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Intelligent Systems Reference Library 179 Mikhail Moshkov Comparative Analysis of Deterministic and Nondeterministic Decision Trees Intelligent Systems Reference Library Volume 179 Series Editors Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland Lakhmi C Jain, Faculty of Engineering and Information Technology, Centre for Artiﬁcial Intelligence, University of Technology, Sydney, NSW, Australia; KES International, Shoreham-by-Sea, UK; Liverpool Hope University, Liverpool, UK The aim of this series is to publish a Reference Library, including novel advances and developments in all aspects of Intelligent Systems in an easily accessible and well structured form The series includes reference works, handbooks, compendia, textbooks, well-structured monographs, dictionaries, and encyclopedias It contains well integrated knowledge and current information in the ﬁeld of Intelligent Systems The series covers the theory, applications, and design methods of Intelligent Systems Virtually 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Indexing: The books of this series are submitted to ISI Web of Science, SCOPUS, DBLP and Springerlink More information about this series at http://www.springer.com/series/8578 Mikhail Moshkov Comparative Analysis of Deterministic and Nondeterministic Decision Trees 123 Mikhail Moshkov Computer, Electrical and Mathematical Science and Engineering Division King Abdullah University of Science and Technology Thuwal, Saudi Arabia ISSN 1868-4394 ISSN 1868-4408 (electronic) Intelligent Systems Reference Library ISBN 978-3-030-41727-7 ISBN 978-3-030-41728-4 (eBook) https://doi.org/10.1007/978-3-030-41728-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on 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published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To my family Preface This book is devoted to the comparative analysis of deterministic, nondeterministic, and strongly nondeterministic decision trees for problems over information systems An information system consists of a universe and a set of attributes deﬁned on the universe Various problems of fault diagnosis, computational geometry, combinatorial optimization, etc., can be represented as problems over appropriate ﬁnite or inﬁnite information systems Each decision table can be interpreted as a problem over corresponding ﬁnite information system Deterministic decision trees are widely used as classiﬁers, as a means of knowledge representation, and as algorithms Nondeterministic decision trees are less known A nondeterministic decision tree can be interpreted as a system of decision rules for a given problem that covers all inputs Strongly nondeterministic decision trees can be considered for problems that have two decisions: and A strongly nondeterministic decision tree accepts only inputs with the decision and can be interpreted as a system of decision rules which covers all inputs with the decision and only such inputs We design tools for study of problems over information systems: lower and upper bounds on complexity and algorithms for construction of deterministic, nondeterministic, and strongly nondeterministic decision trees for decision tables with many-valued decisions We consider two approaches to the study of decision trees for problems over information systems: local, when we can use in decision trees only attributes from the problem representation, and global, when we can use arbitrary attributes from the information system In the frameworks of each approach, we compare the complexity of problem representation and minimum complexities of deterministic, nondeterministic, and strongly nondeterministic decision trees solving problem For the local and global approaches, we describe all possible types of relationships among these four parameters We also study relationships among the following algorithmic problems: problems of computation of the minimum complexity of deterministic, nondeterministic, and strongly nondeterministic decision vii viii Preface trees; problems of construction of decision trees with the minimum complexity; and the problem of solvability of systems of equations over information systems The results presented in this book can be useful for researchers who use decision trees and rules in design and analysis of algorithms, and in data analysis, especially those who work in rough set theory, test theory, and logical analysis of data The book can be used for the creation of courses for graduate students Thuwal, Saudi Arabia November 2019 Mikhail Moshkov Acknowledgements The author wishes to express his deep gratitude to the late Al A Markov, O B Lupanov, Z Pawlak, and S V Yablonskii, and also to thank A Skowron and Ju I Zhuravlev who influenced greatly the author’s views on the subject of the present investigation The author is greatly indebted to Lobachevsky State University of Nizhni Novgorod, University of Warsaw, University of Silesia in Katowice, Stanford University, and King Abdullah University of Science and Technology for their hospitality and support during the preparation of the book The author extend an expression of gratitude to Prof J Kacprzyk, Dr T Ditzinger, and the Series Intelligent Systems Reference Library staff at Springer for their support in making this book possible ix Contents 2 9 10 11 12 13 13 Basic Deﬁnitions and Notation 2.1 Common Notions 2.2 Decision Tables 2.3 Schemes of Decision Trees 2.4 Different Types of Decision Trees for Decision Tables 2.5 Complexity Functions 2.6 Enumerated Signatures References 17 18 18 19 20 21 22 22 Lower Bounds on Complexity of Deterministic Decision Trees for Decision Tables 3.1 Deﬁnitions and Notation 3.2 Auxiliary Statements 25 25 27 Introduction 1.1 Main Directions of Study 1.1.1 Decision Tables with Many-Valued Decisions 1.1.2 Local Approach to Study of Decision Trees 1.1.3 Global Approach to Study of Decision Trees 1.2 Contents of Book 1.2.1 Part I Decision Trees for Decision Tables 1.2.2 Part II Decision Trees for Problems Local Approach 1.2.3 Part III Decision Trees for Problems Global Approach 1.2.4 Index and Notation 1.3 Use of Book References Part I Decision Trees for Decision Tables xi Chapter 25 Algorithmic Problems Global Approach In this chapter, we study algorithmic problems related to the global approach to the investigation of decision trees: problems of computation of the minimum complexity of deterministic, nondeterministic, and strongly nondeterministic decision trees, problems of construction of decision trees with the minimum complexity, and the problem of solvability of systems of equations over information systems We study relationships among these problems We also discuss the notion of a proper weighted depth for which the problems of computation of the minimum complexity of decision trees and problems of construction of decision trees with the minimum complexity are decidable if the problem of solvability of systems of equations over information systems is decidable Some results for deterministic decision trees considered in this chapter were published in [1, 2] 25.1 Some Relationships Among Algorithmic Problems Let τ = (ρ, K , ψ) be a restricted enumerated sccf-triple Let b ∈ {d, a, s} We now define the algorithmic problems Com b (τ ) and Des b (τ ) Problem Com b (τ ): for a given schema z ∈ Στb , it is required to compute the value b ψρ,K (z) Problem Des b (τ ): for a given schema z ∈ Στb , it is required to construct a schema b b and ψ(Γ ) = ψρ,K (z) Γ ∈ Cρ such that (Γ, z) ∈ Rρ,K Theorem 25.1 Let τ = (ρ, K , ψ) be a restricted enumerated sccf-triple, where ρ = (F, k) Then the following statements hold: (a) If the problem E x(τ ) is undecidable and τ is a nondegenerate sccf-triple, then the problems Com d (τ ), Des d (τ ), Com a (τ ), Des a (τ ), Com s (τ ), and Des s (τ ) are undecidable © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4_25 281 282 25 Algorithmic Problems Global Approach (b) If the problem E x(τ ) is undecidable and τ is a degenerate sccf-triple, then the problems Des d (τ ) and Des a (τ ) are undecidable, and the problems Com d (τ ), Com a (τ ), Com s (τ ), and Des s (τ ) are decidable Proof (a) Let τ be a nondegenerate sccf-triple and the problem E x(τ ) be undecidable Let b ∈ {d, a, s} We now show that the problem Com b (τ ) is undecidable Assume the contrary Let us show that the problem E x(τ ) is decidable Since τ is a nondegenerate sccf-triple, there exists an element f i0 ∈ F which is not τ -constant Let α ∈ Ωρ If α = λ, then, evidently, α is a τ -realizable word Let α = λ If α is inconsistent, then, evidently, α is not τ -realizable Let α be a nonempty consistent word and α = ( f i1 , δi1 ) · · · ( f in , δin ) For each δ ∈ E k , we define a schema z δ = (ν, f i0 , f i1 , , f in ) as follows: νδ : E kn+1 → {{0}, {1}} and, for any σ¯ ∈ E kn+1 , if σ¯ = (δ, δi1 , , δin ), then νδ (σ¯ ) = {0}, and if σ¯ = (δ, δi1 , , δin ), then νδ (σ¯ ) = {1} Taking into account that the element f i0 is not τ -constant one can show that z δ ∈ Σρ0−1 (K ) ⊆ Στb Using the decidability of the problem Com b (τ ), for each b δ ∈ E k , we find the value ψρ,K (z δ ) Taking into account that the function ψ has properties Λ3 and Λ4, and f i0 is an element which is not τ -constant one can show b (z δ ) = for any δ ∈ E k Thus, that the word α is not τ -realizable if and only if ψρ,K the problem E x(τ ) is decidable which is impossible Hence the problem Com b (τ ) is undecidable Taking into account that ψ is a computable function we conclude that the problem Des b (τ ) is undecidable (b) Let τ be a degenerate sccf-triple and the problem E x(τ ) be undecidable Since τ is a degenerate sccf-triple, Nρ (T ) = for any table T ∈ Mρ,K Taking into account that the function ψ has the property Λ4 and using Theorem 10.1 we conclude that, for b (z) = holds Hence any b ∈ {d, a, s} and for any schema z ∈ Σρb , the equality ψρ,K d a s the problems Com (τ ), Com (τ ), and Com (τ ) are decidable We denote by Γ0 the schema which contains the root, a terminal node labeled with the number 1, and the edge leaving the root and entering the terminal node Evidently, for any problem schema z ∈ Σρ0−1 (K ), the schema Γ0 is a solution of the problem Des s (τ ) Therefore the problem Des s (τ ) is decidable Let b ∈ {d, a} Let us show that the problem Des b (τ ) is undecidable Assume the contrary We now show that the problem E x(τ ) is decidable Let α ∈ Ωρ If α = λ, then, evidently, the word α is τ -realizable Let α = λ and α = ( f i1 , δ1 ) · · · ( f in , δn ) We now define a mapping ν : E kn → {{0}, {1}} Let σ¯ ∈ E kn If σ¯ = (δ1 , , δn ), then ν(σ¯ ) = {1} If σ¯ = (δ1 , , δn ), then ν(σ¯ ) = {0} Set z = (ν, f i1 , , f in ) Evidently, z ∈ Στb Let Γ ∈ Cρ and the schema Γ is a solution of the problem Des b (τ ) for the problem schema z Taking into account that the function ψ has properties Λ3 and Λ4, and τ is degenerate one can show that Γ consists of the root, a terminal node labeled with a number r and the edge leaving the root and entering the terminal node One can show that r = if and only if the word α is τ -realizable Thus, the problem E x(τ ) is decidable which is impossible Hence the problem Des b (τ ) is undecidable Let τ = (ρ, K , ψ) be a restricted enumerated sccf-triple and b ∈ {d, a, s} We now define the algorithmic problem R b (τ ) 25.1 Some Relationships Among Algorithmic Problems 283 Problem R b (τ ): for given schema z ∈ Στb and schema Γ ∈ Cρ it is required to b recognize if (Γ, z) ∈ Rρ,K Lemma 25.1 Let τ = (ρ, K , ψ) be a restricted enumerated sccf-triple, the problem E x(τ ) be decidable, and b ∈ {s, a, d} Then the problem R b (τ ) is decidable Proof Let z ∈ Στb and Γ ∈ Cρ Since the problem E x(τ ) is decidable, there is an algorithm which, for given z ∈ Στb and Γ ∈ Cρ , constructs the decision table b if and only if Tρ (α(z, P(Γ )), K ) By Theorem 18.1, (Γ, z) ∈ Rρ,K (Γ, Tρ (α(z, P(Γ )), K )) ∈ Rρb It is easy to show that the relation Rρb is decidable Theorem 25.2 Let τ = (ρ, K , ψ) be a restricted enumerated sccf-triple, the problem E x(τ ) be decidable, and b ∈ {s, a, d} Then the problem Com b (τ ) is decidable if and only if the problem Des b (τ ) is decidable Proof Let the problem Des b (τ ) be decidable and z ∈ Στb Using the decidability of b and the problem Des b (τ ) we construct a schema Γ ∈ Cρ such that (Γ, z) ∈ Rρ,K b ψ(Γ ) = ψρ,K (z) Using the computability of the function ψ we compute the value b ψ(Γ ) It coincides with the value ψρ,K (z) Hence the problem Com b (τ ) is decidable b Let the problem Com (τ ) be decidable Using Lemma 25.1 and the fact that the b is decidable We problem E x(τ ) is decidable we conclude that the relation Rρ,K know that the function ψ is computable on the set Cρ One can show that there exists an algorithm which enumerates all elements of the set Cρ Let z ∈ Στb Using b (z) Among the decidability of the problem Com b (τ ) we compute the value ψρ,K b listed schemes from the set Cρ , we choose a schema Γ such that (Γ, z) ∈ Rρ,K and b b ψ(Γ ) = ψρ,K (z) Hence the problem Des (τ ) is decidable 25.2 Proper Weighted Depth Let ρ = (F, k) be an enumerated signature, F = { f i : i ∈ ω}, and ψ be a computable weighted depth of the signature ρ We will say that ψ is a proper weighted depth if, for any nonempty class K of information systems of the signature ρ such that the problem E x(τ ) is decidable for the sccf-triple τ = (ρ, K , ψ), the problems Com b (τ ) and Des b (τ ) are decidable for any b ∈ {d, a, s} For i ∈ ω, we denote ωψ (i) = { j : j ∈ ω, ψ( f j ) = i} Define a partial function Hψ : ω → ω as follows Let i ∈ ω If ωψ (i) is a finite set then Hψ (i) = ωψ (i) If ωψ (i) is an infinite set, then the value of Hψ (i) is indefinite Denote by Arg Hψ the domain of Hψ Lemma 25.2 Let ρ = (F, k) be an enumerated signature, F = { f i : i ∈ ω}, ψ be a computable weighted depth of the signature ρ, Arg Hψ = ω, and the function Hψ be recursive Then ψ is a proper weighed depth 284 25 Algorithmic Problems Global Approach Proof Let b ∈ {d, a, s} Let K be a nonempty class of information systems of the signature ρ such that the problem E x(τ ) is decidable for the sccf-triple τ = (ρ, K , ψ) By Lemma 25.1, the problem R b (τ ) is decidable Since Arg Hψ = ω and the function Hψ is recursive, there exists an algorithm which, for a given number r ∈ ω constructs the set { f i : f i ∈ F, ψ( f i ) ≤ r } Using this fact it is not difficult to show that there exists an algorithm which, for a given number r ∈ ω and a finite nonempty subset D of the set ω, constructs the set Cρ (r, D) of all schemes of decision trees Γ of the signature ρ satisfying the following conditions: h(Γ ) ≤ r , ψ( f i ) ≤ r for any element f i ∈ P(Γ ), each terminal node of the tree Γ is labeled with a number from the set D, and, for any two different complete paths ξ1 and ξ2 of the scheme Γ , π(ξ1 ) = π(ξ2 ) ¯ Let us prove Let z ∈ Στb and z = (ν, f i1 , , f in ) Denote D(z) = δ∈E ¯ kn ν(δ) i that the set Cρ (ψρ,K (z), D(z)) contains a decision tree schema Γ such that (Γ, z) ∈ b b Rρ,K and ψ(Γ ) = ψρ,K (z) One can show that there exists a decision tree schema Γ b b of the signature ρ for which (Γ, z) ∈ Rρ,K , ψ(Γ ) = ψρ,K (z), all terminal nodes are labeled with numbers from the set D(z), and, for any two different complete paths b i (z) ≤ ψρ,K (z) ξ1 and ξ2 of the scheme Γ , π(ξ1 ) = π(ξ2 ) By Lemma 19.1, ψρ,K i i Therefore ψ(Γ ) ≤ ψρ,K (z) Using this inequality we obtain h(Γ ) ≤ ψρ,K (z) and i i ψ( f i ) ≤ ψρ,K (z) for any attribute f i ∈ P(Γ ) Therefore Γ ∈ Cρ (ψρ,K (z), D(z)) b We now describe an algorithm which solves the problem Des (τ ) Let z ∈ i (z) and construct the set D(z) Construct the set Στb Compute the value ψρ,K i Cρ (ψρ,K (z), D(z)) With the help of algorithm which solves the problem R b (τ ) i b we find a decision tree schema Γ ∈ Cρ (ψρ,K (z), D(z)) such that (Γ, z) ∈ Rρ,K i b and ψ(Γ ) = min{ψ(G) : G ∈ Cρ (ψρ,K (z), D(z)), (G, z) ∈ Rρ,K } It is clear that b b ψ(Γ ) = ψρ,K (z) So the problem Des (τ ) is decidable Using Theorem 25.2 we conclude that the problem Com b (τ ) is also decidable Taking into account that b is an arbitrary index from the set {d, a, s} and K is an arbitrary nonempty class of information systems of the signature ρ such that the problem E x(τ ) is decidable for the sccf-triple τ = (ρ, K , ψ), we obtain that ψ is proper weight function Lemma 25.3 Let ρ be an enumerated signature, ψ be a computable weighted depth of the signature ρ, Arg Hψ = ω, and the function Hψ be not recursive Then ψ is not a proper weighed depth Proof In the proof of Lemma 5.41 from [2] it was shown that there exists an information system U of the signature ρ such that, for the sccf-triple τ = (ρ, {U }, ψ), the problem E x(τ ) is decidable but the problem Des d (τ ) is undecidable Therefore ψ is not a proper weighed depth Lemma 25.4 Let ρ be an enumerated signature, ψ be a computable weighted depth of the signature ρ, and Arg Hψ = ω Then ψ is a not proper weighed depth Proof In the proof of Lemma 5.42 from [2] it was shown that there exists an information system U of the signature ρ such that, for the sccf-triple τ = (ρ, {U }, ψ), 25.2 Proper Weighted Depth 285 the problem E x(τ ) is decidable but the problem Com d (τ ) is undecidable Therefore ψ is not a proper weighed depth Theorem 25.3 Let ρ be an enumerated signature and ψ be a computable weighted depth of the signature ρ Then ψ is a proper weighed depth if and only if Arg Hψ = ω and Hψ is a recursive function Proof The considered statement follows immediately from Lemmas 25.2–25.4 Let ρ = (F, k) be an enumerated signature and F = { f i : i ∈ ω} We now consider examples of proper weighted depths of the signature ρ Let ϕ : ω → ω \ {0} be a total recursive nondecreasing function which is unbounded from above Then the ϕ ϕ weighted depth ψ √ such that ψ ( f i ) = ϕ(i) for any i ∈ ω is proper In particular, x x + are positive total recursive nondecreasing functions which x + 1, , and are unbounded from above References Moshkov, M.: Decision Trees Theory and Applications (in Russian) Nizhny Novgorod University Publishers, Nizhny Novgorod (1994) Moshkov, M.: Time complexity of decision trees In: Peters, J.F., Skowron, A (eds.) Transactions on Rough Sets III, Lecture Notes in Computer Science, vol 3400, pp 244–459 Springer, Berlin (2005) Final Remarks The aim of this book is to compare four parameters of problems over information systems: complexity of problem representation and complexity of deterministic, nondeterministic, and strongly nondeterministic decision trees for problem solving First, we created tools for the study of decision trees: lower and upper bounds on complexity and algorithms for construction of decision trees for decision tables with many-valued decisions Next, we studied the local approach to the investigation of decision trees that allows us to use in the decision trees only attributes from the problem representation Finally, we studied the global approach to the investigation of decision trees that allows us to use arbitrary attributes from the considered information system For both approaches, we described all types of relationships among the four parameters of problems We also discussed a number of algorithmic problems related to the local and the global approaches Future study will be devoted to the comparative analysis of deterministic and nondeterministic acyclic programs over various bases © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4 287 Notation A an , 107 AU , 119 AU (α), 119 Aτ , 199 Arg f , 123 α(A), 38 α(z, B), 209 α (t) , 211 α, ˜ 250 B B(f , g, A), 58 Bτbc (n), 211 Btb (z), 215 C Coma (ρ, ψ), 105 Comd (ρ, ψ), 105 Coms (ρ, ψ), 105 Coma (τ ), 281 Comd (τ ), 281 Coms (τ ), 281 ˆ a (τ ), 195 Com ˆ d (τ ), 195 Com ˆ s (τ ), 195 Com Cρ , 19 Cρ0 (T ), 66 Cρd , 71 χ(α), 25 D D1 , 94 D2 , 94 D3 , 94 Dρ , 70 Dρ (η), 80 Des(ρ, ψ), 112 Desa (ρ, ψ), 105 Desd (ρ, ψ), 105 Dess (ρ, ψ), 105 Desa (τ ), 281 Desd (τ ), 281 Dess (τ ), 281 ˆ a (τ ), 195 Des ˆ d (τ ), 195 Des ˆ Dess (τ ), 195 d [α], 41 dn , 95 dρ (T ), 31 dρ (T , m), 31 dm , 168 Δ(T ), 18 Δ(T , 1), 86 Δ(T , α), 155 Δ(T , m), 31 E EV (f ), 95 Ek , 18 Ex(τ ), 196 en , 168 η(D, α), 41 ε2 , 93 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 M Moshkov, Comparative Analysis of Deterministic and Nondeterministic Decision Trees, Intelligent Systems Reference Library 179, https://doi.org/10.1007/978-3-030-41728-4 289 290 F FR, 132 FZ, 129 f −1 , 132 F0 , 168 F1r , 94 F2r , 94 F3r , 94 F4r , 94 f U , 119 G G(α, T ), 85 GHρ (T ), 38 GR, 133 G f , 130 G , 168 G ρ (α), 27 G w , 255 G ϕ ba , 131 G ϕ ba (n), 131 G ψ ab , 130 G ψ ab (n), 130 γ (f ), 119 γD (α), 41 γ(1,m) , 168 γ(2,1,m) , 168 γ(2,2,m) , 176 γ(2,3,w) , 255 γ(3,1,m) , 169 γ(3,2) , 177 γ(3,3,f ) , 178 γ(3,4,f ) , 179 γ(4,2,f ) , 180 γ(4,3,f ) , 181 γ(4,4,f ) , 182 γ(4,5,1) , 256 γ(4,6,f ) , 257 γ(4,7,g) , 258 γ(4,8,q) , 259 γ(4,m) , 170 γ(5,m) , 171 γ(6,1,m) , 171 γ(6,2,f ) , 184 γ(7) , 250 γU , 119 H H V , 114 Hρm (T ), 38 Hψ , 283 Notation a , 94 hU d , 94 hU s , 94 hU h, 21 , 94 hd , 94 hs , 94 I [i]2 , 22 ∞, 210 Iψ (n, T ), 159 Iτ , 160 J Jρ (T ), 67 K Kρ (T ), 67 Kρ (Γ ), 67 Kρ (w), 67 ˜ 128 K, kn , 95 κρ (T , α), 72 L
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