Encoding Challenges for Hard Problems Marijn J.H. Heule Starting at - PowerPoint PPT Presentation

Encoding Challenges for Hard Problems Marijn J.H. Heule Starting at in August Matryoshka workshop June 12, 2019 1/33

Automated Reasoning Has Many Applications security planning and formal verification bioinformatics scheduling train safety exploit term rewriting automated theorem proving generation termination encode automated reasoner decode 2/33

Breakthrough in SAT Solving in the Last 20 Years Satisfiability (SAT) problem: Can a Boolean formula be satisfied? mid ’90s: formulas solvable with thousands of variables and clauses now: formulas solvable with millions of variables and clauses Edmund Clarke: “a key Donald Knuth: “evidently a killer app, technology of the 21st century” because it is key to the solution of so many other problems” [Knuth ’15] [Biere, Heule, vanMaaren, and Walsh ’09] 3/33

Representations Matrix Multiplication The Collatz Conjecture 4/33

Representations Matrix Multiplication The Collatz Conjecture 5/33

The Right Representation is Crucial What makes a problem hard? New angle: does the representation enable efficient reasoning? The famous pigeonhole principle: Can n + 1 pigeons be placed in n holes such that no hole contains multiple pigeons? ◮ Hard for many automated reasoning approaches ◮ Easy for a little kid given the right representation source: pecanpartnership.co.uk/2016/01/05/beware-pigeon-hole- overcoming-stereotypes-build-collaborative-culture 6/33

Artisan Representations (joint work) Architectural 3D Layout Van der Waerden numbers [VSMM ’07] [EJoC ’07] Henriette Bier Edge-matching Puzzles Software Model Synthesis [LaSh ’08] [ICGI ’10, ESE ’13] Sicco Verwer Graceful Graphs Conway’s Game of Life [AAAI ’10] [EJoC ’13] Toby Walsh Willem van der Poel Clique-Width Connect the Pairs [SAT ’13, TOCL ’15] Donald Knuth Stefan Szeider Firewall Verification Pythagorean Triples [SSS ’16] [SAT ’16, CACM ’17] Mohamed Gouda Victor Marek Open Knight Tours Collatz conjecture [Open] Moshe Vardi Scott Aaronson 7/33

Artisan Representations (joint work) Architectural 3D Layout Van der Waerden numbers [VSMM ’07] [EJoC ’07] Henriette Bier Software Model Synthesis Edge-matching Puzzles [ICGI ’10, ESE ’13] [LaSh ’08] Sicco Verwer Graceful Graphs Conway’s Game of Life [AAAI ’10] [EJoC ’13] Toby Walsh Willem van der Poel Clique-Width Connect the Pairs [SAT ’13, TOCL ’15] Donald Knuth Stefan Szeider Firewall Verification Pythagorean Triples [SSS ’16] [SAT ’16, CACM ’17] Mohamed Gouda Victor Marek Open Knight Tours Collatz conjecture [Open] Moshe Vardi Scott Aaronson 7/33

Inprocessing [J¨ arvisalo, Heule, and Biere ’12] How to fix a poor representation fully automatically? Reformulate Reformulate CDCL 8/33

Inprocessing [J¨ arvisalo, Heule, and Biere ’12] How to fix a poor representation fully automatically? Reformulate Reformulate CDCL Example: Bounded Variable Addition [Manthey, Heule, and Biere ’12] Replace ( a ∨ d ) ( a ∨ e ) ( b ∨ d ) ( b ∨ e ) ( c ∨ d ) ( c ∨ e ) by ( x ∨ a ) ( x ∨ b ) ( x ∨ c ) ( x ∨ d ) ( x ∨ e ) adds 1 variable removes 1 clause This technique is crucial for hard bioinformatics problems and turns the naive encoding of AtMostOne into the optimal one. 8/33

Inprocessing [J¨ arvisalo, Heule, and Biere ’12] How to fix a poor representation fully automatically? Reformulate Reformulate CDCL Example: Bounded Variable Addition [Manthey, Heule, and Biere ’12] Replace ( a ∨ d ) ( a ∨ e ) ( b ∨ d ) ( b ∨ e ) ( c ∨ d ) ( c ∨ e ) by ( x ∨ a ) ( x ∨ b ) ( x ∨ c ) ( x ∨ d ) ( x ∨ e ) adds 1 variable removes 1 clause This technique is crucial for hard bioinformatics problems and turns the naive encoding of AtMostOne into the optimal one. 8/33

Matrix Multiplication joint work with Manuel Kauers and Martina Seidl 9/33

Matrix Multiplication: Introduction � � � � � � a 1,1 a 1,2 b 1,1 b 1,2 c 1,1 c 1,2 = a 2,1 a 2,2 b 2,1 b 2,2 c 2,1 c 2,2 c 1,1 = a 1,1 · b 1,1 + a 1,2 · b 2,1 c 1,2 = a 1,1 · b 1,2 + a 1,2 · b 2,2 c 2,1 = a 2,1 · b 1,1 + a 2,2 · b 2,1 c 2,2 = a 2,1 · b 1,2 + a 2,2 · b 2,2 10/33

Matrix Multiplication: Introduction � � � � � � a 1,1 a 1,2 b 1,1 b 1,2 c 1,1 c 1,2 = a 2,1 a 2,2 b 2,1 b 2,2 c 2,1 c 2,2 c 1,1 = M 1 + M 4 − M 5 + M 7 c 1,2 = M 3 + M 5 c 2,1 = M 2 + M 4 c 2,2 = M 1 − M 2 + M 3 + M 6 10/33

Matrix Multiplication: Introduction � � � � � � a 1,1 a 1,2 b 1,1 b 1,2 c 1,1 c 1,2 = a 2,1 a 2,2 b 2,1 b 2,2 c 2,1 c 2,2 . . . where M 1 = ( a 1,1 + a 2,2 ) · ( b 1,1 + b 2,2 ) M 2 = ( a 2,1 + a 2,2 ) · b 1,1 M 3 = a 1,1 · ( b 1,2 − b 2,2 ) M 4 = a 2,2 · ( b 2,1 − b 1,1 ) M 5 = ( a 1,1 + a 1,2 ) · b 2,2 M 6 = ( a 2,1 − a 1,1 ) · ( b 1,1 + b 1,2 ) M 7 = ( a 1,2 − a 2,2 ) · ( b 2,1 + b 2,2 ) 10/33

Matrix Multiplication: Introduction � � � � � � a 1,1 a 1,2 b 1,1 b 1,2 c 1,1 c 1,2 = a 2,1 a 2,2 b 2,1 b 2,2 c 2,1 c 2,2 ◮ This scheme needs 7 multiplications instead of 8. 10/33

Matrix Multiplication: Introduction � � � � � � a 1,1 a 1,2 b 1,1 b 1,2 c 1,1 c 1,2 = a 2,1 a 2,2 b 2,1 b 2,2 c 2,1 c 2,2 ◮ This scheme needs 7 multiplications instead of 8. ◮ Recursive application allows to multiply n × n matrices with O ( n log 2 7 ) operations in the ground ring. 10/33

Matrix Multiplication: Introduction � � � � � � a 1,1 a 1,2 b 1,1 b 1,2 c 1,1 c 1,2 = a 2,1 a 2,2 b 2,1 b 2,2 c 2,1 c 2,2 ◮ This scheme needs 7 multiplications instead of 8. ◮ Recursive application allows to multiply n × n matrices with O ( n log 2 7 ) operations in the ground ring. ◮ Let ω be the smallest number so that n × n matrices can be multiplied using O ( n ω ) operations in the ground domain. 10/33

Matrix Multiplication: Introduction � � � � � � a 1,1 a 1,2 b 1,1 b 1,2 c 1,1 c 1,2 = a 2,1 a 2,2 b 2,1 b 2,2 c 2,1 c 2,2 ◮ This scheme needs 7 multiplications instead of 8. ◮ Recursive application allows to multiply n × n matrices with O ( n log 2 7 ) operations in the ground ring. ◮ Let ω be the smallest number so that n × n matrices can be multiplied using O ( n ω ) operations in the ground domain. ◮ Then 2 ≤ ω < 3. What is the exact value? 10/33

Efficient Matrix Multiplication: Theory ◮ Strassen 1969: ω ≤ log 2 7 ≤ 2.807 11/33

Efficient Matrix Multiplication: Theory ◮ Strassen 1969: ω ≤ log 2 7 ≤ 2.807 ◮ Pan 1978: ω ≤ 2.796 ◮ Bini et al. 1979: ω ≤ 2.7799 ◮ Sch¨ onhage 1981: ω ≤ 2.522 ◮ Romani 1982: ω ≤ 2.517 ◮ Coppersmith/Winograd 1981: ω ≤ 2.496 ◮ Strassen 1986: ω ≤ 2.479 ◮ Coppersmith/Winograd 1990: ω ≤ 2.376 11/33

Efficient Matrix Multiplication: Theory ◮ Strassen 1969: ω ≤ log 2 7 ≤ 2.807 ◮ Pan 1978: ω ≤ 2.796 ◮ Bini et al. 1979: ω ≤ 2.7799 ◮ Sch¨ onhage 1981: ω ≤ 2.522 ◮ Romani 1982: ω ≤ 2.517 ◮ Coppersmith/Winograd 1981: ω ≤ 2.496 ◮ Strassen 1986: ω ≤ 2.479 ◮ Coppersmith/Winograd 1990: ω ≤ 2.376 ◮ Stothers 2010: ω ≤ 2.374 ◮ Williams 2011: ω ≤ 2.3728642 ◮ Le Gall 2014: ω ≤ 2.3728639 11/33

Efficient Matrix Multiplication: Practice ◮ Only Strassen’s algorithm beats the classical algorithm for reasonable problem sizes. 12/33

Efficient Matrix Multiplication: Practice ◮ Only Strassen’s algorithm beats the classical algorithm for reasonable problem sizes. ◮ Want: a matrix multiplication algorithm that beats Strassen’s algorithm for matrices of moderate size. 12/33

Efficient Matrix Multiplication: Practice ◮ Only Strassen’s algorithm beats the classical algorithm for reasonable problem sizes. ◮ Want: a matrix multiplication algorithm that beats Strassen’s algorithm for matrices of moderate size. ◮ Idea: instead of dividing the matrices into 2 × 2-block matrices, divide them into 3 × 3-block matrices. 12/33

Efficient Matrix Multiplication: Practice ◮ Only Strassen’s algorithm beats the classical algorithm for reasonable problem sizes. ◮ Want: a matrix multiplication algorithm that beats Strassen’s algorithm for matrices of moderate size. ◮ Idea: instead of dividing the matrices into 2 × 2-block matrices, divide them into 3 × 3-block matrices. ◮ Question: What’s the minimal number of multiplications needed to multiply two 3 × 3 matrices? 12/33

Efficient Matrix Multiplication: Practice ◮ Only Strassen’s algorithm beats the classical algorithm for reasonable problem sizes. ◮ Want: a matrix multiplication algorithm that beats Strassen’s algorithm for matrices of moderate size. ◮ Idea: instead of dividing the matrices into 2 × 2-block matrices, divide them into 3 × 3-block matrices. ◮ Question: What’s the minimal number of multiplications needed to multiply two 3 × 3 matrices? ◮ Answer: Nobody knows. 12/33

The 3 x 3 Case is Still Open Question: What’s the minimal number of multiplications needed to multiply two 3 × 3 matrices? 13/33

The 3 x 3 Case is Still Open Question: What’s the minimal number of multiplications needed to multiply two 3 × 3 matrices? ◮ naive algorithm: 27 13/33

The 3 x 3 Case is Still Open Question: What’s the minimal number of multiplications needed to multiply two 3 × 3 matrices? ◮ naive algorithm: 27 ◮ padd with zeros, use Strassen twice, cleanup: 25 13/33