Inf-datalog, modal logic and complexities
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 1, pp. 1-21.

Inf-Datalog extends the usual least fixpoint semantics of Datalog with greatest fixpoint semantics: we defined inf-Datalog and characterized the expressive power of various fragments of inf-Datalog in [16]. In the present paper, we study the complexity of query evaluation on finite models for (various fragments of) inf-Datalog. We deduce a unified and elementary proof that global model-checking (i.e. computing all nodes satisfying a formula in a given structure) has 1. quadratic data complexity in time and linear program complexity in space for $CTL$ and alternation-free modal $\mu$-calculus, and 2. linear-space (data and program) complexities, linear-time program complexity and polynomial-time data complexity for $L{\mu }_{k}$ (modal $\mu$-calculus with fixed alternation-depth at most $k$).

DOI : https://doi.org/10.1051/ita:2007043
Classification : 68Q19,  03C13
Mots clés : databases, model-checking, specification languages, performance evaluation
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Foustoucos, Eugénie; Guessarian, Irène. Inf-datalog, modal logic and complexities. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 1, pp. 1-21. doi : 10.1051/ita:2007043. http://www.numdam.org/articles/10.1051/ita:2007043/`

[1] S. Abiteboul, R. Hull and V. Vianu, Foundations of databases. Addison-Wesley (1995). | Zbl 0848.68031

[2] A. Arnold and P. Crubillé, A linear algorithm to solve fixed-point equations on transition systems. Inform. Process. Lett. 29 (1988) 57-66. | MR 974202 | Zbl 0663.68075

[3] A. Arnold and D. Niwiński, Rudiments of $\mu$-calculus. Stud. Logic Found. Math. 146, Elsevier Science, North-Holland, Amsterdam (2001). | MR 1854973 | Zbl 0968.03002

[4] J. Bradfield, Fixpoint alternation: Arithmetic, transition systems, and the binary tree. RAIRO-Theor. Inf. Appl. 33 (1999) 341-356. | Numdam | MR 1748660 | Zbl 0945.68126

[5] A. Browne, E. Clarke, S. Jha, D. Long and W. Marrero, An improved algorithm for the evaluation of fixpoint expressions. Theor. Comput. Sci. 178 (1997) 237-255. | MR 1453852 | Zbl 0901.68118

[6] W. Charatonik, D. Mcallester, D. Niwínski, A. Podelski and I. Walukiewicz, The horn Mu-calculus. LICS (1998) 58-69. | MR 1654520 | Zbl 0945.03541

[7] E.M. Clarke, E.A. Emerson and A.P. Sistla, Automatic Verification of finite-state concurrent systems using temporal logic specifications. ACM TOPLAS 8 (1986) 244-263. | Zbl 0591.68027

[8] R. Cleaveland and B. Steffan, A linear time model-checking algorithm for the alternation-free modal mu-calculus. Formal Method. Syst. Des. 2 (1993) 121-148. | Zbl 0772.68038

[9] E. Emerson, Temporal and modal logic. Handbook of Theoretical Computer Science (1990) 997-1072. | MR 1127201 | Zbl 0900.03030

[10] E. Emerson, Model-Checking and the Mu-Calculus, in Descriptive Complexity and Finite Models, edited by N. Immerman and Ph. Kolaitis, American Mathematical Society (1997). | MR 1451385 | Zbl 0877.03020

[11] E.A. Emerson and C.L. Lei, Efficient model-checking in fragments of the propositional $\mu$-calculus, in Proc. of 1rst Symposium on Logic in Computer Science (1986) 267-278.

[12] E. Foustoucos and I. Guessarian, Complexity of Monadic inf-datalog. Application to temporal logic. Extended abstract in Proceedings 4th Panhellenic Logic Symposium (2003) 95-99.

[13] G. Gottlob and C. Koch, Monadic Datalog and the expressive power of web information extraction languages. Proc. PODS'02 (2002) 17-28.

[14] G. Gottlob, E. Grädel and H. Veith, Datalog LITE: temporal versus deductive reasoning in verification. ACM T. Comput. Log. 3 (2002) 39-74. | MR 1902106

[15] A. Griffault and A. Vincent, The Mec 5 model-checker, CAV'04. Lect. Notes Comput. Sci. 3114 (2004) 488-491. | Zbl 1103.68619

[16] I. Guessarian, E. Foustoucos, T. Andronikos and F. Afrati, On temporal logic versus Datalog. Theor. Comput. Sci. 303 (2003) 103-133. | MR 1990744 | Zbl 1019.03021

[17] M. Jurdzinski, Small progress measures for solving parity games. Proc. STACS'2000 (2000) 290-301. | MR 1781740 | Zbl 0962.68111

[18] M. Jurdzinski, M. Paterson and U. Zwick, A Deterministic subexponential algorithm for solving parity games. Proc. SODA (2006) 117-123. | MR 2368806

[19] D. Kozen, Results on the propositional $\mu$-calculus. Theor. Comput. Sci. 27 (1983) 333-354. | MR 731069 | Zbl 0553.03007

[20] A. Mader, The modal $\mu$-calculus, model-checking, equations systems and Gauss elimination. TACAS 95 (1995) 44-57.

[21] D. Park, Finiteness is $\mu$-ineffable. Theor. Comput. Sci. 3 (1976) 173-181. | MR 437297 | Zbl 0353.02027

[22] H. Seidl, Fast and simple nested fixpoints. Inform. Process. Lett. 59 (1996) 303-308. | MR 1417647 | Zbl 0900.68458

[23] A. Vincent, Conception et réalisation d'un vérificateur de modèles AltaRica. Ph.D. Thesis, LaBRI, University of Bordeaux 1 (2003) http://altarica.labri.fr/Doc/Biblio/Author/VINCENT-A.html

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