Algorithmics of Checking First Result: . . . Whether a Mapping Is - PowerPoint PPT Presentation

Introduction First Problem: . . . Second Problem: . . . Case of Semi- . . . Algorithmics of Checking First Result: . . . Whether a Mapping Is How Efficient Are the . . . Polynomial Mapping . . . Injective, Surjective, Case of Analytical . . . Checking Approximate . . . and/or Bijective Home Page Title Page E. Cabral Balreira 1 , Olga Kosheleva 2 , and Vladik Kreinovich 2 ◭◭ ◮◮ ◭ ◮ 1 Department of Mathematics, Trinity University San Antonio, TX 78212, USA Page 1 of 19 ebalreir@trinity.edu Go Back 2 University of Texas at El Paso Full Screen El Paso, TX 79968, USA olgak@utep.edu, vladik@utep.edu Close Quit

Introduction First Problem: . . . 1. Introduction Second Problem: . . . • States of real-life systems change with time. Case of Semi- . . . First Result: . . . • In some cases, this change comes “by itself”, from laws How Efficient Are the . . . of physics. Polynomial Mapping . . . • Examples: radioactive materials decays, planets go around Case of Analytical . . . each other, etc. Checking Approximate . . . • In other cases, the change comes from our interference. Home Page • E.g., a spaceship changes trajectory after we send a Title Page signal to an engine to perform a trajectory correction. ◭◭ ◮◮ • In many situations, we have equations that describe ◭ ◮ this change, i.e., we know a function f : A → B that: Page 2 of 19 – transform the original state a ∈ A Go Back – into a state f ( a ) ∈ b at a future moment of time. Full Screen • In such situations, the following two natural problems arise. Close Quit

Introduction First Problem: . . . 2. First Problem: Checking Injectivity Second Problem: . . . • The first natural question is: Are the changes reversible? Case of Semi- . . . First Result: . . . • When we erase the value of the variable in a computer, How Efficient Are the . . . by replacing it with 0s, the changes are not reversible. Polynomial Mapping . . . • In such situations, two different original states a � = a ′ Case of Analytical . . . lead to the exact same new state f ( a ) = f ( a ′ ). Checking Approximate . . . • If different states a � = a ′ always lead to different f ( a ) � = Home Page f ( a ′ ), then, in principle, reconstruction is possible. Title Page • In mathematics, mappings f : A → B that map differ- ◭◭ ◮◮ ent elements into different ones are called injective . ◭ ◮ • So the question is: checking whether a given mapping Page 3 of 19 is injective. Go Back Full Screen Close Quit

Introduction First Problem: . . . 3. Second Problem: Checking Surjectivity Second Problem: . . . • The second natural question is: Are all the states b ∈ B Case of Semi- . . . possible as a result of this dynamics. First Result: . . . How Efficient Are the . . . • In other words, is it true that every state b ∈ B can be Polynomial Mapping . . . obtained as f ( a ) for some a ∈ A . Case of Analytical . . . • In mathematical terms, mappings that have this prop- Checking Approximate . . . erty are called surjective. Home Page • We may also want to check whether f is both injective Title Page and surjective, i.e., whether it is a bijection. ◭◭ ◮◮ • In this paper, we analyze these problems from an algo- ◭ ◮ rithmic viewpoint. Page 4 of 19 Go Back Full Screen Close Quit

Introduction First Problem: . . . 4. Case of Semi-Algebraic Mappings Second Problem: . . . • Let’s first the case when A , B are described by finitely Case of Semi- . . . many polynomial (in)equalities w/rational coefficients. First Result: . . . How Efficient Are the . . . • Such sets are called semi-algebraic . Polynomial Mapping . . . • Example: the upper half of the unit circle centered at Case of Analytical . . . the point (0 , 0) is Checking Approximate . . . { ( x 1 , x 2 ) : x 2 1 + x 2 2 = 1 and x 2 ≥ 0 } . Home Page • We also assume that the graph { ( a, f ( a )) : a ∈ A } is Title Page semi-algebraic; such maps are called semi-algebraic . ◭◭ ◮◮ • Example: every polynomial mapping is semi-algebraic. ◭ ◮ • Polynomial mappings are very important: Page 5 of 19 – every continuous f-n on a bounded set can be, w/any Go Back given accuracy, approximated by a polynomial; Full Screen – this means that, in practice, every action can be Close represented by a polynomial mapping. Quit

Introduction First Problem: . . . 5. First Result: Algorithms Are Possible Second Problem: . . . • Proposition: There exists an algorithm, that: Case of Semi- . . . First Result: . . . – given two semi-algebraic sets A and B and a semi- How Efficient Are the . . . algebraic mapping f : A → B , Polynomial Mapping . . . – checks whether f is injective, surjective, and/or bi- Case of Analytical . . . jective. Checking Approximate . . . • Each of the relations a ∈ A , b ∈ B , and f ( a ) = b can be Home Page described by a finite set of polynomial (in)equalities. Title Page • A polynomial is, by definition, a composition of addi- ◭◭ ◮◮ tions and multiplications. ◭ ◮ • Thus, both the injectivity and surjectivity can be de- Page 6 of 19 scribed in terms of the first order language with: Go Back – variables running over real numbers, and Full Screen – elementary formulas coming from addition, multi- plication, and equality: Close Quit

Introduction First Problem: . . . 6. Proof (cont-d) Second Problem: . . . • Reminder: we consider a first order language with: Case of Semi- . . . First Result: . . . – variables running over real numbers, and How Efficient Are the . . . – elementary formulas coming from addition, multi- Polynomial Mapping . . . plication, and equality: Case of Analytical . . . • In this language, injectivity can be described as: Checking Approximate . . . ∀ a ∀ a ′ ∀ b (( a ∈ A & a ′ ∈ A & f ( a ) = b & f ( a ′ ) = b & b ∈ B ) ⇒ Home Page a = a ′ ) . Title Page ◭◭ ◮◮ • Surjectivity can be described as: ◭ ◮ ∀ b ( b ∈ B ⇒ ∃ a ( a ∈ A & f ( a ) = b )) . Page 7 of 19 • For such first order formulas, there is a deciding algo- Go Back rithm designed by a famous logician A. Tarski. Full Screen • Thus, the problems of checking injectivity and surjec- tivity are indeed algorithmically decidable. Close Quit

Introduction First Problem: . . . 7. Remark Second Problem: . . . • One of the main open problems in this area is Jacobian Case of Semi- . . . Conjecture, according to which: First Result: . . . – every polynomial map f : C n → C n from n -dimensional How Efficient Are the . . . Polynomial Mapping . . . complex space into itself � ∂f i Case of Analytical . . . � – for which the Jacobi determinant det equals 1, Checking Approximate . . . ∂x j Home Page – is injective. Title Page • This is an open problem; however: ◭◭ ◮◮ – for any given dimension n and for any given degree ◭ ◮ d of the polynomial, Page 8 of 19 – the corresponding case of this conjecture can be resolved by applying the Tarski algorithm. Go Back Full Screen Close Quit

Introduction First Problem: . . . 8. How Efficient Are the Corresponding Algorithms? Second Problem: . . . Related Results Case of Semi- . . . • Proposition: Checking whether a given polynomial First Result: . . . R n → I R n is injective is NP-hard. mapping f : I How Efficient Are the . . . Polynomial Mapping . . . • Proposition: Checking whether a given polynomial R n → I R n is surjective is NP-hard. Case of Analytical . . . mapping f : I Checking Approximate . . . • Proposition: Checking whether a given polynomial Home Page R n → I R n is bijective is NP-hard. mapping f : I Title Page • Open questions: check whether the following problems ◭◭ ◮◮ are NP-hard: ◭ ◮ – checking whether a given injective polynomial map- Page 9 of 19 ping is also surjective, and Go Back – checking whether a given surjective polynomial map- ping is also injective. Full Screen Close Quit

Introduction First Problem: . . . 9. Proof that Checking Injectivity Is NP-Hard Second Problem: . . . • By definition, a problem is NP-hard if every problem Case of Semi- . . . from the class NP can be reduced to it. First Result: . . . How Efficient Are the . . . • Thus, to prove that this problem is NP-hard, let us Polynomial Mapping . . . reduce a known NP-hard problem to it. Case of Analytical . . . • As such a known problem, we take a subset problem : Checking Approximate . . . – given n + 1 positive integers s 1 , . . . , s n , S , Home Page – check whether there exist values x 1 , . . . , x n ∈ { 0 , 1 } Title Page n � for which s i · x i = S . ◭◭ ◮◮ i =1 ◭ ◮ • For each instance of the subset sum problem, take Page 10 of 19 f ( x 1 , . . . , x n , x n +1 ) = ( x 1 , . . . , x n , P ( x 1 , . . . , x n ) · x n +1 ) , Go Back � n � 2 n i · (1 − x i ) 2 + def � � x 2 Full Screen P ( x 1 , . . . , x n ) = s i · x i − S . i =1 i =1 Close Quit