Abstract
We define an extension of the weak monadic second-order logic of one successor (WS1S) with an infinite family of relations and show its decidability. Analogously to the decision procedure for WS1S, automata are used. But instead of using word automata, we use tree automata that accept or reject words. In particular, we encode a word in a complete leaf labeled tree and restrict the acceptance condition for tree automata to trees that encode words. As applications, we show how this extension can be applied to reason automatically about parameterized families of combinational tree-structured circuits and used to solve certain decision problems.
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References
A. Ayari, D. Basin, and S. Friedrich. Structural and behavioral modeling with monadic logics. In 29th IEEE International Symposium on Multiple-Valued Logic, pages 142–151, 1999.
A. Ayari, D. Basin, and F. Klaedtke. Decision procedures for inductive boolean functions based on alternating automata. In 12th International Conference on Computer-Aided Verification, CAV’’00, volume 1855 of LNCS, pages 170–186, 2000.
D. Basin and N. Klarlund. Automata based symbolic reasoning in hardware verification. The Journal of Formal Methods in Systems Design, 13(3):255–288, 1998.
M. Biehl, N. Klarlund, and T. Rauhe. Algorithms for guided tree automata. In 1st International Workshop on Implementing Automata, WIA’ 96, volume 1260 of LNCS, pages 6–25, 1997.
J. Büchi. Weak second-order arithmetic and finite automata. Zeitschrift der mathematischen Logik und Grundlagen der Mathematik, 6:66–92, 1960.
O. Carton and W. Thomas. The monadic theory of morphic infinite words and generalizations. In 25th International Symposium on Mathematical Foundations of Computer Science, MFCS 2000, volume 1893 of LNCS, pages 275–284, 2000.
H. Comon, M. Dauchet, R. Gilleron, F. Jacquemard, D. Lugiez, S. Tison, and M. Tommasi. Tree automata techniques and applications. Available on http://www.grappa.univ-lille3.fr/tata, 1997.
J. Doner. Tree acceptors and some of their applications. Journal of Computer and System Sciences, 4:406–451, 1970.
J. Elgaard, N. Klarlund, and A. Møller. Mona 1.x: New techniques for WS1S and WS2S. In 10th International Conference on Computer Aided Verification, CAV’98, volume 1427 of LNCS, pages 516–520, 1998.
J. Elgaard, A. Møller, and M. Schwartzbach. Compile-time debugging of C programs working on trees. In 9th European Symposium on Programming, ESOP’00, volume 1782 of LNCS, pages 119–134, 2000.
C. Elgot. Decision problems of finite automata design and related arithmetics. Transactions of the AMS, 98:21–51, 1961.
C. Elgot and M. Rabin. Decidability and undecidability of extensions of second (first) order theory of (generalized) successor. Journal of Symbolic Logic, 31(2): 169–181, 1966.
F. Gécseg and M. Steinby. Tree Automata. Akadémiai Kiadó, Budapest, 1984.
A. Gupta. Inductive Boolean Function Manipulation: A Hardware Verification Methodology for Automatic Induction. PhD thesis, School of Computer Science, Carnegie Mellon University, Pittsburgh, USA, 1994.
A. Gupta and A. Fisher. Parametric circuit representation using inductive boolean functions. In 5th International Conference on Computer-Aided Verification, CAV’93, volume 697 of LNCS, pages 15–28, 1993.
J. Henriksen, J. Jensen, M. Jørgensen, N. Klarlund, B. Paige, T. Rauhe, and A. Sandholm. Mona: Monadic second-order logic in practice. In 1st International Workshop on Tools and Algorithms for the Construction and Analysis of Systems, TACAS’95, volume 1019 of LNCS, pages 89–110, 1996.
J. Hopcroft and J. Ullman. Formal Languages and their Relation to Automata. Addison-Wesley, 1969.
F. Klaedtke. Induktive boolesche Funktionen, endliche Automaten und monadische Logik zweiter Stufe. Diplomarbeit, Institut für Informatik, Albert-Ludwigs-Universität, Freiburg i. Br., 2000. In German.
N. Klarlund and A. Møller. MONA Version 1.3 User Manual. BRICS Notes Series NS-98-3 (2.revision), Department of Computer Science, University of Aarhus, 1998.
A. Meyer. Weak monadic second-order theory of successor is not elementary-recursive. In Logic Colloquium, volume 453 of Lecture Notes in Mathematics, pages 132–154, 1975.
A. Monti and A. Roncato. Completeness results concerning systolic tree automata and E0L languages. Information Processing Letters, 53:11–16, 1995.
J. Thatcher and J. Wright. Generalized finite automata theory with an application to a decision problem in second-order logic. Mathematical Systems Theory, 2:57–81, 1968.
W. Thomas. Languages, automata, and logic. In A. Salomaa and G. Rozenberg, editors, Handbook of Formal Languages, volume 3, chapter 7, pages 389–455. Springer-Verlag, 1997.
B. Trakhtenbrot. Finite automata and the logic of one-place predicates. AMS, Transl., II. Ser., 59:23–55, 1966.
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Klaedtke, F. (2001). Decision Procedure for an Extension of WS1S. In: Fribourg, L. (eds) Computer Science Logic. CSL 2001. Lecture Notes in Computer Science, vol 2142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44802-0_27
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DOI: https://doi.org/10.1007/3-540-44802-0_27
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