uchi games. The worst-case complexity of the obtained algorithms matches the known lower bounds for these logics. This approach provides a uniform, yet complexity-theoretically optimal treatment of satisfiability for branching-time temporal logics. It separates the use of temporal logic machinery from the use of automata thus preserving a syntactical relationship between the input formula and the object that represents satisfiability, i.e. a winning strategy in a parity or B\"uchi game. The games presented here work on a Fischer-Ladner closure of the input formula only. Last but not least, the games presented here come with an attempt at providing tool support for the satisfiability problem of complex branching-time logics like CTL* and CTL+

Separation of Test-Free Propositional Dynamic Logics over Context-Free Languages

For a class L of languages let PDL[L] be an extension of Propositional Dynamic Logic which allows programs to be in a language of L rather than just to be regular. If L contains a non-regular language, PDL[L] can express non-regular properties, in contrast to pure PDL. For regular, visibly pushdown and deterministic context-free languages, the separation of the respective PDLs can be proven by automata-theoretic techniques. However, these techniques introduce non-determinism on the automata side. As non-determinism is also the difference between DCFL and CFL, these techniques seem to be inappropriate to separate PDL[DCFL] from PDL[CFL]. Nevertheless, this separation is shown but for programs without test operators.Comment: In Proceedings GandALF 2011, arXiv:1106.081

A Decision Procedure for CTL* Based on Tableaux and Automata

Abstract. We present a decision procedure for the full branching-time logic CTL ∗ which is based on tableaux with global conditions on infinite branches. These conditions can be checked using automata-theoretic machinery. The decision procedure then consists of a doubly exponential reduction to the problem of solving a parity game. This has advantages over existing decision procedures for CTL ∗ , in particular the automatatheoretic ones: the underlying tableaux only work on subformulas of the input formula. The relationship between the structure of such tableaux and the input formula is given by very intuitive tableau rules. Furthermore, runtime experiments with an implementation of this procedure in the MLSolver tool show the practicality of this approach within the limits of the problem’s computational complexity of being 2EXPTIME-complete.

Rivaroxaban or aspirin for patent foramen ovale and embolic stroke of undetermined source: a prespecified subgroup analysis from the NAVIGATE ESUS trial

Background: Patent foramen ovale (PFO) is a contributor to embolic stroke of undetermined source (ESUS). Subgroup analyses from previous studies suggest that anticoagulation could reduce recurrent stroke compared with antiplatelet therapy. We hypothesised that anticoagulant treatment with rivaroxaban, an oral factor Xa inhibitor, would reduce the risk of recurrent ischaemic stroke compared with aspirin among patients with PFO enrolled in the NAVIGATE ESUS trial. Methods: NAVIGATE ESUS was a double-blinded, randomised, phase 3 trial done at 459 centres in 31 countries that assessed the efficacy and safety of rivaroxaban versus aspirin for secondary stroke prevention in patients with ESUS. For this prespecified subgroup analysis, cohorts with and without PFO were defined on the basis of transthoracic echocardiography (TTE) and transoesophageal echocardiography (TOE). The primary efficacy outcome was time to recurrent ischaemic stroke between treatment groups. The primary safety outcome was major bleeding, according to the criteria of the International Society of Thrombosis and Haemostasis. The primary analyses were based on the intention-to-treat population. Additionally, we did a systematic review and random-effects meta-analysis of studies in which patients with cryptogenic stroke and PFO were randomly assigned to receive anticoagulant or antiplatelet therapy. Findings: Between Dec 23, 2014, and Sept 20, 2017, 7213 participants were enrolled and assigned to receive rivaroxaban (n=3609) or aspirin (n=3604). Patients were followed up for a mean of 11 months because of early trial termination. PFO was reported as present in 534 (7·4%) patients on the basis of either TTE or TOE. Patients with PFO assigned to receive aspirin had a recurrent ischaemic stroke rate of 4·8 events per 100 person-years compared with 2·6 events per 100 person-years in those treated with rivaroxaban. Among patients with known PFO, there was insufficient evidence to support a difference in risk of recurrent ischaemic stroke between rivaroxaban and aspirin (hazard ratio [HR] 0·54; 95% CI 0·22–1·36), and the risk was similar for those without known PFO (1·06; 0·84–1·33; pinteraction=0·18). The risks of major bleeding with rivaroxaban versus aspirin were similar in patients with PFO detected (HR 2·05; 95% CI 0·51–8·18) and in those without PFO detected (HR 2·82; 95% CI 1·69–4·70; pinteraction=0·68). The random-effects meta-analysis combined data from NAVIGATE ESUS with data from two previous trials (PICSS and CLOSE) and yielded a summary odds ratio of 0·48 (95% CI 0·24–0·96; p=0·04) for ischaemic stroke in favour of anticoagulation, without evidence of heterogeneity. Interpretation: Among patients with ESUS who have PFO, anticoagulation might reduce the risk of recurrent stroke by about half, although substantial imprecision remains. Dedicated trials of anticoagulation versus antiplatelet therapy or PFO closure, or both, are warranted. Funding: Bayer and Janssen