# Determinizing discounted-sum automata

Boker U, Henzinger TA. 2011. Determinizing discounted-sum automata. CSL: Computer Science Logic, LIPIcs, vol. 12, 82–96.

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Grant

Series Title

LIPIcs

Abstract

A discounted-sum automaton (NDA) is a nondeterministic finite automaton with edge weights, which values a run by the discounted sum of visited edge weights. More precisely, the weight in the i-th position of the run is divided by lambda^i, where the discount factor lambda is a fixed rational number greater than 1. Discounted summation is a common and useful measuring scheme, especially for infinite sequences, which reflects the assumption that earlier weights are more important than later weights. Determinizing automata is often essential, for example, in formal verification, where there are polynomial algorithms for comparing two deterministic NDAs, while the equivalence problem for NDAs is not known to be decidable. Unfortunately, however, discounted-sum automata are, in general, not determinizable: it is currently known that for every rational discount factor 1 < lambda < 2, there is an NDA with lambda (denoted lambda-NDA) that cannot be determinized. We provide positive news, showing that every NDA with an integral factor is determinizable. We also complete the picture by proving that the integers characterize exactly the discount factors that guarantee determinizability: we show that for every non-integral rational factor lambda, there is a nondeterminizable lambda-NDA. Finally, we prove that the class of NDAs with integral discount factors enjoys closure under the algebraic operations min, max, addition, and subtraction, which is not the case for general NDAs nor for deterministic NDAs. This shows that for integral discount factors, the class of NDAs forms an attractive specification formalism in quantitative formal verification. All our results hold equally for automata over finite words and for automata over infinite words.

Publishing Year

Date Published

2011-08-31

Volume

12

Page

82 - 96

Conference

CSL: Computer Science Logic

Conference Location

Bergen, Norway

Conference Date

2011-09-12 – 2011-09-15

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### Cite this

Boker U, Henzinger TA. Determinizing discounted-sum automata. In: Vol 12. Springer; 2011:82-96. doi:10.4230/LIPIcs.CSL.2011.82

Boker, U., & Henzinger, T. A. (2011). Determinizing discounted-sum automata (Vol. 12, pp. 82–96). Presented at the CSL: Computer Science Logic, Bergen, Norway: Springer. https://doi.org/10.4230/LIPIcs.CSL.2011.82

Boker, Udi, and Thomas A Henzinger. “Determinizing Discounted-Sum Automata,” 12:82–96. Springer, 2011. https://doi.org/10.4230/LIPIcs.CSL.2011.82.

U. Boker and T. A. Henzinger, “Determinizing discounted-sum automata,” presented at the CSL: Computer Science Logic, Bergen, Norway, 2011, vol. 12, pp. 82–96.

Boker, Udi, and Thomas A. Henzinger.

*Determinizing Discounted-Sum Automata*. Vol. 12, Springer, 2011, pp. 82–96, doi:10.4230/LIPIcs.CSL.2011.82.**All files available under the following license(s):**

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2018-12-12

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