Abstract
We study the language inclusion problem \(L_1 \subseteq L_2\) where \(L_1\) is regular. Our approach relies on abstract interpretation and checks whether an overapproximating abstraction of \(L_1\), obtained by successively overapproximating the Kleene iterates of its least fixpoint characterization, is included in \(L_2\). We show that a language inclusion problem is decidable whenever this overapproximating abstraction satisfies a completeness condition (i.e. its loss of precision causes no false alarm) and prevents infinite ascending chains (i.e. it guarantees termination of least fixpoint computations). Such overapproximating abstraction function on languages can be defined using quasiorder relations on words where the abstraction gives the language of all words “greater than or equal to” a given input word for that quasiorder. We put forward a range of quasiorders that allow us to systematically design decision procedures for different language inclusion problems such as regular languages into regular languages or into trace sets of one-counter nets. In the case of inclusion between regular languages, some of the induced inclusion checking procedures correspond to well-known state-of-the-art algorithms like the so-called antichain algorithms. Finally, we provide an equivalent greatest fixpoint language inclusion check which relies on quotients of languages and, to the best of our knowledge, was not previously known.
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Notes
- 1.
Sketch: Given \(\mathcal {A}_1=(Q_1,\delta _1,I_1,F_1,\varSigma )\) and \(\mathcal {A}_2=(Q_2,\delta _2,I_2,F_2,\varSigma )\) define \(\mathcal {A}_3=(Q_1\cup Q_2\cup \{q^{\dag }\}, \delta _3, \{q^{\dag }\}, F_1\cup F_2)\) where \(\delta _3 \) maps \((q^\dag ,a)\) to \(I_1\), \( (q^\dag ,b) \) to \(I_2\) and like \(\delta _1\) or \(\delta _2\) elsewhere. Then, it turns out that \(a \leqq ^r_{{\mathcal {L}(\mathcal {A}_3)}} b \Leftrightarrow a^{-1}{\mathcal {L}(\mathcal {A}_3)} \subseteq b^{-1}{\mathcal {L}(\mathcal {A}_3)} \Leftrightarrow {\mathcal {L}(\mathcal {A}_1)}\subseteq {\mathcal {L}(\mathcal {A}_2)}\).
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Acknowledgements
We would like to thank the reviewers for their insightful feedback that allowed us to find a simpler connection between our work and the antichain algorithms. Pierre Ganty completed this work with the support of the Spanish Ministry of Economy and Competitiveness project No. PGC2018-102210-B-I00, the Madrid Regional Government project No. S2018/TCS-4339 and the Ramón y Cajal fellowship RYC-2016-20281. The work of Francesco Ranzato has been partially funded by the University of Padova, SID2018 project “Analysis of STatic Analyses (ASTA)”, and by the Italian Ministry of Research MIUR, project No. 201784YSZ5 “AnalysiS of PRogram Analyses (ASPRA)”.
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Ganty, P., Ranzato, F., Valero, P. (2019). Language Inclusion Algorithms as Complete Abstract Interpretations. In: Chang, BY. (eds) Static Analysis. SAS 2019. Lecture Notes in Computer Science(), vol 11822. Springer, Cham. https://doi.org/10.1007/978-3-030-32304-2_8
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