Joost Engelfriet
Abstract
We extend a classic result of Büchi, Elgot, and Trakhtenbrot: MSO definable string transductions i.e., string-to-string functions that are definable by an interpretation using monadic second-order (MSO) logic, are exactly those realized by deterministic two-way finite-state transducers, i.e., finite-state automata with a two-way input tape and a one-way output tape. Consequently, the equivalence of two mso definable string transductions is decidable. In the nondeterministic case however, MSO definable string tranductions, i.e., binary relations on strings that are mso definable by an interpretation with parameters, are incomparable to those realized by nondeterministic two-way finite-state transducers. This is a motivation to look for another machine model, and we show that both classes of MSO definable string transductions are characterized in terms of Hennie machines, i.e., two-way finite-state transducers that are allowed to rewrite their input tape, but may visit each position of their input only a bounded number of times.
- AHO,A.V.,HOPCROFT,J.E.,AND ULLMAN, J. D. 1969. A general theory of translation. Math. Syst. Theor. 3, 193-221.]]Google Scholar
- AHO,A.V.AND ULLMAN, J. D. 1970. A characterization of two-way deterministic classes of languages. J. Comput. Syst. Sci. 4, 523-538.]]Google Scholar
- BIRGET, J.-C. 1996. Two-way automata and length-preserving homomorphisms. Math. Syst. Theor. 29, 191-226.]]Google Scholar
- BLOEM,R.AND ENGELFRIET, J. 2000. A comparison of tree transductions defined by monadic second order logic and by attribute grammars. J. Comput. Syst. Sci. 61,1-50.]] Google Scholar
- BOUCHI, J. R. 1960. Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlagen Math. 6,66-92.]]Google Scholar
- BOUCHI, J. R. 1962. On a decision method in restricted second order arithmetic. In Proc. Int. Congr. Logic, Methodology and Philosophy of Sciences 1960. Stanford University Press, Stanford, CA.]]Google Scholar
- CHOMSKY, N. 1956. Three models for the description of language. IRE Transactions on Information Theory 2, 113-124.]]Google Scholar
- CHYTIL,M.P.AND J' AKL, V. 1977. Serial composition of 2-way finite-state transducers and simple programs on strings. In Automata, Languages and Programming, Fourth Colloquium, A. Salomaa and M. Steinby, Eds. Lect. Notes Comput. Sci., vol. 52. Springer Verlag, Berlin, 135-147.]] Google Scholar
- CLARKE, E. M., GRUMBERG,O.,AND PELED, D. A. 1999. Model Checking. MIT Press, Cambridge, MA, USA.]] Google Scholar
- COURCELLE, B. 1991. The monadic second-order logic of graphs V: On closing the gap between definability and recognizability. Theor. Comput. Sci. 80, 153-202.]] Google Scholar
- COURCELLE, B. 1992. The monadic second-order logic of graphs VII: Graphs as relational structures. Theor. Comput. Sci. 101, 1, 3-33.]] Google Scholar
- COURCELLE, B. 1994. Monadic second-order definable graph transductions: a survey. Theor. Comput. Sci. 126,53-75.]] Google Scholar
- COURCELLE, B. 1997. The expression of graph properties and graph transformations in monadic second-order logic. In Handbook of graph grammars and computing by graph transformation, vol. 1: Foundations, G. Rozenberg, Ed. World Scientific Publishing Co., 313-400.]] Google Scholar
- COURCELLE,B.AND ENGELFRIET, J. 1995. A logical characterization of the sets of hypergraphs defined by hyperedge replacement grammars. Math. Syst. Theor. 28, 515-552.]]Google Scholar
- DONER, J. 1970. Tree acceptors and some of their applications. J. Comput. Syst. Sci. 4, 406-451.]]Google Scholar
- EBINGER, W. 1995. Logical definability of trace languages. In The Book of Traces, V. Diekert and G. Rozenberg, Eds. World Scientific, 382-390. Appendix to Chapter 10.]]Google Scholar
- ELGOT, C. C. 1961. Decision problems of finite automata design and related arithmetics. Trans. AMS 98,21-52.]]Google Scholar
- ENGELFRIET, J. 1977. Top-down tree transducers with regular look-ahead. Math. Syst. Theor. 10, 289-303.]]Google Scholar
- ENGELFRIET, J. 1982. Three hierarchies of transducers. Math. Syst. Theor. 15,95-125.]]Google Scholar
- ENGELFRIET, J. 1991a. A characterization of context-free NCE graph languages by monadic second-order logic on trees. In Graph Grammars and Their Application to Computer Science, H. Ehrig, H.-J. Kreowski, and G. Rozenberg, Eds. Lect. Notes Comput. Sci., vol. 532. Springer Verlag, Berlin, 311-327.]] Google Scholar
- ENGELFRIET, J. 1991b. Iterated stack automata and complexity classes. Inf. Comput. 95,21-75.]] Google Scholar
- ENGELFRIET, J. 1997. Context-free graph grammars. In Handbook of Formal Languages,G. Rozenberg and A. Salomaa, Eds. Vol. 3: Beyond Words. Springer Verlag, Berlin, 125-213.]] Google Scholar
- ENGELFRIET,J.AND HEYKER, L. M. 1991. The string generating power of context-free hypergraph grammars. J. Comput. Syst. Sci. 43, 328-360.]] Google Scholar
- ENGELFRIET,J.AND HOOGEBOOM, H. J. 1999. Two-way finite state transducers and monadic secondorder logic. In 26-th International Colloquium on Automata, Languages and Programming, J. Wiedermann, P. van Emde Boas, and M. Nielsen, Eds. Lect. Notes Comput. Sci., vol. 1644. Springer Verlag, Berlin, 311-320.]] Google Scholar
- ENGELFRIET,J.AND MANETH, S. 1999. Macro tree transducers, attribute grammars, and MSO definable tree translations. Inf. Comput. 154,34-91.]] Google Scholar
- ENGELFRIET,J.AND VAN OOSTROM, V. 1997. Logical description of context-free graph languages. J. Comput. Syst. Sci. 55, 489-503.]] Google Scholar
- FISCHER, M. J. 1969. Two characterizations of the context-sensitive languages. In IEEE Conference Record of 10th Annual Symposium on Switching and Automata Theory. 149-156.]]Google Scholar
- GREIBACH, S. A. 1978a. Hierarchy theorems for two-way finite state transducers. Acta Inf. 11, 89-101.]]Google Scholar
- GREIBACH, S. A. 1978b. One way finite visit automata. Theor. Comput. Sci. 6, 175-221.]]Google Scholar
- GREIBACH, S. A. 1978c. Visits, crosses, and reversals for nondeterministic off-line machines. Inf. and Con. 36, 174-216.]]Google Scholar
- GRIFFITHS, T. V. 1968. The unsolvability of the equivalence problem for -free nondeterministic generalized machines. J. ACM 15, 409-413.]] Google Scholar
- GURARI, E. M. 1982. The equivalence problem for deterministic two-way sequential transducers is decidable. SIAM J. Comput. 11, 3, 448-452.]]Google Scholar
- HENNIE, F. C. 1965. One-tape, off-line turing machine computations. Inf. and Con. 8, 553-578.]]Google Scholar
- HODGES, W. 1993. Model Theory. Encyclopedia of Mathematics and its Applications, vol. 42. Cambridge University Press.]]Google Scholar
- HOOGEBOOM,H.J.AND TEN PAS, P. 1997. Monadic second-order definable text languages. ACM Trans. Comput. Syst. 30, 335-354.]]Google Scholar
- HOPCROFT,J.E.AND ULLMAN, J. D. 1967. An approach to a unified theory of automata. The Bell System Technical Journal 46, 1793-1829. Also in: IEEE Conference Record of 8th Annual Sym-posium on Switching and Automata Theory, Austin, Texas, 1967, pages 140-147.]]Google Scholar
- HOPCROFT,J.E.AND ULLMAN, J. D. 1979. Introduction to Automata Theory, Language, and Computation. Addison-Wesley, Reading, Mass.]] Google Scholar
- IMMERMAN, N. 1999. Descriptive Complexity. Springer Verlag, Berlin.]]Google Scholar
- KIEL, D. 1975. Two-way a-transducers and AFL. J. Comput. Syst. Sci. 10,88-109.]]Google Scholar
- KLEENE, S. C. 1956. Representation of events in nerve nets and finite automata. In Automata Studies. Annals of Mathematics Studies, vol. 34. Princeton University Press, Princeton, N.J., 3-42.]]Google Scholar
- KURSHAN, R. P. 1994. Computer-Aided Verification of Coordinated Processes. Princeton University Press, Princeton, N.J.]] Google Scholar
- LAPOIRE, D. 1998. Recognizability equals monadic second-order definability for sets of graphs of bounded tree-width. In 15th Annual Symposium on Theoretical Aspects of Computer Science. Lect. Notes Comput. Sci., vol. 1373. Springer Verlag, Berlin, 618-628.]] Google Scholar
- LAUTEMANN, C., MCKENZIE, P., SCHWENTICK,T.,AND VOLLMER, H. 1999. The descriptive complexity approach to LOGCFL. In 16th Symposium on Theoretical Aspects of Computer Science, C. Meinel and S. Tison, Eds. Lect. Notes Comput. Sci., vol. 1563. Springer Verlag, Berlin, 444-454. Full version: Electronic Colloquium on Computational Complexity, Report TR98-059.]]Google Scholar
- MCCULLOCH,W.S.AND PITTS, W. 1943. A logical calculus of the ideas imminent in nervous activity. Bull. Math. Bioph. 5, 115-133.]]Google Scholar
- MCNAUGHTON,R.AND PAPERT, S. 1971. Counter-free automata. MIT Press, Cambridge, MA.]] Google Scholar
- MYHILL, J. 1957. Finite automata and the representation of events. Tech. Rep. WADD TR-57-624, Wright Patterson AFB, Ohio.]]Google Scholar
- NERODE, A. 1958. Linear automata transformation. Proc. AMS 9, 541-544.]]Google Scholar
- NIJHOLT, A. 1982. The equivalence problem for LL- and LR-regular grammars. J. Comput. Syst. Sci. 24, 149-161.]]Google Scholar
- PIXTON, D. 1996. Regularity of splicing languages. Discr. Appl. Math. 69, 101-124.]] Google Scholar
- RABIN, M. O. 1963. Real-time computation. Isr. J. Math 1, 203-211.]]Google Scholar
- RABIN, M. O. 1969. Decidability of second-order theories and automata on infinite trees. Trans. AMS 141,1-35.]]Google Scholar
- RABIN, M. O. 1977. Decidable theories. In Handbook of Mathematical Logic, J. Barwise, Ed. Studies in Logic and the Foundations of Mathematics, vol. 30. North-Holland, 595-629.]]Google Scholar
- RABIN,M.O.AND SCOTT, D. 1959. Finite automata and their decision problems. IBM J. Res. Dev. 3, 114-125. Also in: E.F. Moore, Ed., Sequential Machines: Selected Papers. Addison-Wesley, Reading, MA, 1964, pages 63-91.]]Google Scholar
- RAJLICH, V. 1975. Bounded-crossing transducers. Inf. and Con. 27, 329-335.]]Google Scholar
- SEESE, D. 1992. Interpretability and tree automata: a simple way to solve algorithmic problems on graphs closely related to trees. In Tree Automata and Languages, M. Nivat and A. Podelski, Eds. Elsevier Science Publishers, Amsterdam, 83-114.]]Google Scholar
- SHEPHERDSON, J. C. 1959. The reduction of two-way automata to one-way automata. IBM J. Res. Dev. 3, 198-200. Also in: E.F. Moore, Ed., Sequential Machines: Selected Papers. Addison-Wesley, Reading, MA, 1964, pages 92-97.]]Google Scholar
- THATCHER,J.W.AND WRIGHT, J. B. 1968. Generalized finite automata theory with an application to a decision problem of second-order logic. Math. Syst. Theor. 2,57-82.]]Google Scholar
- THOMAS, W. 1997a. Ehrenfeucht games, the composition method, and the monadic theory of ordinal words. In Structures in Logic and Computer Science-A Selection of Essays in Honor of A. Ehrenfeucht, J. Mycielski, G. Rozenberg, and A. Salomaa, Eds. Lect. Notes Comput. Sci., vol. 1261. Springer Verlag, Berlin, 118-143.]] Google Scholar
- THOMAS, W. 1997b. Languages, automata, and logic. In Handbook of Formal Languages, G. Rozenberg and A. Salomaa, Eds. Vol. 3: Beyond Words. Springer Verlag, Berlin, 389-455.]] Google Scholar
- TRAKHTENBROT, B. A. 1962. Finite automata and the logic of one-place predicates. Siberian Mathematical Journal 3, 103-131 (in Russian). English translation: American Mathematical Society Translations, Series 2, 59 (1966), 23-55.]]Google Scholar
- YU, S. 1997. Regular languages. In Handbook of Formal Languages, G. Rozenberg and A. Salomaa, Eds. Vol. 1: Word, Language, Grammar. Springer Verlag, Berlin, 41-110.]] Google Scholar
Index Terms
- MSO definable string transductions and two-way finite-state transducers
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Reviews
Robert Stewart Roos
Thirty years ago, Buchi, Elgot, and Trakhtenbrot showed that monadic second-order (MSO) logical expressions quantified over sets of string positions define precisely the set of regular languages. Engelfriet and Hoogeboom extended this equivalence to deterministic transductions defined by MSOs and two-way deterministic generalized sequential machines (2DGSMs). In the nondeterministic case, MSO-definable string relations are shown to be equivalent in power to nondeterministic Hennie machines, two-way nondeterministic transducers than can overwrite their input tapes but are permitted only a bounded number of visits to each cell. The proofs of these results introduce much machinery that is interesting in its own right. To choose just two examples, one of the intermediate steps in the proof involves demonstrating that the two-way generalized sequential machines of Aho and Ullman are equivalent to 2GMs in which moves are described using MSO expressions. At another point the authors show that each nondeterministic Hennie transducer is equivalent to the composition of two finite-visit nondeterministic 2GSMs. Readers unfamiliar with earlier results in this area will find all the relevant background information, together with examples, in this rich and clearly written paper, with a list of more than sixty references if more detail is desired. Evidently the authors took much care to present this material in a fashion that makes it accessible to a large audience.
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