Abstract
Restarting automata have been introduced as a formal model for the linguistic technique of analysis by reduction, which can be used to check the correctness of natural language sentences. In order to study quantitative aspects of restarting automata, we introduce the concept of a weighted restarting automaton. Such an automaton is given through a pair \((M,\omega )\), where M is a restarting automaton on some input alphabet \(\Sigma \), and \(\omega \) is a weight function that assigns an element of a given semiring S to each transition of M. Thus, \((M,\omega )\) defines a function \(f_\omega ^M:\Sigma ^*\rightarrow S\) that associates an element of S to each input word over \(\Sigma \). By looking at different semirings S and different weight functions \(\omega \), various quantitative aspects of the behavior of M can be expressed through these functions. We are interested in the syntactic and semantic properties of these functions, e.g., their growth rates and the closure properties under various operations.
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Bollig B, Gastin P, Monmege B, Zeitoun M (2010) Pebble weighted automata and transitive closure logics. In: Abramsky S, Gavoille C, Kirchner C, Meyer auf der Heide F, Spirakis PG (eds) ICALP 2010, Part II, Lecture notes in computer science, Springer, Heidelberg, vol 6199, pp 587–598
Book RV, Otto F (1993) String-rewriting systems. Texts and monographs in computer science. Springer, New York
Chatterjee K, Doyen L, Henzinger TA (2009) Probabilistic weighted automata. In: Bravetti M, Zavattaro G (eds) CONCUR 2009, Proc., Lecture Notes in Computer Science, vol 5710, Springer, Heidelberg, pp 244–258
Dahlhaus E, Warmuth MK (1986) Membership for growing context-sensitive grammars is polynomial. J Comput Syst Sci 33:456–472
Droste M, Götze D (2013) The support of nested weighted automata. In: Bensch S, Drewes F, Freund R, Otto F (eds) NCMA 2013, Proc., Oesterreichische Computer Gesellschaft, Wien, books@ocg.at, Band 294, pp 101–116
Droste M, Kuich W (2009) Semirings and formal power series. In: Droste M, Kuich W, Vogler H (eds) Handbook of weighted automata, monographs in theoretical computer science. Springer, Heidelberg, pp 3–28
Droste M, Meinecke I (2011) Weighted automata and regular expressions over valuation monoids. Int J Found Comput Sci 22:1829–1844
Droste M, Kuich W, Vogler H (2009) Handbook of weighted automata. Monographs in theoretical computer science. Springer, Heidelberg
Golan JS (1999) Semirings and their applications. Kluwer Academic Publishers, Dordrecht
Hebisch U, Weinert HJ (1998) Semirings: algebraic theory and applications in computer science. World Scientific, Singapore
Hundeshagen N (2013) Relations and transductions realized by restarting automata. Doctoral thesis, Universität Kassel
Hundeshagen N, Otto F (2012) Characterizing the rational functions by restarting transducers. In: Dediu AH, Martín-Vide C (eds) LATA 2012, Proc., Lecture notes in computer science, vol 7183. Springer, Heidelberg, pp 325–336
Jančar P, Mráz F, Plátek M, Vogel J (1995) Restarting automata. In: Reichel H (ed) FCT, Lecture notes in computer science, vol 965. Springer, Heidelberg, pp 283–292
Jančar P, Mráz F, Plátek M, Vogel J (1997) On restarting automata with rewriting. In: Pǎun G, Salomaa A (eds) New trends in formal languages, Lecture notes in computer science, vol 1218. Springer, Heidelberg, pp 119–136
Jančar P, Mráz F, Plátek M, Vogel J (1998) Different types of monotonicity for restarting automata. In: Arvind V, Ramanujam S (eds) Foundations of software technology and theoretical computer science, Lecture notes in computer science, vol 1530. Springer, Heidelberg, pp 343–354
Jančar P, Mráz F, Plátek M, Vogel J (1999) On monotonic automata with a restart operation. J Autom Lang Comb 4(4):287–311
Jurdziński T, Loryś K, Niemann G, Otto F (2004) Some results on RWW- and RRWW-automata and their relation to the class of growing context-sensitive languages. J Autom Lang Comb 9(4):407–437
Kirsten D (2009) The support of a recognizable series over a zero-sum free, commutative semiring is recognizable. In: Diekert V, Nowotka D (eds) DLT 2009, Lecture Notes in Computer Science, vol 5583. Springer, Heidelberg, pp 326–333
Kirsten D (2011) The support of a recognizable series over a zero-sum free, commutative semiring is recognizable. Acta Cybern 20:211–221
McNaughton R, Narendran P, Otto F (1988) Church–Rosser Thue systems and formal languages. J ACM 35(2):324–344
Niemann G, Otto F (2000) Restarting automata, Church–Rosser languages, and representations of r.e. languages. In: Rozenberg G, Thomas W (eds) Developments in Language Theory—Foundations, Applications, and Perspectives, DLT 1999, Proc., World Scientific, Singapore, pp 103–114
Otto F (2006) Restarting automata. In: Ésik Z, Martín-Vide C, Mitrana V (eds) Recent advances in formal languages and applications, studies in computational intelligence, vol 25. Springer, Heidelberg, pp 269–303
Salomaa A, Soittola M (1978) Automata-theoretic aspects of formal power series. Texts and monographs in computer science. Springer, New York
Schützenberger MP (1961) On the definition of a family of automata. Inform Control 4(2–3):245–270
Straňáková M (2000) Selected types of pg-ambiguity: processing based on analysis by reduction. In: Sojka P, Kopeček I, Pala K (eds) Text, speech and dialogue, Lecture notes in computer science, vol 1902. Springer, Heidelberg, pp 139–144
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Communicated by M. Droste, Z. Esik and K. Larsen.
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Otto, F., Wang, Q. Weighted restarting automata. Soft Comput 22, 1067–1083 (2018). https://doi.org/10.1007/s00500-016-2164-4
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DOI: https://doi.org/10.1007/s00500-016-2164-4