# Second-Order Finite Automata

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12159)

## Abstract

Traditionally, finite automata theory has been used as a framework for the representation of possibly infinite sets of strings. In this work, we introduce the notion of second-order finite automata, a formalism that combines finite automata with ordered decision diagrams, with the aim of representing possibly infinite sets of sets of strings. Our main result states that second-order finite automata can be canonized with respect to the second-order language they represent. Using this canonization result, we show that sets of sets of strings represented by second-order finite automata are closed under the usual Boolean operations, such as union, intersection, difference and even under a suitable notion of complementation. Additionally, emptiness of intersection and inclusion are decidable.

We provide two algorithmic applications for second-order automata. First, we show that they can be used to show that several width and size minimization problems for deterministic and nondeterministic ODDs are solvable in fixed-parameter tractable time when parameterized by the width of the input ODD. In particular, our results imply FPT algorithms for corresponding width and size minimization problems for ordered binary decision diagrams with a fixed variable ordering. The previous best algorithms for these problems were exponential on the size of the input ODD even for ODDs of constant width. Second, we show that for each $$k$$ and $$w$$ one can count the number of distinct functions accepted by ODDs of width $$w$$ and length $$k$$ in time $$h_{\varSigma }(w)\cdot k^{O(1)}$$. This improves exponentially on the time necessary to explicitly enumerate all distinct functions, which take time exponential in both the width parameter $$w$$ and in the length $$k$$ of the ODDs.

## References

1. 1.
Bollig, B.: On the width of ordered binary decision diagrams. In: Zhang, Z., Wu, L., Xu, W., Du, D.-Z. (eds.) COCOA 2014. LNCS, vol. 8881, pp. 444–458. Springer, Cham (2014).
2. 2.
Bollig, B.: On the minimization of (complete) ordered binary decision diagrams. Theory Comput. Syst. 59(3), 532–559 (2016)
3. 3.
Bouajjani, A., Habermehl, P., Rogalewicz, A., Vojnar, T.: Abstract regular tree model checking. Electron. Notes Theoret. Comput. Sci. 149(1), 37–48 (2006)
4. 4.
Bozapalidis, S., Kalampakas, A.: Graph automata. Theoret. Comput. Sci. 393(1–3), 147–165 (2008)
5. 5.
Courcelle, B.: On recognizable sets and tree automata. In: Algebraic Techniques, pp. 93–126. Elsevier (1989)Google Scholar
6. 6.
Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)Google Scholar
7. 7.
Courcelle, B., Durand, I.: Verifying monadic second order graph properties with tree automata. In: Rhodes, C. (ed.) 3rd European Lisp Symposium, ELS, pp. 7–21. ELSAA (2010)Google Scholar
8. 8.
Ebbinghaus, H.D., Flum, J.: Finite automata and logic: a microcosm of finite model theory. In: Finite Model Theory, pp. 107–118. Springer, Heidelberg (1995).
9. 9.
Ergün, F., Kumar, R., Rubinfeld, R.: On learning bounded-width branching programs. In: Maass, W. (ed.) Proceedings of the Eighth Annual Conference on Computational Learning Theory, COLT, pp. 361–368. ACM (1995)Google Scholar
10. 10.
Forbes, M.A., Kelley, Z.: Pseudorandom generators for read-once branching programs, in any order. In: Thorup, M. (ed.) 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pp. 946–955. IEEE (2018)Google Scholar
11. 11.
Giammarresi, D., Restivo, A.: Recognizable picture languages. Int. J. Pattern Recogn. Artif. Intell. 6(02n03), 241–256 (1992)Google Scholar
12. 12.
Godefroid, P.: Using partial orders to improve automatic verification methods. In: Clarke, E.M., Kurshan, R.P. (eds.) CAV 1990. LNCS, vol. 531, pp. 176–185. Springer, Heidelberg (1991).
13. 13.
Hopcroft, J.: An $$n \log n$$ algorithm for minimizing states in a finite automaton. In: Theory of Machines and Computations, pp. 189–196. Elsevier (1971)Google Scholar
14. 14.
Priese, L.: Automata and concurrency. Theoret. Comput. Sci. 25(3), 221–265 (1983)
15. 15.
Thomas, W.: Automata theory on trees and partial orders. In: Bidoit, M., Dauchet, M. (eds.) CAAP 1997. LNCS, vol. 1214, pp. 20–38. Springer, Heidelberg (1997).