Real Quantifier Elimination by Computation of Comprehensive Gr - PowerPoint PPT Presentation

Real Quantifier Elimination by Computation of Comprehensive Gr¨ obner Systems Ryoya Fukasaku 1 Hidenao Iwane 2 Yosuke Sato 1 1 Tokyo University of Science 2 National Institute of Informatics / Fujitsu Laboratories Ltd Tuesday 7th July 2015 1 / 35

Contents of My Talks 1 Motivation Todai Robot Project Quantified Formula with Many Equalities Real Quantifier Elimination by Computation of Comprehensive Gr¨ obner Systems 2 Real Root Counting Real Root Counting Theorem Charateristic Polynomial 3 Comprehensive Gr¨ obner System Algebraic Partition Comprehensive Gr¨ obner System 4 Main Algorithm 5 Computation Data 6 Conclusion 2 / 35

Motivation 3 / 35

Todai Robot Project The motivation of our work has its roots in “Todai Robot Project”. “Todai Robot Project” is the ongoing research project of artificial intelligence. The purpose of “Todai Robot Project” is to develop software which automatically produces an answer sheet for an entrance examination of “Todai”. University of Tokyo is known as “Todai” in Japan. University of Tokyo is the highest rank university in Japan. We have to obtain a sufficient score to pass by using our software. 4 / 35

Todai Robot Project How does our software solve math problems? Syntatic Parsing Math Problem Anaphora Resolution Discourse Analysis Natural Language Processing Semantic Representation Formula Rewriting Math Knowledge Base Input of Solver Gr¨ obner Basis Computer Algebra Quantifier Elimination (QE) etc. Answer 5 / 35

Todai Robot Project How does our software solve math problems? Syntatic Parsing “Find the radius r Math Problem of a circle c s.t. Anaphora Resolution the area is 4 π .” Discourse Analysis Natural Language Processing Dictionary: circle c : C p C q Semantic Representation area of c : A p C q Find( r ) [ @ c p C p c q^ radius of c : R p C q A p c q “ 4 π Ñ R p c q “ r q ]. Formula Rewriting Find( r )[ @ s p s ą 0 ^ Math Knowledge Base π s 2 “ 4 π Ñ s “ r q ]. Input of Solver Gr¨ obner Basis Computer Algebra Quantifier Elimination (QE) etc. r “ 2. Answer 6 / 35

Quantified Formula with Many Equalities Our software often generates a quantified formula with many equalities. Example △ ABC is inscribed in a circle with the radius 1, tan p = CAB q = m and tan p = ABC q “ n . However m , n ě 3. Let S be the area of △ ABC . (1) Represent S on the terms of m , n . Let φ 1 be x 0 x 3 ´ x 0 x 4 ` x 1 x 2 ´ x 1 x 3 ´ x 2 x 5 ` x 4 x 5 ě 0, φ 2 be p x 5 ´ x 0 q x 4 ´ x 2 ´ p x 3 ´ x 2 q x 1 ´ x 0 ě 0, A φ 3 be p x 5 ´ x 0 qpp 1 { 2 q x 0 ` p 1 { 2 q x 5 ` x 7 q ` p x 3 ´ x 2 qpp 1 { 2 q x 2 ` p 1 { 2 q x 3 ´ x 6 q “ 0, φ 4 be p x 1 ´ x 5 qpp 1 { 2 q x 5 ` p 1 { 2 q x 1 ´ x 7 q ` p x 4 ´ x 3 qpp 1 { 2 q x 3 ` p 1 { 2 q x 4 ´ x 6 q “ 0, 1 φ 5 be pp x 7 ´ x 0 q 2 ` p x 6 ´ x 2 q 2 q 1 { 2 “ 1, φ 6 be | x 0 x 3 ´ x 0 x 4 ` x 1 x 2 ´ x 1 x 3 ´ x 2 x 5 ` x 4 x 5 |{pp x 1 ´ x 0 qp x 5 ´ x 0 q ` p x 4 ´ x 2 qp x 3 ´ x 2 qq “ m , S C φ 7 be | x 0 x 3 ´ x 0 x 4 ` x 1 x 2 ´ x 1 x 3 ´ x 2 x 5 ` x 4 x 5 |{pp x 0 ´ x 5 qp x 1 ´ x 5 q ` p x 2 ´ x 3 qp x 4 ´ x 3 qq “ n , B φ 8 be m ě 3 ^ n ě 3, φ 9 be | x 5 ´ x 0 x 4 ´ x 2 ` x 3 ´ x 2 x 1 ´ x 0 |{ 2 “ S and φ a be D x 0 D x 1 D x 2 D x 3 D x 4 D x 5 D x 6 D x 7 p Ź 1 ď i ď 9 φ i q . φ can be not solved within 1 hour by the existing QE software SyNRAC@Maple, RegularChains@Maple, Resolve@Mathematica, Reduce@Mathematica, QEPCAD and RedLog@Reduce. 7 / 35

Quantified Formula with Many Equalities We need to establish a practical implementation of QE for a quantified formula with many equalities. We improve the following work: 1998: Weispfenning, V. : A New Approach to Quantifier Elimination for Real Algebra. We call for short “comprehensive Gr¨ obner system” “CGS” and “real QE by computation of CGSs” “CGS-QE”. 8 / 35

Real QE by Computation of CGSs CGS-QE is a special QE method for the input formula D ¯ x pp Ź i f i “ 0 q ^ p Ź i p i ą 0 q ^ p Ź i q i ­“ 0 qq , X s . where ¯ X “ X 1 , . . . , X n , ¯ Y “ Y 1 , . . . , Y m , f i , p i , q i P K r ¯ Y , ¯ CGS-QE uses “Real Root Counting Theorem (Pedersen)” and “CGS”. In Section Real Root Counting , we modify “Real Root Counting Theorem” for improving CGS-QE. In Section Comprehensive Gr¨ obner System , we show its definition. 9 / 35

Real Root Counting 10 / 35

Notations R denotes a real closed field, C its algebraic closed extension and K a computable subfield of R . Let ¯ X be variables X 1 , . . . , X n . T p ¯ X q denotes the set of all terms consisting of variables in ¯ X . In this section, let I be a zero dimensional ideal in K r ¯ X s . c P R n |@ f P I f p ¯ c P C n |@ f P I f p ¯ Let V R p I q “ t ¯ c q “ 0 u , V C p I q “ t ¯ c q “ 0 u . 11 / 35

Real Root Counting Theorem Let v 1 , . . . , v d be the basis of the residue class ring A “ K r ¯ X s{ I . For p P A and each i , j p 1 ď i , j ď d q , we consider the followings: Let Q p , i , j be the trace of a linear map A Ñ A by f ÞÑ pv i v j a for a P A . Let M I p be a symmetric matrix p M I p q p i , j q “ Q p , i , j . The signature of M I p is denoted σ p M I p q . Pedersen σ p M I p q “ # pt ¯ c P V R p I q| p p ¯ c qą 0 uq´ # pt ¯ c P V R p I q| p p ¯ c qă 0 uq . Corollary σ p M I 1 q “ # p V R p I qq . We can compute σ p M I Remark p q by computing the number of the sign cheanges of the coefficients of the chracteristic polynomial of M I p . 12 / 35

In CGS-QE, by using the obvious equivalent relations “ p ą 0 ô D z z 2 p “ 1” and “ q ­“ 0 ô D w wq “ 1” we reduce “the degree of a charateristic polynomial”. 13 / 35

Real Root Counting Theorem Let p 1 , . . . , p s P K r ¯ X s and ¯ Z be new variables Z 1 , . . . , Z s . s p s ´ 1 y be an ideal in K r ¯ X , ¯ Let J “ I ` x Z 2 1 p 1 ´ 1 , . . . , Z 2 Z s . Corollary # p V R p J qq “ 2 s # pt ¯ c P V R p I q| p 1 p ¯ c q ą 0 , . . . , p s p ¯ c q ą 0 uq . Let I 1 be the elimination ideal J X K r ¯ X s . Corollary I 1 “ x 1 y _ p i is invertible in K r ¯ X s{ I 1 for i “ 1 , . . . , s . X s{ I 1 for i “ 1 , . . . , s . i in K r ¯ We assume that p i has the inverse p 1 Corollary J “ I 1 ` x Z 2 1 ´ p 1 1 , . . . , Z 2 s ´ p 1 s y . 14 / 35

Real Root Counting Theorem s ´ p s y with p 1 , . . . , p s P K r ¯ Let J “ I ` x Z 2 1 ´ p 1 , . . . , Z 2 X s . Let B I “ t t 1 , . . . , t k u Ă T p ¯ X q be a basis of K r ¯ X s{ I and B J “ t t 1 Z e 1 1 Z e 2 s , . . . , t k Z e 1 1 Z e 2 2 ¨ ¨ ¨ Z e s 2 ¨ ¨ ¨ Z e s s |p e 1 , e 2 , . . . , e s qPt 0 , 1 u s u . # Then B J forms a basis of K r ¯ X , ¯ Z s{ J . For g P K r ¯ X s , we consider the followings: M J g denote a symmetric matrix such as the matrix of Pedersen for J , g and χ J g its characteristic polynomial. We consider also M I g and χ I g simiraly as M J g and χ J g . Theorem χ J g p 2 s X q “ c ś p e 1 , e 2 ,..., e s qPt 0 , 1 u s χ I s p X q (a non-zero constant c ) . gp e 1 1 p e 2 2 ¨¨¨ p es ( 7 See the proceedings) 15 / 35

Charateristic Polynomial Example We consider I “ xp x 2 1 ´ x 2 2 qp x 1 ` 2 x 2 ´ 1 q , p 3 x 1 ` x 2 ´ 1 q 2 y , p 1 “ 2 p 2 ´ 1 y , I 1 “ J X Q r x 1 , x 2 s x 1 ´ x 2 and p 2 “ x 1 ` x 2 . Let J “ I ` x z 2 1 p 1 ´ 1 , z 2 and ą be a term order such that z 1 ą z 2 ą x 1 ą x 2 . t 25 x 2 2 ´ 20 x 2 ` 4 , x 1 ` 2 x 2 ´ 1 , 9 z 2 2 ´ 25 x 2 ´ 5 , z 2 1 ´ 75 x 2 ` 35 u is a Gr¨ obner basis of J w.r.t. ą . I 1 “ x 25 x 2 2 ´ 20 x 2 ` 4 , x 1 ` 2 x 2 ´ 1 y . Let p 1 1 “ 15 Y ´ 7 , p 1 2 “ 5 Y ` 1. χ I 2 p X q χ I 2 p X q χ I 1 p 2 p X q χ I p 1 p 2 p X q has a degree 24, p 2 1 p 2 p 1 p 2 p 2 whereas χ I 1 1 p X q χ I 1 1 p X q χ I 1 2 p X q χ I 1 2 p X q has a degree 8. p 1 p 1 p 1 1 p 1 The original CGS-QE computes χ I 2 p X q χ I 2 p X q χ I 1 p 2 p X q χ I p 1 p 2 p X q . p 2 1 p 2 p 1 p 2 p 2 16 / 35

Charateristic Polynomial We compute the saturation ideal I 1 of I w.r.t. polynomials p 1 ,. . ., p s . x s{ I 1 is smaller than it of K r ¯ The dimension of K r ¯ x s{ I . We can reduce the degree of our charateristic polynomial. By using a primary decomposition of I , we can certainly remove the unnecessary portion from I . For parametric polynomial ideals, this computation or even factorization of a polynomial becomes a significantly heavy computation. Using the relation q ­“ 0 ô D W Wq “ 1, we further can reduce the degree of a charateristic polynomial. 17 / 35

Comprehensive Gr¨ obner System 18 / 35

Notations Let ¯ X be main variables X 1 , . . . , X n . Let ¯ Y be parameters Y 1 , . . . , Y m . Given a term order, LM p f q , LT p f q , LC p f q denotes the leading monomial, the leading term, the leading coefficient of a polynomial f , respectively. 19 / 35

Algebraic Partition Algebraic Partition Let S be a subset of an affine space C n for some natural number n . A finite set t S 1 , . . . , S t u of non-empty subsets of S is called an algebraic partition of S if it satisfies the properties 1 , 2 , 3 : 1 Y t i “ 1 S i “ S . 2 S i X S j “ H if i ­“ j . 3 For each i , S i “ V C p I 1 qz V C p I 2 q for some ideals I 1 , I 2 of K r ¯ Y s . Each S i is called a segment. We identify each S i with its defining formula. 20 / 35