## Abstract

We present global and local algorithms for evaluating minimal fixed points of dependency graphs, a general problem in fixed-point computation and model checking. Our algorithms run in linear-time, matching the complexity of the best existing algorithms for similar problems, and are simple to understand. The main novelty of our global algorithm is that it does not use the counter and “reverse list” data structures commonly found in existing linear-time global algorithms. This distinction plays an essential role in allowing us to easily derive our local algorithm from our global one. Our local algorithm is distinguished from existing linear-time local algorithms by a combination of its simplicity and suitability for direct implementation.

We also provide linear-time reductions from the problems of computing minimal and maximal fixed points in Boolean graphs to the problem of minimal fixed-point evaluation in dependency graphs. This establishes dependency graphs as a suitable framework in which to express and compute alternation-free fixed points.

Finally, we relate HORNSAT, the problem of Horn formula satisfiability, to the problem of minimal fixed-point evaluation in dependency graphs. In particular, we present straightforward, linear-time reductions between these problems for both directions of reducibility. As a result, we derive a linear-time local algorithm for HORNSAT, the first of its kind as far as we are aware.

Research supported in part by NSF grants CCR-9505562 and CCR-9705998, and AFOSR. grants F49620-95-1-0508 and F49620-9C-1-0087.

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## References

A. Arnold and P. Crubille. A linear algorithm to solve fixed-point equations on transition systems.

*Information Processing Letters*, 29:57–66, September 1988.H. R. Andersen. Model checking and boolean graphs.

*Theoretical Computer Science*, 126(1), 1994.G. S. Bhat and R. Cleaveland. Efficient model checking via the equational Μ-calculus. In E. M. Clarke, editor,

*11th Annual Symposium on Logic in Computer Science (LICS '96)*, pages 304–312, New Brunswick, NJ, July 1996. Computer Society Press.E. M. Clarke and E. A. Emerson. Design and synthesis of synchronization skele-tons using branching-time temporal logic. In D. Kozen, editor,

*Proceedings of the Workshop on Logic of Programs*, Yorktown Heights, volume 131 of*Lecture Notes in Computer Science*, pages 52–71. Springer-Verlag, 1981.E. M. Clarke, E. A. Emerson, and A. P. Sistla. Automatic verification of finite-state concurrent systems using temporal logic specifications.

*ACM TOPLAS*, 8(2), 1986.R. Cleaveland, M. Klein, and B. Steffen. Faster model checking for the modal mu-calculus. In G.v. Bochmann and D.K. Probst, editors,

*Proceedings of the Fourth International Conference*on*Computer Aided Verification (CAV '92)*, Vol. 663 of*Lecture Notes in Computer Science*, pages 410–422. Springer-Verlag, 1992.R. Cleaveland. Tableau-based model checking in the propositional mu-calculus.

*Acta Informatica*, 27:725–747, 1990.R. Cleaveland and B. U. Steffen. A linear-time model checking algorithm for the alternation-free modal mu-calculus.

*Formal Methods in System Design*, 2:121–147, 1993.E. M. Clarke and J. M. Wing. Formal methods: State of the art and future directions.

*ACM Computing Surveys*, 28(4), December 1996.W. F. Dowling and J. H. Gallier. Linear-time algorithms for testing the satisfiability of propositional horn formulae.

*Journal of Logic Programming*, 3:267–284, 1984.L. Henschen and L. Wos. Unit refutations and horn sets.

*Journal of the ACM*, 21:590–605, 1974.D. Kozen. Results on the propositional /gm-calculus.

*Theoretical Computer Science*, 27:333–354, 1983.K.G. Larsen. Proof systems for Hennessy-Milner logic with recursion.

*Lecture Notes In Computer Science, Springer Verlag*, 299, 1988. in Proceedings of 13th Colloquium on Trees in Algebra and Programming 1988.K.G. Larsen. Efficient local correctness checking.

*Lecture Notes In Computer Science, Springer Verlag*, 663, 1992. in Proceedings of the 4th Workshop on Computer Aided Verification, 1992.X. Liu, C. R. Ramakrishnan, and S. A. Smolka. Fully local and efficient evaluation of alternating fixed points. In

*Proceedings of the Fourth International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS '98), Lecture Notes in Computer Science*. Springer-Verlag, 1998.V.R. Pratt. A decidable mu-calculus. In

*Proceedings of the 22nd IEEE Ann. Symp. on Foundations of Computer Science*, Nashville, Tennessee, pages 421–427, 1981.J. P. Queille and J. Sifakis. Specification and verification of concurrent systems in Cesar. In

*Proceedings of the International Symposium in Programming*, volume 137 of*Lecture Notes in Computer Science*, Berlin, 1982. Springer-Verlag.S. K. Shukla, H. B. Hunt III, and D. J. Rosenkrantz. HORNSAT, model checking, verification and games. In R. Alur and T. A. Henzinger, editors,

*Computer Aided Verification (CAV '96)*, volume 1102 of*Lecture Notes in Computer Science*, pages 99–110, New Brunswick, New Jersey, July 1996. Springer-Verlag.C. Stirling and D. Walker. Local model checking in the modal mu-calculus.

*Theoretical Computer Science*, 89(1), 1991.B. Vergauwen and J. Lewi. Efficient local correctness checking for single and alternating boolean equation systems. In

*Proceedings of ICALP'94*, pages 304–315. LNCS 820, 1994.

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Liu, X., Smolka, S.A. (1998). Simple linear-time algorithms for minimal fixed points. In: Larsen, K.G., Skyum, S., Winskel, G. (eds) Automata, Languages and Programming. ICALP 1998. Lecture Notes in Computer Science, vol 1443. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0055040

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DOI: https://doi.org/10.1007/BFb0055040

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