---
_id: '5439'
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. '
alternative_title:
- IST Austria Technical Report
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. IST
Austria; 2015. doi:10.15479/AT:IST-2015-335-v1-1
apa: Boker, U., Henzinger, T. A., & Otop, J. (2015). The target discounted-sum
problem. IST Austria. https://doi.org/10.15479/AT:IST-2015-335-v1-1
chicago: Boker, Udi, Thomas A Henzinger, and Jan Otop. The Target Discounted-Sum
Problem. IST Austria, 2015. https://doi.org/10.15479/AT:IST-2015-335-v1-1.
ieee: U. Boker, T. A. Henzinger, and J. Otop, The target discounted-sum problem.
IST Austria, 2015.
ista: Boker U, Henzinger TA, Otop J. 2015. The target discounted-sum problem, IST
Austria, 20p.
mla: Boker, Udi, et al. The Target Discounted-Sum Problem. IST Austria, 2015,
doi:10.15479/AT:IST-2015-335-v1-1.
short: U. Boker, T.A. Henzinger, J. Otop, The Target Discounted-Sum Problem, IST
Austria, 2015.
date_created: 2018-12-12T11:39:20Z
date_published: 2015-05-18T00:00:00Z
date_updated: 2023-02-23T10:08:48Z
day: '18'
ddc:
- '004'
- '512'
- '513'
department:
- _id: ToHe
doi: 10.15479/AT:IST-2015-335-v1-1
file:
- access_level: open_access
checksum: 40405907aa012acece1bc26cf0be554d
content_type: application/pdf
creator: system
date_created: 2018-12-12T11:53:55Z
date_updated: 2020-07-14T12:46:55Z
file_id: '5517'
file_name: IST-2015-335-v1+1_report.pdf
file_size: 589619
relation: main_file
file_date_updated: 2020-07-14T12:46:55Z
has_accepted_license: '1'
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
page: '20'
publication_identifier:
issn:
- 2664-1690
publication_status: published
publisher: IST Austria
pubrep_id: '335'
related_material:
record:
- id: '1659'
relation: later_version
status: public
status: public
title: The target discounted-sum problem
type: technical_report
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2015'
...