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Effective Quantifier Elimination Over Real Closed Fields

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Effective Quantifier Elimination Over Real Closed Fields EFFECTIVE QUANTIFIER ELIMINATION OVER REAL CLOSED FIELDS N. Vorobjov 1 Tarski (1931, 1951): Elementary algebra and geometry is decidable. More generally, there is an algorithm for quan- tifier elimination in the first order theory of the reals. Boolean formula F (x1;:::; xº) with atoms of the kind f > 0, where f 2 R[x0; x1;:::; xº], xi = (xi;1; : : : ; xi;ni). Φ(x0) := Q1x1Q2x2 ¢ ¢ ¢ QºxºF (x0; x1;:::; xº), where Qi 2 f9; 8g, Qi 6= Qi+1. Theorem 1. There is a quantifier-free formula Ψ(x0) such that Ψ(x0) , Φ(x0) over R, n n i.e., fx0 2 R 0j Ψ(x0)g = fx0 2 R 0j Φ(x0)g. Theorem remains true if R is replaced by any real closed field R, e.g., real algebraic numbers, Puiseux series over R in an infinitesimal. 2 Theorem 2. If atoms f 2 Z[x0; x1;:::; xº], then there is an algorithm which eliminates quanti- fiers from Φ(x0), i.e., for a given Φ(x0) pro- duces Ψ(x0). Corollary 3. The first order theory of the reals is decidable, i.e., there is an algorithm which for a given closed formula Q1x1Q2x2 ¢ ¢ ¢ QºxºF (x1;:::; xº) decides whether it’s true or false. 3 Integer coefficients are mentioned here to be able to handle the problem on a Turing ma- chine. If we are less picky about the model of computation, for example allow exact arith- metic operations and comparisons in a real closed field R, then Theorem ?? and Corol- lary ?? are also true for formulae over R. In what follows we will use exactly this approach. 3-1 Let R be a real closed field. Definition 4. A set X ½ Rn is called semialgebraic if X is representable in a form X = fx 2 RnjF (x)g, where F (x) is a quantifier- free formula. n Corollary 5. Let X = fx0 2 R 0jΦ(x0)g, where Φ(x0) := Q1x1Q2x2 ¢ ¢ ¢ QºxºF (x0; x1;:::; xº). Then X is semialgebraic. 4 Examples: ² Feasibility of a system of polynomial equa- tions and inequalities (= is a semialgebraic set empty?) ² Representation of the closure (or interior, or singular locus, or tubular neighbourhood, etc.) of a semi- algebraic set by a Boolean combination of polynomial inequalities ² Polynomial optimization: find the set of points of local minima of a polynomial function on a semialgebraic set. 5 There is no need to explain to logicians the usefulness of quantifier elimination. Instead I list a few straightforward examples of how it can be used in “ordinary” mathematics. Semialgebraic set in all the examples initially can be naturally represented by prenex formu- lae with quantifiers (sometimes involving "=± language. 5-1 Complexity: Tarski’s quantifier elimination algorithm: “non-elementary” (can’t be bounded from above by any tower of exponentials of a finite height). Problem: construct an efficient algorithm Possible approach: cylindrical (algebraic) cell decomposition (mid-70s, W¨uthrich,Collins) 6 Definition 6. Cylindrical cell is defined by in- duction as follows. 1. Cylindrical 0-cell in Rn is an isolated point. Cylindrical 1-cell in R is an open interval (a; b) ½ R. 2. For n ¸ 2 and 0 · k < n a cylindrical (k + 1)-cell B in Rn is either a graph of a continuous bounded function f : C ! R, where C is a cylindrical a cylindrical (k+1)- cell in Rn¡1, or else a set of the form n f(x1; : : : ; xn) 2 R j (x1; : : : ; xn¡1) 2 C and f(x1; : : : ; xn¡1) < xn < g(x1; : : : ; xn¡1)g; where C is a cylindrical k-cell in Rn¡1, and f; g : C ! R are continuous bounded func- tions such that f(x1; : : : ; xn¡1) < g(x1; : : : ; xn¡1) for all points (x1; : : : ; xn¡1) 2 C. 7 Cylindrical cell decomposition is a very use- ful tool when we study o-minimal structures. Since I’ll need this concept also in my other lecture, let me give a definition. We first de- fine a cylindrical cell. One can notice that any cylindrical cell is homeomorphic to an open ball of the matching dimension (semialgebraically homeomorphic in the case of an arbitrary R). We will give the definition only for the bounded case. 7-1 Definition 7. Cylindrical cell decomposition D of a subset A ½ Rn is defined by induction as follows. 1. If n = 1, then D is a finite family of pair- wise disjoint cylindrical cells (i.e., isolated points and intervals) whose union is A. 2. If n ¸ 2, then D is a finite family of pair- wise disjoint cylindrical cells in Rn whose union is A and there is a cylindrical cell de- composition D0 of ¼(A) such that ¼(C) is its cell for each C 2 D, where ¼ : Rn ! Rn¡1 is the projection map onto the co- ordinate subspace of x1; : : : ; xn¡1. We say that D0 is induced by D. Definition 8. Let B ½ A ½ Rn and D be a CCD of A. Then D is compatible with B if for any C 2 D we have either C ½ B or C \ B = ; (i.e., some subset D0 ½ D is a CCD of B). 8 Example: 9 Theorem 9. (Collins, W¨uthrich). If X ½ Rn is a semialgebraic set, then there is an algorithm for computing CCD of Rn compatible with X. Moreover, CCD is semialgebraic, i.e., each cell is described by a system of polynomial equa- tions and inequalities. Technical tools: resultants and sub-resultants. Compare with classical elimination theory (van-der-Waerden, Modern Algebra) and Chevalley’s Theorem. Corollary 10. There is an algorithm for quan- tifier elimination. 10 The corollary is almost obvious from the defini- tion of the CCD. One reasons by induction on quantifiers starting from Qº and moving out- ward. On each step we use the relation (de- scribed in the definition) between the decom- position of the space of all current variables and the decomposition of the subspace of cur- rent free variables. 10-1 Complexity: Let Φ(x0) involve s atoms of the kind f > 0, where f 2 R[x0;:::; xº], n := n0 + ¢ ¢ ¢ + nº, and deg(f) < d. Then the complexity of constructing CCD and the corresponding quantifier elimination has an upper bound n (sd)O(1) : Similar upper bound on the number of cells, degrees and quantities of polynomials describ- ing CCD and quantifier-free formula equivalent to Φ. If coefficients of polynomials f are integer of bit-sizes less than M, then the (Turing ma- chine) complexity does not exceed n MO(1)(sd)O(1) : 11 Davenport & Heintz (1988) (similar results by Weispfenning (1988)) found a (parametric) example of Φ of degrees · 4, with two free variables and n1 · 2; : : : ; nº · 2 (i.e., each quantifier is responsible for at most two variables) defining in R2 a semialgebraic set consisting of n 22 isolated points. ) Any CCD for Φ can’t contain lesser num- ber of cells. ) Lower complexity bound for any CCD-based algorithm is doubly exponential. Fisher & Rabin (1974): lower bound for deci- sion methods is exponential in the number of quantifier alternations. 12 Simplest case: 9x 2 R (f(x) = 0); where f 2 R[x], deg(f) < d. Sturm’s Theorem (1835): Euclidean algorithm for the division of f by its derivative f 0. 0 Let g1 = f; g2 = f , and g3; : : : ; gt be the se- quence of the negatives of the remainders in the Euclidean algorithm. For any x 2 R let V (x) be the number of sign changes in g1(x); g2(x); : : : ; gt(x). Theorem 11. The number of distinct real roots of f in [a; b] ½ R is S(f; f 0) := V (a) ¡ V (b): Note: sign(f(1)) =sign(leading coefficient a0 of f) d sign(f(¡1)) =sign(a0x at ¡1). 13 Note that over an algebraically closed field the answer is always “Yes”. Sturm’s theorem provides an algorithm to de- cide the existential formula over any real closed field. Now we want to generalize this algorithm to be able to handle systems of many equations and inequalities (in one variable, for time being). 13-1 Ben-Or/Kozen/Reif algorithm (1986): Definition 12. Let f1; : : : ; fk 2 R[x], deg(fi) < d. Consistent sign assignment is a string σ = (σ1; : : : ; σk) where σi 2 f>; <; =g, such that the system f1σ10; : : : ; fkσk0 has a solution in R. Input: f1; : : : ; fk 2 R[x]. Output: The list of all consistent sign assignments. Prepare the input: Make all fi squarefree and relatively prime. Then sets of roots of polynomials fi are disjoint, in particular any consistent sign assignment has at most one =. Moreover, we can add a new polynomial f so that all con- sistent assignments for f1; : : : ; fk can be recon- structed from all consistent assignments of < and > for f1; : : : ; fk at roots of f. 14 Case k = 0: f = 0 Sturm’s Theorem. Case k = 1: f; f1 C1 := fxj f = 0 ^ f1 > 0g C1 := fxj f = 0 ^ f1 < 0g Obviously, the total number of roots of f is 0 S(f; f ) = jC1j + jC1j. (Here and in the sequel S corresponds to the interval [¡1; 1].) 0 Lemma 13. S(f; f f1) = jC1j ¡ jC1j. Sturm + Lemma ) Ã !Ã ! Ã ! 0 1 1 jC1j S(f; f ) = 0 1 ¡1 jC1j S(f; f f1) 15 Case k = 2: f; f1; f2 C2 := fxjf = 0 ^ f2 > 0g C2 := fxjf = 0 ^ f2 < 0g 0 1 0 1 1 1 1 1 jC1 \ C2j B C B C B1 ¡1 1 ¡1C BjC \ C jC B C B 1 2 C = @1 1 ¡1 ¡1A @jC1 \ C2jA 1 ¡1 ¡1 1 jC1 \ C2j 0 1 S(f; f 0) B 0 C B S(f; f f1) C = B 0 C @ S(f; f f2) A 0 S(f; f f1f2) 16 In case k we get a matrix Ak which is a tensor product of k copies of A1 and is non-singular.
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