Technion, Spring semester 2013 238900-13 The millennium question over the reals, the complex numbers and other general structures. Research seminar 238900-13 Johann A. Makowsky ∗ ∗ Faculty of Computer Science, Technion - Israel Institute of Technology, Haifa, Israel janos@cs.technion.ac.il Graph polynomial project: http://www.cs.technion.ac.il/ ∼ janos/RESEARCH/gp-homepage.html File:title 1
Technion, Spring semester 2013 238900-13-1 Lecture 1: Summary • Introducing the topic. • Register machines for arbitrary rings R . • Basic observations concerning unit cost over R . • Defining decidability DEC R over R . • Defining P R and NP R over R . • Proving computability of NP R ⊂ DEC R over the real and complex num- bers using quantifier elimination (QE) . • What is QE? File:lec-1 2
Technion, Spring semester 2013 238900-13-2 Lecture 2: Summary We shall consider two cases where P R � = NP R . 0n one case we add the FORTRAN-function: sin( x ), in the oder case we disregard multiplication and the order. • Adding sin( x ) to R . R = � R , + , × , <, sin x, 0 , 1 � . We define P sin , NP sin and DEC sin in the natural way. • Disregarding multiplication and order and test only for equality: R = � R , + , 0 , 1 � We define P lin , NP lin and DEC lin in the natural way. Theorem: (Klaus Meer) P sin � = NP sin and P lin � = NP lin . Klaus Meer, • A note on a P � = NP for a restricted class of real machines, Journal of Complexity 8 (1992), 451-453 • Real number models under various sets of operations, Journal of Complexity 9 (1993), 366-372 File:lec-2 3
Technion, Spring semester 2013 238900-13-2 Lecture 2: Summary (contd) We shall also discuss adding other FORTRAN functions: exp , log , sin • Adding exp( x ) to R . R = � R , + , × , <, exp( x ) , 0 , 1 � . We define P exp and NP exp in the natural way. • Adding two functions: F 1 = { exp( x ) , sin( x ) } or F 2 = { exp( x ) , log( x ) } to R . R = � R , + , × , <, exp( x ) , log( x ) , sin( x )0 , 1 � . We define P F 1 , P F 2 , NP F 1 and NP F 2 in the natural way. Theorem: (Mihai Prunescu) P exp � = NP exp and P F 1 � = NP F 1 . Note: P F 2 � = NP F 2 remains open . Mihai Prunescu, • P � = NP for a the reals with Various Analytic Functions. Journal of Complexity 17 (2001), 17-26 File:lec-2 4
Technion, Spring semester 2013 238900-13-2 Exercise: Computability of Z in R We want to decide whether the sets Z ⊂ R and Q ⊂ R are computable. • The decision problems for the sets Z and Q are computable in BSS over the field R , but there is no bound on the length of the computation. • Are the problems ( R , Z ) and ( R , Q ) in NP R ? • The problems ( R , Z ) and ( R , Q ) are in P sin . In fact they are computable in constant time. We use the fact that sin( k · π ) = 0 for k ∈ Z . File:lec-2 5
Technion, Spring semester 2013 238900-13-2 The set A Meer We look at the set ( k · t A Meer = A = { t ∈ [0 , 2 π ] : ∃ k ∈ N 2 π ∈ N ) } . We note that t • t ∈ A iff 2 π ∈ Q ∩ [0 , 1], hence A is countable and dense in [0 , 2 π ]. • The problem ( R , [0 , 2 π ]) is in P sin . Here we use a constant a = 2 π . • We study the problem ( R , A Meer ), respectively ([0 , 2 π ] , A Meer ). File:lec-2 6
Technion, Spring semester 2013 238900-13-2 ([0 , 2 π ] , A Meer ) is in NP sin • Input x ∈ R always has size 1. • We show that using a single guess ([0 , 2 π ] , A Meer ) can be solved in constant time. 1. GUESS k ∈ R . 2. TEST k ≥ 0 and then sin( kπ ) = 0. 3. Two yes show that k ∈ N . 4. TEST sin( k · x 2 ) = 0. 5. Yes shows that x ∈ A . Q.E.D. File:lec-2 7
Technion, Spring semester 2013 238900-13-2 Real-analytic functions S.G. Krantz and H.R. Parks, A Primer for Real-Analytic Functions, Birkh¨ auser, 2002 (2nd edition) A function f : R → R is real-analytic on an open set U ⊆ R if for all x ∈ U • f has derivatives of all orders at x , and • for every a ∈ U there is a neighborhood a ∈ V ⊂ U such that for all x ∈ V f agrees with its Taylor series, i.e., f ( n ) ( a ) � ( x − a ) n f ( x ) = n ! n Examples: polynomials, 1 x , sin , cos , exp , log are real analytic. � 0 x ∈ Q The function f ( x ) = x ∈ R − Q is not real-analytic. 1 File:lec-2 8
Technion, Spring semester 2013 238900-13-2 Properties of real-analytic functions Proposition: (Classical) • The set of real-analytic functions on U is closed under scalar multiplication, pointwise addition multiplication and composition. • The reciprocal of an analytic function that is nowhere zero is analytic. • The inverse of an invertible analytic function whose derivative is nowhere zero is analytic. • Assume r n are distinct zeroes of f and lim r n = r is in a connected component D r the domain D of f . Then f ( x ) = 0 for all x ∈ D r . Hence, if f is not constant on D r , it has at most countable many zeroes in D r . File:lec-2 9
Technion, Spring semester 2013 238900-13-2 ([0 , 2 π ] , A Meer ) is not in DEC sin hence not P sin . There is no deterministic program using sin( x ) which always terminates and decides ([0 , 2 π ] , A Meer ). • We proceed by contradiction. • We do not allow division, and discuss later what effect divsion has. • Assume we have a program which decides ([0 , 2 π ] , A Meer ). It may have a fixed number of constants, c 1 , . . . , c s ∈ R . File:lec-2 10
Technion, Spring semester 2013 238900-13-2 Evaluating paths in the computation tree Let γ be a path in the (unwound) computation tree of a program over R and sin. • γ evaluates a term T γ ( x, c 1 , . . . , c s ). • T γ ( x, c 1 , . . . , c s ) represents a a real f γ ( x ) function which is real-analytic. • Since the program always terminates, there are at most countably many paths γ . • We can replace functions f γ ( x ) which are identically 0 by constant as- signments. • Let So there at most countably many values x ∈ R for which f γ ( x ) = 0. File:lec-2 11
Technion, Spring semester 2013 238900-13-2 Heading for the contradiction We now look at the set B = { t ∈ [0 , 2 π ] : ∃ γ ( f γ ( t ) = 0) } We note: • B is countable. • Let t 0 ∈ [0 , 2 π ] − B . So for all γ we have f γ ( t 0 ) � = 0. • Since each path γ is finite there is some open set U ( t 0 ) ⊆ R such that our program gives the saem answer for all inputs x ∈ U ( t 0 ), i.e., U ( t 0 ) ⊆ A or U ( t 0 ) ⊆ [0 , 2 π ] − A . • But this is impossible since A is countable and dense in [0 , 2 π ]. Q.E.D. File:lec-2 12
Technion, Spring semester 2013 238900-13-2 Meer’s Theorem We have shown Theorem: (K. Meer 1992) Let F be any set of real-analytic functions such that ( R , Z ) ∈ NP F . Then P F � = NP F . Problem: What limitations does the requirement ( R , Z ) ∈ NP F impose? • For a set D ⊆ R the problem is in NP F iff it is existentially first order de- finable using addition, multiplication, order, constants and unary function symbols for functions from F . • For F = { exp( x ) } we have ( R , Z ) �∈ NP F . This is so, because every set D ⊆ R with ( R , D ) ∈ NP exp has only finitely many connected components. L. van den Dries and C. Miller, The field of reals with restricted analytic functions and exponentiation, Israel Journal of Mathematics (1994). File:lec-2 13
Technion, Spring semester 2013 238900-13-2 The role of division Our proof shows really: Theorem: (Meer-Prunescu) Let D ⊆ R and R − D be dense in [ a, b ] ⊂ R . Then ([ a, b ] , D ) �∈ DEC F for any set F of real-analytic functions. • This allows to include division, although division is not a total function. • To obtain Then P F � = NP F we still need to show that ([ a, b ] , D ) ∈ NP F . • To show that P exp � = NP exp Prunescu uses a different approach. File:lec-2 14
Technion, Spring semester 2013 238900-13-2 Semi-algebraic sets Let D ⊂ R n . • D is semi-algebraic if it is the solution set of a quantifierfree first order formula over the ordered field of R . • A function f : R n → R is semi-algebraic, if its graph is a semi-algebraic set. • A function f is essentially non-semi-algebraic if for no open set U ⊆ R n the function f | U is semi-algebraic. • Let F be a set of real-analytic functions. f is semi-analytic if it is the solution set of a quantifierfree first order formula over the ordered field of R with function symbols for functions from F . • f is sub-analytic if it is the solution set of a first order formula over the ordered field of R with function symbols for functions from F . • A real-analytic function is tame on U if it has no analytical singularities on the boundaries of U . • A total real-analytic function f is always tame. File:lec-2 15
Technion, Spring semester 2013 238900-13-2 Prunescu’s Theorem Theorem: (M. Prunescu 2001) Let F be a set of real-analytic tame functions containing at least one function which is essentially non semi-algebraic. Then P F � = NP F . Comments: • F 0 = { exp } is tame. F 1 = { exp , log } is not tame. F 2 = { exp , log | (1 , ∞ ) } is tame. • A P runescu = { ( x, y, z ) ∈ R 3 : y > 0 ∧ z = y · exp( x y ) } • A P runescu ∈ NP F i for i = 0 , 1 , 2. • A P runescu �∈ P F 0 . A P runescu ∈ P F 1 in constant time. A P runescu �∈ P F 2 but A P runescu ∈ DEC F 2 . File:lec-2 16
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