10.15479/AT:IST-2015-335-v1-1
Boker, Udi
Udi
Boker
Henzinger, Thomas A
Thomas A
Henzinger0000−0002−2985−7724
Otop, Jan
Jan
Otop
The target discounted-sum problem
IST Austria Technical Report
IST Austria
2015
2018-12-12T11:39:20Z
2019-11-14T08:43:06Z
technical_report
https://research-explorer.app.ist.ac.at/record/5439
https://research-explorer.app.ist.ac.at/record/5439.json
2664-1690
589619 bytes
application/pdf
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.