--- _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' ...