Symmetry in SAT:
an overview

August 27th, 2018
Theory and Practice of SAT solving

Oaxaca, Mexico


 

Jo Devriendt, KU Leuven

Intro

Everyone knows symmetry:

"something does not change under a set of transformations"
- Wikipedia

Symmetry :=

Permutation of variable assignments that preserves satisfaction

In combinatorial solving

Symmetry induces symmetry classes:

Symmetry classes tend to get huge -> search problem

"......calculating......"

In combinatorial solving

In combinatorial solving

Goal: investigate only asymmetrical cases

                    !!

Contents

  1. Intro
  2. SAT Prelims
  3. "Classic" symmetry breaking
  4. The pigeonhole problem
  5. "Recent" symmetry breaking
  6. Non-breaking approaches
  7. Bonus: symmetry, local search & maxSAT

"Interesting research question"

In SAT:

Syntactic symmetry :=
literal permutation that preserves the CNF

a \vee \neg b \\ b \vee \neg c \\ c \vee \neg a
a¬bb¬cc¬aa \vee \neg b \\ b \vee \neg c \\ c \vee \neg a
a \mapsto b \\ \neg a \mapsto \neg b \\ b \mapsto c \\ \neg b \mapsto \neg c \\ c \mapsto a \\ \neg c \mapsto \neg a \\
ab¬a¬bbc¬b¬cca¬c¬aa \mapsto b \\ \neg a \mapsto \neg b \\ b \mapsto c \\ \neg b \mapsto \neg c \\ c \mapsto a \\ \neg c \mapsto \neg a \\
(a\;b\;c)
(a  b  c)(a\;b\;c)

static ↔ dynamic

other ↔ breaking

Shatter
BreakID

CDCLSym

SymChaff
Symmetric learning

?

In SAT literature:

Adaptive prefix-assignment

Terminology

  • variable x
    • set of all variables X
  • literal l
  • clause c
  • (propositional) formula φ
  • (variable) assignment α
    • α(l) is the truth value of l in α
  • symmetry σ
    • σ(...) is the symmetrical image of ...
  • symmetry group Σ
    • Σ(...) is the orbit of ... under Σ
    • generator set gen(Σ)

3. "Classic" symmetry breaking

Symmetry breaking formulas:
Crawford [1]

Given: φ, Σ

Find: symmetry breaking formula sbf
that invalidates symmetrical assignments

\varphi
φ\varphi
\varphi \cup sbf
φsbf\varphi \cup sbf

Symmetry breaking formulas:
Crawford [1]

Core idea: sbf encodes
"α is lexicographically smaller than σ(α)"
for σ ϵ Σ

x_1 \leq \sigma(x_1) \\ x_1 = \sigma(x_1) \Rightarrow x_2 \leq \sigma(x_2) \\ (x_1 = \sigma(x_1) \wedge x_2 = \sigma(x_2)) \Rightarrow x_3 \leq \sigma(x_3) \\ \ldots
x1σ(x1)x1=σ(x1)x2σ(x2)(x1=σ(x1)x2=σ(x2))x3σ(x3)x_1 \leq \sigma(x_1) \\ x_1 = \sigma(x_1) \Rightarrow x_2 \leq \sigma(x_2) \\ (x_1 = \sigma(x_1) \wedge x_2 = \sigma(x_2)) \Rightarrow x_3 \leq \sigma(x_3) \\ \ldots

parameter: total order on X

Symmetry breaking formulas:
Crawford [1]

Sound

Complete

Huge: O(|X|²|Σ|)

Core idea: sbf encodes
"α is lexicographically smaller than σ(α)"
for all σ ϵ Σ

\varphi \cup sbf
φsbf\varphi \cup sbf

Symmetry breaking:
Shatter [2]

  • construct sbf for -much smaller- gen(Σ)
  • "chain encoding"
  • improved clausal encoding

Sound

Incomplete

Feasible: O(|X||gen(Σ)|)

\varphi \cup sbf
φsbf\varphi \cup sbf

Detecting symmetry:
Saucy [3]

Sparse graph automorphism detection

  • Graph construction from CNF:
    • node for each literal and clause
    • edge between literals occurring in clause
    • edge between literal and negation
  • No polynomial algorithm known
  • Output: generators to automorphism group

Static symmetry breaking:
Shatter+Saucy

Propositional description

off-the-shelf SAT solver

SAT/UNSAT

Graph automorphism detection

\neg a \vee b
¬ab\neg a \vee b

Add symmetry breaking formulas

a \vee \neg b \\ b \vee \neg c \\ c \vee \neg a
a¬bb¬cc¬aa \vee \neg b \\ b \vee \neg c \\ c \vee \neg a
(a \; b \; c)
(a  b  c)(a \; b \; c)

4. The pigeonhole problem

Pigeonhole!

Do n pigeons fit in n-1 holes?

  • Proof-theoretic problem
  • Simple but large symmetry group
    • composition of "pigeon interchangeability"
      and "hole interchangeability"
    • 1 symmetry class
\forall p\colon \bigvee_{h} x_{ph} \\ \forall h\colon \forall p\neq p'\colon \neg x_{ph} \vee \neg x_{p'h}
p​:hxphh​:pp​:¬xph¬xph\forall p\colon \bigvee_{h} x_{ph} \\ \forall h\colon \forall p\neq p'\colon \neg x_{ph} \vee \neg x_{p'h}
x_{11}
x11x_{11}
x_{21}
x21x_{21}
x_{31}
x31x_{31}
x_{41}
x41x_{41}
x_{12}
x12x_{12}
x_{22}
x22x_{22}
x_{32}
x32x_{32}
x_{42}
x42x_{42}
x_{13}
x13x_{13}
x_{23}
x23x_{23}
x_{33}
x33x_{33}
x_{43}
x43x_{43}

Pigeonhole!

Seems ok?

Original Shatter experiment:

Pigeonhole!

Modest gains...
Better results in original paper?

Own Shatter experiment:

Pigeonhole!

  • Propositional encoding reduces "pigeon interchangeability"
    to "row interchangeability"
  • Shatter's sbf's are complete [4] with correct choice of
    • gen(Σ)
    • variable order




       
  • |full sbf| = O(n²)
x_{11}
x11x_{11}
x_{21}
x21x_{21}
x_{31}
x31x_{31}
x_{41}
x41x_{41}
x_{12}
x12x_{12}
x_{22}
x22x_{22}
x_{32}
x32x_{32}
x_{42}
x42x_{42}
x_{13}
x13x_{13}
x_{23}
x23x_{23}
x_{33}
x33x_{33}
x_{43}
x43x_{43}

Pigeonhole!

x_{11}
x11x_{11}
x_{21}
x21x_{21}
x_{31}
x31x_{31}
x_{41}
x41x_{41}
x_{12}
x12x_{12}
x_{22}
x22x_{22}
x_{32}
x32x_{32}
x_{42}
x42x_{42}
x_{13}
x13x_{13}
x_{23}
x23x_{23}
x_{33}
x33x_{33}
x_{43}
x43x_{43}
  • Propositional encoding reduces "pigeon interchangeability"
    to "row interchangeability"
  • Shatter's sbf's are complete [4] with correct choice of
    • gen(Σ)
    • variable order




       
  • |full sbf| = O(n²)

5. "Recent" symmetry breaking

Symmetry breaking:
BreakID [5]

Core idea: detect "row swap" symmetries

  1. Search σ1 , σ2 ϵgen(Σ) that form 2 subsequent row swaps
    • forms initial 3-rowed variable matrix M
  2. Apply every σ ϵ gen(Σ) to all detected rows r ϵ M so far
    • images σ(r) disjoint of M are candidates to extend M
    • test if swap r ↔ σ(r) is a symmetry by syntactical check on φ
    • if success, extend M with σ(r)
  3. Use Saucy to extend gen(Σ) with new symmetry generators by fixing all variable nodes with variable in M, first row excepted

*Approximative algorithm

Symmetry breaking:
BreakID [5]

Core idea: detect "row swap" symmetries

Detect row interchangeability subgroups?

Caveat!

\neg x_1 \vee \sigma(x_1)
¬x1σ(x1)\neg x_1 \vee \sigma(x_1)
  • First clause in sbf for σ is binary:

     
  • x is stabilized by Σ iff Σ(x)={x}
  • Given Σ with smallest non-stabilized x, for each x' ϵ Σ(x):
     

    is clause of sbf under some σ ϵ Σ
\neg x \vee x'
¬xx\neg x \vee x'

Symmetry breaking:
BreakID [5]

Core idea: maximize number of binary sbf clauses

  • Create stabilizer chain of Σ:

     
  • Σi is the stabilizer subgroup of Σi-1 stabilizing the next non-stabilized variable in ordering
    • Σi have different smallest non-stabilized variables xi
  • For each x' ϵ Σi(xi)


    is a clause of some sbf
\Sigma \supset \Sigma_1 \supset \Sigma_2 \supset \ldots \supset 1
ΣΣ1Σ21\Sigma \supset \Sigma_1 \supset \Sigma_2 \supset \ldots \supset 1
\neg x_i \vee x'
¬xix\neg x_i \vee x'

Symmetry breaking:
BreakID [5]

Core idea: exploit binary sbf clauses

\sigma(x)=\neg x
σ(x)=¬x\sigma(x)=\neg x

Symmetry breaking:
BreakID [5]

Core idea: exploit binary sbf clauses

  • Approximative implementation
    • which adapts the variable order!
       
  • Works well for polarity symmetry σ where for all x:

    as sbf is equivalent to unit clause

    and their number is maximized through adopted variable order.
\neg x_1
¬x1\neg x_1

Complete algorithm?

Symmetry breaking:
CDCLSym [6]

Core idea: generate sbf dynamically

  • Keep track of reducer symmetries where σ(α)<α
    • by watching smallest variable s.t. σ(v)≠v
  • Generate clause from sbf forcing α≤σ(α)

    Additionally: try Bliss instead of Saucy

Use clauses for propagation?

Not only generator symmetries?

Symmetry breaking:
On completeness

  • Pigeon interchangeability can be completely broken with polynomial sbf
  • How about edge interchangeability?
    • E.g., find coloring of complete graph (Ramsey numbers)
    • Recent intrest [11] [14]
  • How about general interchangeability over arbitrary high dimensional relations?

Tractable sbf for edge interchangeability?

Symmetry breaking:
Prefix breaking [7]

Core idea: enumerate asymmetrical assignments to variable prefix

High-level constraints

Incremental / parallel SAT solver

Graph  representation

\forall x\colon \exists y\colon \\ \varphi(x,y)
x&NegativeThinSpace;:y&NegativeThinSpace;:φ(x,y)\forall x\colon \exists y\colon \\ \varphi(x,y)

Symmetry class representative enumeration

Variable prefix

Clausal encoding

6. Non-breaking approaches

Symmetry handling:
SymChaff [8]

Core idea: search decisions consider row interchangeability

x_{11}
x11x_{11}
x_{21}
x21x_{21}
x_{31}
x31x_{31}
x_{41}
x41x_{41}
x_{12}
x12x_{12}
x_{22}
x22x_{22}
x_{32}
x32x_{32}
x_{42}
x42x_{42}
x_{13}
x13x_{13}
x_{23}
x23x_{23}
x_{33}
x33x_{33}
x_{43}
x43x_{43}
  • Only for row interchangeability symmetry
  • Keep track of row-interchangeable variables
    • interchangeability reduces under previous choices
  • Use cardinality decision points over row-interchangeable variables

Cardinality decision of 1 over first column:

Symmetry handling:
SymChaff [8]

  • Only for row interchangeability symmetry
  • Keep track of row-interchangeable variables
    • interchangeability reduces under previous choices
  • Use cardinality decision points over row-interchangeable variables

Core idea: search decisions consider row interchangeability

0
00
0
00
0
00
1
11
x_{12}
x12x_{12}
x_{22}
x22x_{22}
x_{32}
x32x_{32}
x_{42}
x42x_{42}
x_{13}
x13x_{13}
x_{23}
x23x_{23}
x_{33}
x33x_{33}
x_{43}
x43x_{43}

Cardinality decision of 1 over first column:

Symmetry handling:
SymChaff [8]

Strong performance on pigeonhole

Symmetry handling:
Symmetric learning [9]

  • Learnt clauses are logical consequences of φ
  • Whenever c is a consequence of φ, so is σ(c)
  • Problem: Σ(c) is huge
    • Learn only σ(c) for σ ϵ gen(Σ)

Core idea: consider symmetrical learned clauses

Symmetry handling:
Symmetric explanation learning [10]

  • Learn σ(c) that propagate at least once
    • symmetries typically permute only a fraction of the literals
    • if c is unit, σ(c) has a good chance of being unit as well
    • explanation clauses are unit ;-)

Core idea: consider useful symmetrical explanation clauses

  • For each σ ϵ gen(Σ), whenever c propagates, store σ(c) in a separate clause store θ
    • Propagation on θ happens only if standard unit propagation is at a fixpoint
    • Whenever a σ(c) ϵ θ propagates, upgrade it to a "normal" learned clause
    • After backjump over c's propagation level, clear σ(c) from θ
  • Every learned σ(c) is useful & original
  • Transitive effect: track σ'(σ(c)) when σ(c) propagates

Symmetry handling:
Symmetric explanation learning [10]

Core idea: consider useful symmetrical explanation clauses

Symmetry handling:
Symmetric explanation learning [10]

What is "complete" symmetrical learning?

Can it be done efficiently?

Caveat: performance on larger instances

Research trends:

  • Symmetry detection on propositional level is hard
    • not a solved problem, cfr. pigeonhole
    • papers often assume high-level symmetry input [7] [8]
  • Sbf construction based on canonical labeling [7] [11]
  • Dynamical approaches often perform lazy clause generation [6] [10] [12]
  • Use computational group algebra to detect symmetry group structure [5] [13]

The influence of the variable order on an sbf?

Proof checking and symmetrical learning?

Thanks for listening!
Questions?

[1] Symmetry-Breaking Predicates for Search Problems (1996) Crawford et al.

[2] Efficient Symmetry-Breaking for Boolean Satisfiability (2003) Aloul et al.

[3] Symmetry and Satisfiability: An Update (2010) Katebi et al.

[4] Breaking row and column symmetries in matrix models (2002) Flener et al.

[5] Improved Static Symmetry Breaking for SAT (2016) Devriendt et al.

[6] CDCLSym: Introducing Effective Symmetry Breaking in SAT Solving (2018) Metin et al.

[7] An Adaptive Prefix-Assignment Technique for Symmetry Reduction (2017) Juntilla et al.

[8] Symchaff: exploiting symmetry in a structure-aware satisfiability solver (2009) Sabharwal

[9] Enhancing clause learning by symmetry in SAT solvers (2010) Benhamou

[10] Symmetric explanation learning: Effective dynamic symmetry handling for SAT (2017) Devriendt et al.

[11] Breaking Symmetries in Graphs: The Nauty Way (2016) Codish et al.

[12] Symmetries, almost symmetries, and lazy clause generation (2014) Chu et al.

[13] Breaking symmetries in all-different problems (2005) Puget

[14] The quest for perfect and compact symmetry breaking for graph problems (2016) Heule

7. Bonus: symmetry, local search & maxSAT

Bonus: symmetry, local search & maxSAT

  • (Satisfying) assignments now have an associated score
  • Local search "moves" from one to the other based on structure-preserving transformations

Bonus: symmetry, local search & maxSAT

  • (Satisfying) assignments now have an associated score
  • Local search "moves" from one to the other based on structure-preserving transformations
  • Designing local moves is typically done by hand...

Bonus: symmetry, local search & maxSAT

  • (Satisfying) assignments now have an associated score
  • Local search "moves" from one to the other based on structure-preserving transformations
  • Designing local moves is typically done by hand...

Symmetries form moves!
Can be automatically detected!

Bonus: symmetry, local search & maxSAT

Symmetry-based local search in weighted maxSAT?

Copy of Symmetry in SAT: an overview

By krr

Copy of Symmetry in SAT: an overview

talk for Theory and Practice of Satisfiability Solving

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