Abstract
Every univariate rational series over an algebraically closed field is shown to be realised by some polynomially ambiguous unary weighted automaton. Unary weighted automata over algebraically closed fields thus always admit polynomially ambiguous equivalents. On the other hand, it is shown that this property does not hold over any other field of characteristic zero, generalising a recent observation about unary weighted automata over the field of rational numbers.
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Notes
We do not consider automata over other than unary alphabets in this article – we thus leave the relation between polynomially ambiguous and unrestricted weighted finite automata over algebraically closed fields open for general alphabets.
We get xt = Atf = P− 1JtPf for some matrices \(J,P \in \overline {\mathbb {F}}\) and all \(t \in \mathbb {N}\), where J is in Jordan canonical form and P is invertible. Now, an easy combinatorial argument can be employed to observe that each entry of Jt takes the form \({t \choose k} \lambda ^{t - k}\) for some eigenvalue λ of A over \(\overline {\mathbb {F}}\) and \(k \in \mathbb {N}\) that is smaller than the algebraic multiplicity of λ. The entries of P, P− 1, and f are constants (they do not depend on t). See also [10, Subsection 3.3.2].
The binomial coefficient \({t \choose k}\) is a nonnegative integer, so it should be interpreted as a sum of \({t \choose k}\) ones in \(\overline {\mathbb {F}}\).
The determinant of the Casorati matrix is usually called the Casoratian and is a discrete counterpart of the Wrońskian, which is important for the theory of linear differential equations.
More precisely, one only distinguishes between zero and nonzero weights and interprets this distinction over the Boolean semiring.
The only possible source of ambiguity in such automata is related to the fact that an automaton may have multiple states with nonzero initial weights.
One may view this system over \(\mathbb {N}\) as a system over the field of rational or complex numbers as well.
In fact, we could just note that \(\mathcal {A}\) and \(\mathcal {J}\) are evidently similar [24]. The observation established can be rephrased as a well-known fact that similar automata are always equivalent.
Although the automaton \(\mathcal {J}\) does not have to be trim in general, it can be turned into a trim automaton by possibly removing several states. It is clear that the nature of strongly connected components is not spoiled by this process, and that any of the two automata is polynomially ambiguous if and only if the other automaton is. Theorem 3.2 thus can be invoked.
Note that the automaton \(\mathcal {F}\) in Fig. 2 is not polynomially ambiguous by Theorem 3.2, as its graph consists of a single strongly connected component with two vertices, which does not take the form of a directed cycle.
Their existence follows from separability of irreducible polynomials over fields of characteristic zero (see, e.g., S. Roman [23, Corollary 1.6.3]).
For instance, as (3 + 4i)2 = − 7 + 24i ≡ 3 + 4i (mod 5), it follows that there is no \(n \in \mathbb {N} \setminus \{0\}\) such that the real part of (3 + 4i)n equals 5n, which has to hold whenever νn = 1.
References
Barloy, C., Fijalkow, N., Lhote, N., Mazowiecki, F.: A robust class of linear recurrence sequences. In: Computer Science Logic, CSL 2020 article 9 (2020)
Bell, J., Smertnig, D.: Noncommutative rational Pólya series. Selecta Mathematica 27(3), article 34 (2021)
Bell, P. C.: Polynomially ambiguous probabilistic automata on restricted languages. In: Automata, Languages and Programming, ICALP 2019 article 105 (2019)
Berstel, J., Reutenauer, C.: Noncommutative Rational Series with Applications. Cambridge University Press, Cambridge (2011)
Chattopadhyay, A., Mazowiecki, F., Muscholl, A., Riveros, C.: Pumping lemmas for weighted automata. Logical Methods in Computer Science 17(3), article 7 (2021)
Droste, M., Gastin, P.: Aperiodic weighted automata and weighted first-order logic. In: Mathematical Foundations of Computer Science, MFCS 2019 article 76 (2019)
Droste, M., Kuich, W.: Semirings and Formal Power Series. In: Droste, M., Kuich, W., Vogler, H (eds.) Handbook of Weighted Automata, pp 3–28. Springer, Heidelberg (2009)
Droste, M., Kuich, W., Vogler, H. (eds.): Handbook of Weighted Automata. Springer, Heidelberg (2009)
Droste, M., Kuske, D.: Weighted Automata. In: Pin, J.-É. (ed.) Handbook of Automata Theory, vol. 1, pp 113–150. European Mathematical Society, Zürich (2021)
Elaydi, S.: An Introduction to Difference Equations, 3rd edn. Springer, New York (2005)
Ésik, Z., Kuich, W.: Finite Automata. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata, pp 69–104. Springer, Heidelberg (2009)
Hungerford, T. W.: Algebra. Springer, New York (1974)
Kalman, D.: The generalized Vandermonde matrix. Math. Mag. 57(1), 15–21 (1984)
Kirsten, D.: A Burnside approach to the termination of Mohri’s algorithm for polynomially ambiguous min-plus-automata. RAIRO – Theoret. Inform. Appl. 42(3), 553–581 (2008)
Kirsten, D., Lombardy, S.: Deciding unambiguity and sequentiality ofpolynomially ambiguous min-plus automata. In: Symposium on Theoretical Aspects of Computer Science, STACS 2009, pp 589–600 (2009)
Klimann, I., Lombardy, S., Mairesse, J., Prieur, C.: Deciding unambiguity and sequentiality from a finitely ambiguous max-plus automaton. Theor. Comput. Sci. 327(3), 349–373 (2004)
Lombardy, S., Sakarovitch, J.: Sequential? Theoret. Comput. Sci. 356, 224–244 (2006)
Maletti, A., Nasz, T., Stier, K., Ulbricht, M.: Ambiguity hierarchies for weighted tree automata. In: Implementation and Application ofAutomata, CIAA 202, pp 140–151 (2021)
Mazowiecki, F., Riveros, C.: Copyless cost-register automata: structure, expressiveness, and closure properties. J. Comput. Syst. Sci. 100, 1–29 (2019)
Minc, H.: Nonnegative Matrices. Wiley, New York (1988)
Paul, E.: On finite and polynomial ambiguity of weighted tree automata. In: Developments in Language Theory, DLT 2016, pp 368–379 (2016)
Ravikumar, B., Ibarra, O. H.: Relating the type of ambiguity of finite automata to the succinctness of their representation. SIAM J. Comput. 18(6), 1263–1282 (1989)
Roman, S.: Field Theory, 2nd edn. Springer, New York (2006)
Sakarovitch, J.: Elements of Automata Theory. Cambridge University Press, Cambridge (2009)
Sakarovitch, J.: Rational and Recognisable Power Series. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata, pp 105–174. Springer, Heidelberg (2009)
Schützenberger, M. -P.: On the definition of a family of automata. Inf. Control. 4(2–3), 245–270 (1961)
Weber, A., Seidl, H.: On the degree of ambiguity of finite automata. Theor. Comput. Sci. 88(2), 325–349 (1991)
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I would like to thank the anonymous reviewers for their valuable comments.
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The work was supported by the grant VEGA 1/0601/20.
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Kostolányi, P. Polynomially Ambiguous Unary Weighted Automata over Fields. Theory Comput Syst 67, 291–309 (2023). https://doi.org/10.1007/s00224-022-10107-7
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DOI: https://doi.org/10.1007/s00224-022-10107-7