TY - JOUR

T1 - Decision trees for regular factorial languages

AU - Moshkov, Mikhail

N1 - KAUST Repository Item: Exported on 2022-09-14
Acknowledgements: Research reported in this publication was supported by King Abdullah University of Science and Technology (KAUST), Saudi Arabia . The author is grateful to the anonymous reviewers for useful remarks and suggestions.

PY - 2022/6/14

Y1 - 2022/6/14

N2 - In this paper, we study arbitrary regular factorial languages over a finite alphabet Σ. For the set of words L(n) of the length n belonging to a regular factorial language L, we investigate the depth of decision trees solving the recognition and the membership problems deterministically and nondeterministically. In the case of recognition problem, for a given word from L(n), we should recognize it using queries each of which, for some i∈{1,…,n}, returns the ith letter of the word. In the case of membership problem, for a given word over the alphabet Σ of the length n, we should recognize if it belongs to the set L(n) using the same queries. For a given problem and type of trees, instead of the minimum depth h(n) of a decision tree of the considered type solving the problem for L(n), we study the smoothed minimum depth H(n)=max{h(m):m≤n}. With the growth of n, the smoothed minimum depth of decision trees solving the problem of recognition deterministically is either bounded from above by a constant, or grows as a logarithm, or linearly. For other cases (decision trees solving the problem of recognition nondeterministically, and decision trees solving the membership problem deterministically and nondeterministically), with the growth of n, the smoothed minimum depth of decision trees is either bounded from above by a constant or grows linearly. As corollaries of the obtained results, we study joint behavior of smoothed minimum depths of decision trees for the considered four cases and describe five complexity classes of regular factorial languages. We also investigate the class of regular factorial languages over the alphabet {0,1} each of which is given by one forbidden word.

AB - In this paper, we study arbitrary regular factorial languages over a finite alphabet Σ. For the set of words L(n) of the length n belonging to a regular factorial language L, we investigate the depth of decision trees solving the recognition and the membership problems deterministically and nondeterministically. In the case of recognition problem, for a given word from L(n), we should recognize it using queries each of which, for some i∈{1,…,n}, returns the ith letter of the word. In the case of membership problem, for a given word over the alphabet Σ of the length n, we should recognize if it belongs to the set L(n) using the same queries. For a given problem and type of trees, instead of the minimum depth h(n) of a decision tree of the considered type solving the problem for L(n), we study the smoothed minimum depth H(n)=max{h(m):m≤n}. With the growth of n, the smoothed minimum depth of decision trees solving the problem of recognition deterministically is either bounded from above by a constant, or grows as a logarithm, or linearly. For other cases (decision trees solving the problem of recognition nondeterministically, and decision trees solving the membership problem deterministically and nondeterministically), with the growth of n, the smoothed minimum depth of decision trees is either bounded from above by a constant or grows linearly. As corollaries of the obtained results, we study joint behavior of smoothed minimum depths of decision trees for the considered four cases and describe five complexity classes of regular factorial languages. We also investigate the class of regular factorial languages over the alphabet {0,1} each of which is given by one forbidden word.

UR - http://hdl.handle.net/10754/674913

UR - https://linkinghub.elsevier.com/retrieve/pii/S2590005622000510

UR - http://www.scopus.com/inward/record.url?scp=85132510055&partnerID=8YFLogxK

U2 - 10.1016/j.array.2022.100203

DO - 10.1016/j.array.2022.100203

M3 - Article

SN - 2590-0056

VL - 15

SP - 100203

JO - Array

JF - Array

ER -