GiNaCRA:A C++ Library for Real Algebraic Computations Work-In-Progress Presentation NFM 2011 Contribution Ulrich Loup Erika Abrah´am´ AlgoSyn February 2011 AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Naming GiNaC C++ library for symbolic computations GiNaCRA { GiNaC Real Algebra package http://ginacra.sourceforge.net/ RealAlgebra E.g. 9x; y ( x42y − 2y 4 < 1 _ y − x ≥ 0 ) ^ x2 − y > 0 Ulrich Loup 2/ 17 AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Outline Introduction Features and Functioning Outlook Ulrich Loup 3/ 17 The quadratic case (Virtual substitution solver) p 2 −b± b2−4ac ax + bx − c = 0 () x 2 f 2a g; a; b; c 2 Z: I Real solutions: rationals with square roots Arbitrary degree polynomials (I-RiSC) Pk i i=0 ci x = 0; k 2 N; ci 2 Z; 0 ≤ i ≤ k; ck 6= 0: I Real solutions: real algebraic numbers AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Number representations in exact solving over the reals Linear real arithmetic (Yices/Z3, OpenSMT,...) a ax − b = 0 () x = b ; a; b 2 Z: I Real solutions: rational numbers Introduction Ulrich Loup 4/ 17 Arbitrary degree polynomials (I-RiSC) Pk i i=0 ci x = 0; k 2 N; ci 2 Z; 0 ≤ i ≤ k; ck 6= 0: I Real solutions: real algebraic numbers AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Number representations in exact solving over the reals Linear real arithmetic (Yices/Z3, OpenSMT,...) a ax − b = 0 () x = b ; a; b 2 Z: I Real solutions: rational numbers The quadratic case (Virtual substitution solver) p 2 −b± b2−4ac ax + bx − c = 0 () x 2 f 2a g; a; b; c 2 Z: I Real solutions: rationals with square roots Introduction Ulrich Loup 4/ 17 AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Number representations in exact solving over the reals Linear real arithmetic (Yices/Z3, OpenSMT,...) a ax − b = 0 () x = b ; a; b 2 Z: I Real solutions: rational numbers The quadratic case (Virtual substitution solver) p 2 −b± b2−4ac ax + bx − c = 0 () x 2 f 2a g; a; b; c 2 Z: I Real solutions: rationals with square roots Arbitrary degree polynomials (I-RiSC) Pk i i=0 ci x = 0; k 2 N; ci 2 Z; 0 ≤ i ≤ k; ck 6= 0: I Real solutions: real algebraic numbers Introduction Ulrich Loup 4/ 17 AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Where is exact solving for real algebra desired? Verification of, e.g., I probabilistic programs ( Friedrich Gretz's topic) or I hybrid systems. ; Synthesis of, e.g., I parameters for hybrid systems. Nonlinear optimization in, e.g., I designing a railway infrastructure ( Jacob Sp¨onemann'stopic). ... ; Introduction Ulrich Loup 5/ 17 AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations State of the art C++ libraries implementing real root computations Libreduce (based on Reduce), CoCoALib (based on CoCoA), Givaro, SYNAPS Software implementing algebraic number arithmetic Singular, KANT, PARI/GP Software implementing polynomial arithmetic GiNaC, Maple, MATLAB, Mathematica,... Introduction Ulrich Loup 6/ 17 Features and Functioning AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Representing a real algebraic number Interval representation p; ( l; r ) |{z} | {z } 2 Z[x] exactly one root of p in (l; r) Example 2 p x − 2 p − 2 2 ( )( ) R -3 -2 -1 0 1 2 3 Features and Functioning Ulrich Loup 8/ 17 AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Demonstration A life cycle of a real algebraic number I Real root isolation I Refinement of isolating intervals I Arithmetic and relational operations I Sign determination with polynomials Features and Functioning Ulrich Loup 9/ 17 AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Mathematical ingredients (1/2): Cauchy bound Let Pk i I p = i=0 ci x with k 2 N, ci 2 Z, 0 ≤ i ≤ k, ck 6= 0. then for all roots a 2 R of p k X ci jaj ≤ ck i=0 Example 2 −2 1 The Cauchy bound of p = x − 2 is 1 + 1 = 3. Indeed, p j 2j ≤ 3: Features and Functioning Ulrich Loup 10/ 17 Example Given p = x2 − 2 in , then 2 and #s−3 = 2, =) x − 2 has . AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Mathematical ingredients (2/2): Sturm's Theorem Let I (l; r) be an interval with l; r 2 Q, 0 0 I s = (p; p ; −rem(p; p );:::; c) with rem(p1; p2) being the remainder when dividing p1 by p2 and c constant, 0 0 I sa = (p(a); p (a); −rem(p; p )(a);:::; c) with a 2 Q, and I #sa be the number of sign changes in sa (0 not counted), then the p has #sl − #sr distinct roots in (l; r). Features and Functioning Ulrich Loup 11/ 17 AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Mathematical ingredients (2/2): Sturm's Theorem Let I (l; r) be an interval with l; r 2 Q, 0 0 I s = (p; p ; −rem(p; p );:::; c) with rem(p1; p2) being the remainder when dividing p1 by p2 and c constant, 0 0 I sa = (p(a); p (a); −rem(p; p )(a);:::; c) with a 2 Q, and I #sa be the number of sign changes in sa (0 not counted), then the p has #sl − #sr distinct roots in (l; r). Example Given p = x2 − 2 in (−3; 0), then 2 s = (x − 2; 2x; 2) s−3 = (7; −6; 2) s0 = (−2; 0; 2) 2 and #s−3 = 2, #s0 = 1 =) x − 2 has one root in (−3; 0). Features and Functioning Ulrich Loup 11/ 17 AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Mathematical ingredients (2/2): Sturm's Theorem Let I (l; r) be an interval with l; r 2 Q, 0 0 I s = (p; p ; −rem(p; p );:::; c) with rem(p1; p2) being the remainder when dividing p1 by p2 and c constant, 0 0 I sa = (p(a); p (a); −rem(p; p )(a);:::; c) with a 2 Q, and I #sa be the number of sign changes in sa (0 not counted), then the p has #sl − #sr distinct roots in (l; r). Example Given p = x2 − 2 in (−3; 3), then 2 s = (x − 2; 2x; 2) s−3 = (7; −6; 2) s3 = (7; 6; 2) 2 and #s−3 = 2, #s3 = 0 =) x − 2 has two roots in (−3; 3). Features and Functioning Ulrich Loup 11/ 17 AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Isolating a real root Divide and conquer 1. Calculate Cauchy bound C(p) and set l := −C(p), r := C(p). 2. If p(l) = 0, stop with output l. If p(r) = 0, stop with output r. 3. If #sl − #sr = 0, stop. 4. If #sl − #sr = 1, stop with output (l; r). 5. If #sl − #sr > 1, goto 1. with both r−l 5.1 r = l + 2 and r−l 5.2 l = l + 2 . Features and Functioning Ulrich Loup 12/ 17 Operations on real algebraic numbers I arithmetic operations: +, −, ∗, = I relational operations: = I sign determination w.r.t. other univariate polynomials I common real roots of a set of univariate polynomials AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Main contributions in a nutshell Class-oriented approach I real algebraic numbers I special types of univariate and multivariate polynomials I open intervals including interval arithmetic Features and Functioning Ulrich Loup 13/ 17 AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Main contributions in a nutshell Class-oriented approach I real algebraic numbers I special types of univariate and multivariate polynomials I open intervals including interval arithmetic Operations on real algebraic numbers I arithmetic operations: +, −, ∗, = I relational operations: = I sign determination w.r.t. other univariate polynomials I common real roots of a set of univariate polynomials Features and Functioning Ulrich Loup 13/ 17 Standalone Reliable I only non-standard library I CppUnit test suite with GiNaC many test cases I independent of closed-source I uses C++ standard library software wherever possible AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Some more features Open Source (GNU LGPLv3) Powerful interface I accessible and extensible by I uncomplicated and efficient everyone communication with a C++ I usable in other application non-proprietary projects I simple console application Features and Functioning Ulrich Loup 14/ 17 AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Some more features Open Source (GNU LGPLv3) Powerful interface I accessible and extensible by I uncomplicated and efficient everyone communication with a C++ I usable in other application non-proprietary projects I simple console application Standalone Reliable I only non-standard library I CppUnit test suite with GiNaC many test cases I independent of closed-source I uses C++ standard library software wherever possible Features and Functioning Ulrich Loup 14/ 17 Outlook AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Towards satisfiability modulo theories (SMT) ' Boolean abstraction unsatisfiable satisfiable SAT-solver UNSAT Constraint set Reason unsatisfiable Theory solver SAT satisfiable Outlook Ulrich Loup 16/ 17 AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations Towards satisfiability modulo theories (SMT) ' Boolean abstraction unsatisfiable satisfiable SAT-solver UNSAT Constraint set Reason unsatisfiable Theory solver SAT satisfiable Prototype: Virtual Substitution Solver Planed: I-RiSC{I ncremental Computation of the Realization of Sign Conditions Outlook Ulrich Loup 16/ 17 AlgoSyn GiNaCRA:A C++ Library for Real Algebraic Computations To conclude GiNaCRA { GiNaC Real Algebra package http://ginacra.sourceforge.net/ Support: Ulrich Loup RWTH Aachen University Computer Science 2 Theory of Hybrid Systems http://www-i2.informatik.rwth-aachen.de/i2/hybrid/ Outlook Ulrich Loup 17/ 17.