(Mostly Real) Quantifier Elimination Thomas Sturm AVACS Autumn - - PowerPoint PPT Presentation
(Mostly Real) Quantifier Elimination Thomas Sturm AVACS Autumn School, Oldenburg, Germany, October 1, 2015 http://www.mpi-inf.mpg.de/~sturm/ Overview Introduction Definitions Virtual Substitution Variants of Quantifier Elimination Software
Introduction · 2/42
Introduction · 3/42
Introduction · 3/42
◮ all variables in ϕ are quantified decision problem ◮ only existential quantifiers satisfiability problem
Introduction · 3/42
Automated Reasoning heuristics learning model-based construction . . . Algebraic Model Theory definable sets elementary extensions substructure completeness . . .
Introduction · 4/42
Definitions · 5/42
Definitions · 6/42
Definitions · 6/42
Definitions · 6/42
1 + (2x2 + 2x3)x1 + (x2 2 + 2x3x2 + x2 3 )
Definitions · 7/42
1 + (2x2 + 2x3)x1 + (x2 2 + 2x3x2 + x2 3 )
◮ R ∈ L= = {≤, <, =, ≥, >, =} as discussed ◮ f a recursive polynomial in some variables x1, . . . , xn as above ◮ L= is closed under negation: For R ∈ L= there is ¯
Definitions · 7/42
◮ −
◮ Eliminate ¬ using de Morgan’s law and closure property of L w.r.t. negation, e.g.:
Definitions · 8/42
◮ −
◮ Eliminate ¬ using de Morgan’s law and closure property of L w.r.t. negation, e.g.:
Definitions · 8/42
Definitions · 9/42
Virtual Substitution · 10/42
◮ ψ is quantifier-free ◮ the variables of ψ are a subset of quantified (bound) variables X = {x1, . . . , xn}
Virtual Substitution · 11/42
◮ The variables in ψ are among x1 and V1 := (X \ {x1}) ∪ U. ◮ All variables from V1 will play the same role now, say, V1 = {v1, . . . , vk}.
◮ Intuitively, ∃x is like a big disjunction over all real numbers. ◮ Could there be a finite E set of terms t such that
Virtual Substitution · 12/42
◮ Sets of satisfying values for x1 in f(x1) R 0 are
Virtual Substitution · 13/42
◮ Sets of satisfying values for x1 in f(x1) R 0 are
Virtual Substitution · 13/42
◮ Sets of satisfying values for x1 in f(x1) R 0 are
◮ if bi ∈ R, then f(bi) = 0 ◮ Set of satisfying values for x1 in ψ has the same form.
Virtual Substitution · 13/42
◮ Sets of satisfying values for x1 in f(x1) R 0 are
◮ if bi ∈ R, then f(bi) = 0 ◮ Set of satisfying values for x1 in ψ has the same form.
Virtual Substitution · 13/42
◮ Sets of satisfying values for x1 in f(x1) R 0 are
◮ if bi ∈ R, then f(bi) = 0 ◮ Set of satisfying values for x1 in ψ has the same form.
◮ Idea: E = all b2 or b2 − ε and ∞.
Virtual Substitution · 13/42
1 + b(v1, . . . , vk)x1 + c(v1, . . . , vk) yields candidate solutions
t
Virtual Substitution · 14/42
1 + b(v1, . . . , vk)x1 + c(v1, . . . , vk) yields candidate solutions
t
Virtual Substitution · 14/42
Virtual Substitution · 15/42
◮ Method generalizes to arbitrary degrees (in principle long known). ◮ first implementation will be available this year (PhD thesis by M. Košta). ◮ For higher degrees, t will be way more abstract.
Virtual Substitution · 15/42
◮ Method generalizes to arbitrary degrees (in principle long known). ◮ first implementation will be available this year (PhD thesis by M. Košta). ◮ For higher degrees, t will be way more abstract.
Virtual Substitution · 15/42
i ∈ Z[y]
g2
g1 g2 + f0 ≤ 0 ≡ f1g1g2 + f0g2 2 ≤ 0
Virtual Substitution · 16/42
i ∈ Z[y]
g2
g1 g2 + f0 ≤ 0 ≡ f1g1g2 + f0g2 2 ≤ 0
g1+g2 √g3 g4
1 + g∗ 2
4
Virtual Substitution · 16/42
i ∈ Z[y]
g2
g1 g2 + f0 ≤ 0 ≡ f1g1g2 + f0g2 2 ≤ 0
g1+g2 √g3 g4
1 + g∗ 2
4
g∗
1+g∗ 2
√g3 g∗
4
1 2 − g∗ 2 2g3 = 0 ∧ g∗ 1g∗ 2 ≤ 0
Virtual Substitution · 16/42
i ∈ Z[y]
g2
g1 g2 + f0 ≤ 0 ≡ f1g1g2 + f0g2 2 ≤ 0
g1+g2 √g3 g4
1 + g∗ 2
4
g∗
1+g∗ 2
√g3 g∗
4
1 2 − g∗ 2 2g3 = 0 ∧ g∗ 1g∗ 2 ≤ 0 g∗
1+g∗ 2
√g3 g∗
4
1 2 − g∗ 2 2g3 ≥ 0 ∧ g∗ 1g∗ 4 ≤ 0) ∨ (g∗ 1 2 − g∗ 2 2g3 ≤ 0 ∧ g∗ 2g∗ 4 ≤ 0)
Virtual Substitution · 16/42
i ∈ Z[y]
g2
g1 g2 + f0 ≤ 0 ≡ f1g1g2 + f0g2 2 ≤ 0
g1+g2 √g3 g4
1 + g∗ 2
4
g∗
1+g∗ 2
√g3 g∗
4
1 2 − g∗ 2 2g3 = 0 ∧ g∗ 1g∗ 2 ≤ 0 g∗
1+g∗ 2
√g3 g∗
4
1 2 − g∗ 2 2g3 ≥ 0 ∧ g∗ 1g∗ 4 ≤ 0) ∨ (g∗ 1 2 − g∗ 2 2g3 ≤ 0 ∧ g∗ 2g∗ 4 ≤ 0)
Virtual Substitution · 16/42
i ∈ Z[y]
g2
g1 g2 + f0 ≤ 0 ≡ f1g1g2 + f0g2 2 ≤ 0
g1+g2 √g3 g4
1 + g∗ 2
4
g∗
1+g∗ 2
√g3 g∗
4
1 2 − g∗ 2 2g3 = 0 ∧ g∗ 1g∗ 2 ≤ 0 g∗
1+g∗ 2
√g3 g∗
4
1 2 − g∗ 2 2g3 ≥ 0 ∧ g∗ 1g∗ 4 ≤ 0) ∨ (g∗ 1 2 − g∗ 2 2g3 ≤ 0 ∧ g∗ 2g∗ 4 ≤ 0)
Virtual Substitution · 16/42
Virtual Substitution · 17/42
◮ more local substitution of test points With the elimination of the next quantifiers ◮ even improves upper bound on asymptotic worst-case complexity
Virtual Substitution · 17/42
Virtual Substitution · 18/42
Variants of Quantifier Elimination · 19/42
Variants of Quantifier Elimination · 20/42
Variants of Quantifier Elimination · 21/42
Software · 22/42
◮ interactive system, QE and decision for many domains, normal forms,
◮ interfaces to Qepcad B, Gurobi, Mathematica, Z3, . . . ◮ more than 300 citations of applications in the literature:
◮ Redlog development since 1992 as part of the CAS Reduce [Hearn, 1968] ◮ Reduce/Redlog open-source (free-BSD) on Sourceforge since 12/2008
◮ 48,318 downloads since 12/2008 (7,496 in 2014), 500+ SVN commits per year
Software · 23/42
◮ Presburger Arithmetic ◮ weak quantifier elimination for the full linear theory ◮ weak quantifier elimination also for higher degrees (special cases)
◮ experimental
◮ language of rings only
◮ language of rings with unary differential operator ◮ computation in differentially closed field (A. Robinson, Blum)
Software · 24/42
◮ linear formulas over p-adic fields for p prime ◮ optionally uniform in p ◮ used e.g. for solving parametric systems of congruences over the integers
◮ Malcev-type term algebras (with functions instead of relations)
◮ two-sided queues over the other theories (2-sorted) ◮ Implemented at present for queues of reals
◮ generalization of SAT solving ◮ quantified propositional calculus, i.e., parametric QSAT (aka QBF) solving
Software · 25/42
◮ Qepcad B (Hong and Brown)
◮ The computer algebra system Mathematica
◮ The computer algebra system Maple
◮ The computer algebra system Risa/Asir (originally by Fujitsu)
◮ Some prototypes in Japan
◮ Specialized implementations of CAD in SMT solvers (z3) ◮ Specialized implementations of virtual substitutions for SMT (SMT-RAT)
Software · 26/42
Applications in Geometry and Verification · 27/42
1 = u2 1 + (u2 − x1)2
2
2 < 22
6 = (u1 − x5)2 + (u2 − x6)2 ∧ u1x6 − u2x5 − u2 + x6 = 0
2 < (x5 − 1)2 + x2 6 B=(1,0) M X Y W A=(−1,0) C=(u1,u2) V
Applications in Geometry and Verification · 28/42
1 = u2 1 + (u2 − x1)2
2
2 < 22
6 = (u1 − x5)2 + (u2 − x6)2 ∧ u1x6 − u2x5 − u2 + x6 = 0
2 < (x5 − 1)2 + x2 6 B=(1,0) M X Y W A=(−1,0) C=(u1,u2) V ◮ ϕ ≡ ∀x6∀x5∀x4∀x3∀x2∀x1∀r
1 = u2 1 + (u2 − x1)2
2
2 < 22
6 = (u1 − x5)2 + (u2 − x6)2 ∧ u1x6 − u2x5 − u2 + x6 = 0
2 < (x5 − 1)2 + x2 6 B=(1,0) M X Y W A=(−1,0) C=(u1,u2) V ◮ ϕ ≡ ∀x6∀x5∀x4∀x3∀x2∀x1∀r
1 − 2u1 + u2 2 − 3 = 0
2=4
Applications in Geometry and Verification · 28/42
1 = u2 1 + (u2 − x1)2
2
2 < 22
6 = (u1 − x5)2 + (u2 − x6)2 ∧ u1x6 − u2x5 − u2 + x6 = 0
2 < (x5 − 1)2 + x2 6 B=(1,0) M X Y W A=(−1,0) C=(u1,u2) V ◮ ϕ ≡ ∀x6∀x5∀x4∀x3∀x2∀x1∀r
1 − 2u1 + u2 2 − 3 = 0
2=4
◮ CAD (0.9 s): ∀u1∀u2( Θ −
Applications in Geometry and Verification · 28/42
Applications in Geometry and Verification · 29/42
Applications in Geometry and Verification · 29/42
Applications in Geometry and Verification · 30/42
◮ 584 disjuncts, 33365 atomic formulas, depth 13, some still containing ∃c5 ◮ first 33 disjuncts automatically simplify to c2 2 − 30c2 − 75 ≤ 0 for c1 > 0, c2 > 0. ◮ ⇒ no collision for c2 = v ≤ 32
Applications in Geometry and Verification · 30/42
CAD for Satistfiability Checking · 31/42
CAD for Satistfiability Checking · 32/42
CAD for Satistfiability Checking · 32/42
CAD for Satistfiability Checking · 32/42
CAD for Satistfiability Checking · 32/42
◮ projection operator
CAD for Satistfiability Checking · 33/42
◮ projection operator
◮ Projections of critical points
CAD for Satistfiability Checking · 33/42
◮ projection operator
◮ Projections of critical points
◮ The zeros of the gi are real
CAD for Satistfiability Checking · 33/42
◮ projection operator
◮ Projections of critical points
◮ The zeros of the gi are real
◮ Their computation is
CAD for Satistfiability Checking · 33/42
◮ Add points
CAD for Satistfiability Checking · 34/42
◮ Add points
◮ This yields a decomposition
CAD for Satistfiability Checking · 34/42
◮ Add points
◮ This yields a decomposition
◮ We want to lift this
CAD for Satistfiability Checking · 34/42
◮ Add points
◮ This yields a decomposition
◮ We want to lift this
◮ By the way: How many cells
CAD for Satistfiability Checking · 34/42
◮ compute univariate
CAD for Satistfiability Checking · 35/42
◮ compute univariate
◮ compute zeros and points
CAD for Satistfiability Checking · 35/42
◮ compute univariate
◮ compute zeros and points
◮ this yields test points
CAD for Satistfiability Checking · 35/42
3
3
3
3
◮ Finitely many test points
CAD for Satistfiability Checking · 36/42
◮ Finitely many test points
◮ R |
CAD for Satistfiability Checking · 36/42
◮ Finitely many test points
CAD as a Complete Decision Procedure · 37/42
◮ Finitely many test points
◮ ∀x∃yϕ(f1, f2):
CAD as a Complete Decision Procedure · 37/42
◮ Finitely many test points
◮ ∀x∃yϕ(f1, f2):
◮ ∃x∀yϕ(f1, f2):
CAD as a Complete Decision Procedure · 37/42
◮ Finitely many test points
◮ ∀x∃yϕ(f1, f2):
◮ ∃x∀yϕ(f1, f2):
◮ The innermost variable y was
CAD as a Complete Decision Procedure · 37/42
◮ Given ϕ(f1, f2) essentially all the algorithmic work we have done is valid for
CAD as a Complete Decision Procedure · 38/42
◮ Given ϕ(f1, f2) essentially all the algorithmic work we have done is valid for
◮ Furthermore, even for arbitrary quantification QxQ′y (in that order).
CAD as a Complete Decision Procedure · 38/42
◮ Given ϕ(f1, f2) essentially all the algorithmic work we have done is valid for
◮ Furthermore, even for arbitrary quantification QxQ′y (in that order). ◮ This indicates that the CAD procedure is somewhat an overkill.
CAD as a Complete Decision Procedure · 38/42
◮ Given ϕ(f1, f2) essentially all the algorithmic work we have done is valid for
◮ Furthermore, even for arbitrary quantification QxQ′y (in that order). ◮ This indicates that the CAD procedure is somewhat an overkill. ◮ On the other hand, the asymptotic worst complexity 22O(n)
CAD as a Complete Decision Procedure · 38/42
◮ Given ϕ(f1, f2) essentially all the algorithmic work we have done is valid for
◮ Furthermore, even for arbitrary quantification QxQ′y (in that order). ◮ This indicates that the CAD procedure is somewhat an overkill. ◮ On the other hand, the asymptotic worst complexity 22O(n)
◮ Asymptotically better bounds with refined complexity parameters.
CAD as a Complete Decision Procedure · 38/42
◮ Given ϕ(f1, f2) essentially all the algorithmic work we have done is valid for
◮ Furthermore, even for arbitrary quantification QxQ′y (in that order). ◮ This indicates that the CAD procedure is somewhat an overkill. ◮ On the other hand, the asymptotic worst complexity 22O(n)
◮ Asymptotically better bounds with refined complexity parameters. ◮ In practice, for general input, CAD is the best we have.
CAD as a Complete Decision Procedure · 38/42
◮ Given ϕ(f1, f2) essentially all the algorithmic work we have done is valid for
◮ Furthermore, even for arbitrary quantification QxQ′y (in that order). ◮ This indicates that the CAD procedure is somewhat an overkill. ◮ On the other hand, the asymptotic worst complexity 22O(n)
◮ Asymptotically better bounds with refined complexity parameters. ◮ In practice, for general input, CAD is the best we have. ◮ Until now, we have not used and did not “really know” the cells – only test points.
CAD as a Complete Decision Procedure · 38/42
CAD for Quantifier Elimination · 39/42
◮ Given ψ(x1, . . . , xk) = Qk+1xk+1 . . . Qrxrϕ(x1, . . . , xk, xk+1, . . . , xr).
CAD for Quantifier Elimination · 39/42
◮ Given ψ(x1, . . . , xk) = Qk+1xk+1 . . . Qrxrϕ(x1, . . . , xk, xk+1, . . . , xr). ◮ x1, . . . xk are parameters.
CAD for Quantifier Elimination · 39/42
◮ Given ψ(x1, . . . , xk) = Qk+1xk+1 . . . Qrxrϕ(x1, . . . , xk, xk+1, . . . , xr). ◮ x1, . . . xk are parameters. ◮ Construct CAD with projection order xr → · · · → xk+1 → xk → · · · → x1.
CAD for Quantifier Elimination · 39/42
◮ Given ψ(x1, . . . , xk) = Qk+1xk+1 . . . Qrxrϕ(x1, . . . , xk, xk+1, . . . , xr). ◮ x1, . . . xk are parameters. ◮ Construct CAD with projection order xr → · · · → xk+1 → xk → · · · → x1.
◮ Consider the finite set C ⊆ Pot(Rk) of cells in parameter space,
CAD for Quantifier Elimination · 39/42
◮ Given ψ(x1, . . . , xk) = Qk+1xk+1 . . . Qrxrϕ(x1, . . . , xk, xk+1, . . . , xr). ◮ x1, . . . xk are parameters. ◮ Construct CAD with projection order xr → · · · → xk+1 → xk → · · · → x1.
◮ Consider the finite set C ⊆ Pot(Rk) of cells in parameter space,
◮ For each c ∈ C with test point tc ∈ Rn−k we can decide ψ(tc) and collect
CAD for Quantifier Elimination · 39/42
◮ Given ψ(x1, . . . , xk) = Qk+1xk+1 . . . Qrxrϕ(x1, . . . , xk, xk+1, . . . , xr). ◮ x1, . . . xk are parameters. ◮ Construct CAD with projection order xr → · · · → xk+1 → xk → · · · → x1.
◮ Consider the finite set C ⊆ Pot(Rk) of cells in parameter space,
◮ For each c ∈ C with test point tc ∈ Rn−k we can decide ψ(tc) and collect
◮ Assume that for c ∈ C we have a quantifier-free description formula
CAD for Quantifier Elimination · 39/42
◮ Given ψ(x1, . . . , xk) = Qk+1xk+1 . . . Qrxrϕ(x1, . . . , xk, xk+1, . . . , xr). ◮ x1, . . . xk are parameters. ◮ Construct CAD with projection order xr → · · · → xk+1 → xk → · · · → x1.
◮ Consider the finite set C ⊆ Pot(Rk) of cells in parameter space,
◮ For each c ∈ C with test point tc ∈ Rn−k we can decide ψ(tc) and collect
◮ Assume that for c ∈ C we have a quantifier-free description formula
c∈TRUECELLS ∆c.
CAD for Quantifier Elimination · 39/42
24
24
24
25
◮ The approach of the original Collins article (1975). ◮ Idea: Produce sufficiently many polynomials during projection. ◮ Technically one adds “lots of derivatives.”
◮ Consider a single polynomial f = x3 − 12x2 + 44x − 48.
CAD for Quantifier Elimination · 40/42
◮ The approach of the original Collins article (1975). ◮ Idea: Produce sufficiently many polynomials during projection. ◮ Technically one adds “lots of derivatives.”
◮ Consider a single polynomial f = x3 − 12x2 + 44x − 48. ◮ f > 0 describes ]2, 4[ ∪ ]6, ∞[, f = 0 describes {2, 4, 6} .
CAD for Quantifier Elimination · 40/42
◮ The approach of the original Collins article (1975). ◮ Idea: Produce sufficiently many polynomials during projection. ◮ Technically one adds “lots of derivatives.”
◮ Consider a single polynomial f = x3 − 12x2 + 44x − 48. ◮ f > 0 describes ]2, 4[ ∪ ]6, ∞[, f = 0 describes {2, 4, 6} . ◮ f cannot describe exclusively ]2, 4[ or {4}.
CAD for Quantifier Elimination · 40/42
◮ The approach of the original Collins article (1975). ◮ Idea: Produce sufficiently many polynomials during projection. ◮ Technically one adds “lots of derivatives.”
◮ Consider a single polynomial f = x3 − 12x2 + 44x − 48. ◮ f > 0 describes ]2, 4[ ∪ ]6, ∞[, f = 0 describes {2, 4, 6} . ◮ f cannot describe exclusively ]2, 4[ or {4}. ◮ f = 0 ∧ f ′ = 3x2 − 24x + 44 < 0 describes {4}.
CAD for Quantifier Elimination · 40/42
◮ The approach of the original Collins article (1975). ◮ Idea: Produce sufficiently many polynomials during projection. ◮ Technically one adds “lots of derivatives.”
◮ Consider a single polynomial f = x3 − 12x2 + 44x − 48. ◮ f > 0 describes ]2, 4[ ∪ ]6, ∞[, f = 0 describes {2, 4, 6} . ◮ f cannot describe exclusively ]2, 4[ or {4}. ◮ f = 0 ∧ f ′ = 3x2 − 24x + 44 < 0 describes {4}. ◮ f > 0 ∧ f ′′ = 6x − 24 < 0 describes ]2, 4[.
CAD for Quantifier Elimination · 40/42
◮ The approach of the original Collins article (1975). ◮ Idea: Produce sufficiently many polynomials during projection. ◮ Technically one adds “lots of derivatives.”
◮ Consider a single polynomial f = x3 − 12x2 + 44x − 48. ◮ f > 0 describes ]2, 4[ ∪ ]6, ∞[, f = 0 describes {2, 4, 6} . ◮ f cannot describe exclusively ]2, 4[ or {4}. ◮ f = 0 ∧ f ′ = 3x2 − 24x + 44 < 0 describes {4}. ◮ f > 0 ∧ f ′′ = 6x − 24 < 0 describes ]2, 4[. ◮ Isn’t this somehow Rolle’s Theorem? Yes it is!
CAD for Quantifier Elimination · 40/42
◮ The approach of the original Collins article (1975). ◮ Idea: Produce sufficiently many polynomials during projection. ◮ Technically one adds “lots of derivatives.”
◮ Consider a single polynomial f = x3 − 12x2 + 44x − 48. ◮ f > 0 describes ]2, 4[ ∪ ]6, ∞[, f = 0 describes {2, 4, 6} . ◮ f cannot describe exclusively ]2, 4[ or {4}. ◮ f = 0 ∧ f ′ = 3x2 − 24x + 44 < 0 describes {4}. ◮ f > 0 ∧ f ′′ = 6x − 24 < 0 describes ]2, 4[. ◮ Isn’t this somehow Rolle’s Theorem? Yes it is!
CAD for Quantifier Elimination · 40/42
◮ PhD thesis of Brown (1999).
CAD for Quantifier Elimination · 41/42
◮ PhD thesis of Brown (1999). ◮ Use extended language with predicates like
CAD for Quantifier Elimination · 41/42
◮ PhD thesis of Brown (1999). ◮ Use extended language with predicates like
◮ Predicate is false if f has less than n roots.
CAD for Quantifier Elimination · 41/42
◮ PhD thesis of Brown (1999). ◮ Use extended language with predicates like
◮ Predicate is false if f has less than n roots.
◮ f = x3 − 12x2 + 44x − 48 revisited:
CAD for Quantifier Elimination · 41/42
◮ PhD thesis of Brown (1999). ◮ Use extended language with predicates like
◮ Predicate is false if f has less than n roots.
◮ f = x3 − 12x2 + 44x − 48 revisited:
◮ In several variables one could obtain, e.g.,
CAD for Quantifier Elimination · 41/42
◮ PhD thesis of Brown (1999). ◮ Use extended language with predicates like
◮ Predicate is false if f has less than n roots.
◮ f = x3 − 12x2 + 44x − 48 revisited:
◮ In several variables one could obtain, e.g.,
CAD for Quantifier Elimination · 41/42
◮ PhD thesis of Brown (1999). ◮ Use extended language with predicates like
◮ Predicate is false if f has less than n roots.
◮ f = x3 − 12x2 + 44x − 48 revisited:
◮ In several variables one could obtain, e.g.,
CAD for Quantifier Elimination · 41/42
◮ virtual substitution for real quantifier elimination and some variants
◮ software: Redlog and other ◮ other theories
◮ applications in geometry, verification, . . . ◮ cylindrical algebraic decomposition (CAD)
Summary · 42/42