---
_id: '1659'
abstract:
- lang: eng
text: 'The target discounted-sum problem is the following: Given a rational discount
factor 0 < λ < 1 and three rational values a, b, and t, does there exist
a finite or an infinite sequence w ε(a, b)∗ or w ε(a, b)w, such that Σ|w| i=0
w(i)λi equals t? The problem turns out to relate to many fields of mathematics
and computer science, and its decidability question is surprisingly hard to solve.
We solve the finite version of the problem, and show the hardness of the infinite
version, linking it to various areas and open problems in mathematics and computer
science: β-expansions, discounted-sum automata, piecewise affine maps, and generalizations
of the Cantor set. We provide some partial results to the infinite version, among
which are solutions to its restriction to eventually-periodic sequences and to
the cases that λ λ 1/2 or λ = 1/n, for every n ε N. We use our results for solving
some open problems on discounted-sum automata, among which are the exact-value
problem for nondeterministic automata over finite words and the universality and
inclusion problems for functional automata.'
acknowledgement: 'A technical report of the article is available at: https://repository.ist.ac.at/335/'
author:
- first_name: Udi
full_name: Boker, Udi
id: 31E297B6-F248-11E8-B48F-1D18A9856A87
last_name: Boker
- first_name: Thomas A
full_name: Henzinger, Thomas A
id: 40876CD8-F248-11E8-B48F-1D18A9856A87
last_name: Henzinger
orcid: 0000−0002−2985−7724
- first_name: Jan
full_name: Otop, Jan
id: 2FC5DA74-F248-11E8-B48F-1D18A9856A87
last_name: Otop
citation:
ama: 'Boker U, Henzinger TA, Otop J. The target discounted-sum problem. In: *LICS*.
Logic in Computer Science. IEEE; 2015:750-761. doi:10.1109/LICS.2015.74'
apa: 'Boker, U., Henzinger, T. A., & Otop, J. (2015). The target discounted-sum
problem. In *LICS* (pp. 750–761). Kyoto, Japan: IEEE. https://doi.org/10.1109/LICS.2015.74'
chicago: Boker, Udi, Thomas A Henzinger, and Jan Otop. “The Target Discounted-Sum
Problem.” In *LICS*, 750–61. Logic in Computer Science. IEEE, 2015. https://doi.org/10.1109/LICS.2015.74.
ieee: U. Boker, T. A. Henzinger, and J. Otop, “The target discounted-sum problem,”
in *LICS*, Kyoto, Japan, 2015, pp. 750–761.
ista: 'Boker U, Henzinger TA, Otop J. 2015. The target discounted-sum problem. LICS.
LICS: Logic in Computer ScienceLogic in Computer Science 750–761.'
mla: Boker, Udi, et al. “The Target Discounted-Sum Problem.” *LICS*, IEEE,
2015, pp. 750–61, doi:10.1109/LICS.2015.74.
short: U. Boker, T.A. Henzinger, J. Otop, in:, LICS, IEEE, 2015, pp. 750–761.
conference:
end_date: 2015-07-10
location: Kyoto, Japan
name: 'LICS: Logic in Computer Science'
start_date: 2015-007-06
date_created: 2018-12-11T11:53:19Z
date_published: 2015-07-01T00:00:00Z
date_updated: 2020-01-21T13:18:24Z
day: '01'
department:
- _id: ToHe
doi: 10.1109/LICS.2015.74
language:
- iso: eng
month: '07'
oa_version: None
page: 750 - 761
project:
- _id: 25EE3708-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '267989'
name: Quantitative Reactive Modeling
- _id: 25832EC2-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: S 11407_N23
name: Rigorous Systems Engineering
- _id: 25F42A32-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z211
name: The Wittgenstein Prize
publication: LICS
publication_identifier:
eisbn:
- '978-1-4799-8875-4 '
issn:
- '1043-6871 '
publication_status: published
publisher: IEEE
publist_id: '5491'
quality_controlled: '1'
related_material:
record:
- id: '5439'
relation: earlier_version
status: public
series_title: Logic in Computer Science
status: public
title: The target discounted-sum problem
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2015'
...