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Background Message Passing Modifying the PMPF Conclusions

An Ising Model inspired Extension of the Product-based MP Framework for SAT

Oliver Gableske1

1Institute of Theoretical Computer Science

Ulm University Germany

• liver@gableske.net

https://www.gableske.net

SAT 2014, 17.07.2014

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Background Message Passing Modifying the PMPF Conclusions

Background and Motivation (1)

General context Message Passing algorithms used in CNF-SAT solving Message Passing algorithms used to provide biases for variables in a given CNF formula β(v) ∈ [−1.0, +1.0] a variable bias estimates the marginal assignment in all solutions realize a variable ordering heuristic (via absolute value of bias) realize a value ordering heuristic (via sign of bias)

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Background Message Passing Modifying the PMPF Conclusions

Background and Motivation (2)

Message Passing algorithms are comprised of two parts

1

MP framework

governs the overall process to compute biases defines the message types defines how the message are updated (iterations) defines how long the updates are performed (convergence) defines how equilibrium messages are used to compute biases

2

MP heuristic

influences the overall process to compute biases specifies messages specifies equations to compute biases

apply an MP heuristic in an MP framework in order to compute biases

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Background Message Passing Modifying the PMPF Conclusions

Background and Motivation (3)

Last years paper explained, that all product-based MP algorithms (e.g. BP, SP, EM variants) currently available for SAT can be represented with the same MP framework (PMPF) in conjunction with the respective MP heuristics equations It furthermore showed, that all MP heuristics can be combined into a single heuristic (ρσPMP) which interpolates between the “original” equations (ρ = 0, σ = 0 → BP equations; ρ = 1, σ = 0 → SP equations, . . . ) The final result was, that all MP algorithms can be realized by applying the same MP heuristic (ρσPMP) in the same MP framework (PMPF)

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Background Message Passing Modifying the PMPF Conclusions

Background and Motivation (4)

Current situation we have a rather flexible MP heuristic (ρσPMP)

we have to implement only one heuristic to “get” all of the product-based MP algorithms allows parameter tuning (adapt ρ, σ for a given type of CNF formula) allows us to understand “theoretical connections” between the various MP heuristics

we have a totally static MP framework (PMPF)

the PMPF does influence the MP algorithm but we cannot adapt it

One possible goal to improve the situation: extend the PMPF in a “reasonable” (theoretically motivated) way increase its flexibility in practice increase its usefulness in theory

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Background Message Passing Modifying the PMPF Conclusions

Message Passing on a conceptual level (1)

Example F = (v1 ∨ v2 ∨ v3) ∧ (v1 ∨ ¯ v2 ∨ v3) ∧ ( ¯ v1 ∨ ¯ v2 ∨ ¯ v3) It is helpful to understand F as a factor graph.

v

1

v

2

v

3

c1 c2 c3

Undirected, bipartite graph Two types of nodes (variable nodes (circles), clause nodes (squares)) Two types of edges (positive edges (solid), negative edges (dashed)) Edges constitute literal occurrences

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Background Message Passing Modifying the PMPF Conclusions

Message Passing on a conceptual level (2)

Example F = (v1 ∨ v2 ∨ v3) ∧ (v1 ∨ ¯ v2 ∨ v3) ∧ ( ¯ v1 ∨ ¯ v2 ∨ ¯ v3)

v

1

v

2

v

3

c1 c2 c3

MP algorithm sends around messages along the edges clauses and variables “negotiate” about possible assignments assume variable v is contained in clause c as literal l MP framework defines two types of messages.

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Background Message Passing Modifying the PMPF Conclusions

Message Passing on a conceptual level (3)

Example F = (v1 ∨ v2 ∨ v3) ∧ (v1 ∨ ¯ v2 ∨ v3) ∧ ( ¯ v1 ∨ ¯ v2 ∨ ¯ v3)

v

1

c1 c2 c3 d d d

1 2 3

• 1. Disrespect Messages (from variable nodes towards clause nodes):

δ(l, c) ∈ [0.0, 1.0] The chance that l will not satisfy c Intuitive meaning of δ(l, c) ≈ 1.0: Variable v tells clause c that it cannot satisfy it via its occurrence l.

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Background Message Passing Modifying the PMPF Conclusions

Message Passing on a conceptual level (4)

Example F = (v1 ∨ v2 ∨ v3) ∧ (v1 ∨ ¯ v2 ∨ v3) ∧ ( ¯ v1 ∨ ¯ v2 ∨ ¯ v3)

v

1

c1 c2 c3 w w w

1 2 3

• 2. Warning Messages (from clause nodes towards variable nodes):

ω(c, v) ∈ [0.0, 1.0] The chance that no other literal in c can satisfy c Intuitive meaning of ω(c, v) ≈ 1.0: Clause c is telling variable v that it needs it to be satisfied.

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Background Message Passing Modifying the PMPF Conclusions

Message Passing on a conceptual level (5)

Example F = (v1 ∨ v2 ∨ v3) ∧ (v1 ∨ ¯ v2 ∨ v3) ∧ ( ¯ v1 ∨ ¯ v2 ∨ ¯ v3) The MP framework defines the warning computation as ω(c, v) =

• l∈c\{v,¯

v}

δ(l, c)

v

1

v

2

v

3

c1 d1 d3 w d1

2

. = d3

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Background Message Passing Modifying the PMPF Conclusions

Message Passing on a conceptual level (6)

The MP framework defines the cavity freedom value computations.

• 1. Cavity freedom of l to satisfy c.

[0.0, 1.0] ∋ S(l, c) =

• d∈C¬l

[1 − ω(d, Var(l))]

• 2. Cavity freedom of l to not satisfy c.

[0.0, 1.0] ∋ U (l, c) =

• d∈Cl\{c}

[1 − ω(d, (V ar)(l))]

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Background Message Passing Modifying the PMPF Conclusions

Message Passing on a conceptual level (7)

In summary, the MP framework defines, that δ values are used to compute the ω values ω values are used to compute the S, U values However, the MP algorithms do not update the values arbitrarily. MP framework governs the overall process to update values. following a random clause permutation π ∈ Sm update one clause at a time (clause-update). ∀i ∈ {1, . . . , m} :

1

∀l ∈ cπ(i) : update δ(l, cπ(i))

2

∀v ∈ Var(cπ(i)) : update ω(cπ(i), v)

3

∀l ∈ cπ(i) : update S(l, cπ(i)), U (l, cπ(i))

updating all clauses once is called an iteration (z = 0, 1, 2, . . .) multiple iterations form a cycle (y = 1, 2, 3, . . .) the cycle of iterations is over in case an abort condition holds

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Background Message Passing Modifying the PMPF Conclusions

Message Passing on a conceptual level (8)

Conceptually, a cycle y of iterations z = 0, 1, . . . , ∗ looks as follows. The MP heuristic “links” the consecutive iterations. For example, the Belief Propagation (BP) heuristic defines

y zδBP(l, c) = y z−1UBP(l, c) y z−1UBP(l, c) + y z−1SBP(l, c)

• =

U U + S

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Background Message Passing Modifying the PMPF Conclusions

Message Passing on a conceptual level (9)

The MP framework defines how to compute the biases using the y

∗ω(c, v).

1 Compute the variable freedom to be assigned to true (T ) or false (F)

yT (v) =

• c∈C−

v

[1 − y

∗ω(c, v)] yF(v) =

• c∈C+

v

[1 − y

∗ω(c, v)]

2 Compute magnetization values using T and F

yµ+(v), yµ−(v), yµ±(v) ∈ [0.0, 1.0]

These give yµ(v) = yµ+(v) + yµ−(v) + yµ±(v)

3 Compute the biases

yβ+(v) = yµ+(v) yµ(v) yβ−(v) = yµ−(v) yµ(v) yβ(v) = yβ+(v) − yβ−(v)

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Background Message Passing Modifying the PMPF Conclusions

Message Passing on a conceptual level (10)

Here, all computations are defined by the MP framework except the µ. The exact µ equations are defined by the MP heuristic. The Belief Propagation (BP) heuristic defines

yµ+ BP(v) = yTBP(v) yµ− BP(v) = yFBP(v) yµ± BP(v) = 0

Using the BP heuristic in the MP framework then results in

yβBP(v) = yTBP(v) − yFBP(v) yTBP(v) + yFBP(v)

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Background Message Passing Modifying the PMPF Conclusions

Heuristics vs. Frameworks (1)

In summary, we have the following set of equations (defined by either the MP framework or the MP heuristic). During the iterations we compute disrespect messages δ(l, c) warning messages ω(l, c) cavity freedom values S(l, c), U (l, c) After convergence we compute variable freedom values T (v), F(v) variable magnetization values µ+(v), µ−(v), µ±(v), µ(v) variable bias value β+(v), β−(v), β(v) If we want to extend the MP framework we must not touch the MP heuristic equations we must influence the equations for the iterations

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Background Message Passing Modifying the PMPF Conclusions

Heuristics vs. Frameworks (2)

The focus lies on the waring message, currently defined as ω(c, v) =

• l∈c\{v,¯

v}

δ(l, c) How can we modify this equation in a meaningful way? Check Statistical Mechanics and the Ising Model

has served in the derivation of the SP equations provides concepts that have been ignored so far

• However. . .

quite substantial theory

In the following, modifications of ω are presented

but the “bottom-up” explanations of the underlying ideas are omitted

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Background Message Passing Modifying the PMPF Conclusions

Site Agitation with Arbitrary Coupling Strength (1)

Originally, the MP framework defines ω(c, v) =

• l∈c\{v,¯

v}

δ(l, c) all literals have the same “weight” in the warning computation ignores arbitrary coupling strength in spin glasses Introducing literal-weights. . . as a set Ψ = {ψi,l|ψi,l ∈ [0, ∞), ci ∈ F, l ∈ ci} ω(ci, v, Ψ) =

• l∈ci\{v,¯

v}

ψi,l

• δ(l, ci, Ψ)

ψi,l = 1 results in no modification ψi,l > 1 will increase the factor for the product ψi,l < 1 will decrease the factor for the product

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Background Message Passing Modifying the PMPF Conclusions

Site Agitation with Arbitrary Coupling Strength (2)

What can we do with the ψi,l values? In theory we can study the effects of Ψ → ∞ or Ψ → 0 on an MP algorithm’s behavior

using Ψ → ∞ forces the MP algorithm to simulate

a literal-occurrence based variable ordering that acts conflict avoiding β(v) =

|C+

v |−|C− v |

|Cv|

using Ψ → 0 forces the MP algorithm to simulate

a literal-occurrence based variable ordering that acts conflict seeking β(v) =

|C−

v |−|C+ v |

|Cv|

In practice we can gradually interpolate between conflict avoiding/seeking behavior use these values to realize feedback from the search to the MP alg.

bi-directional flow of information is important for performance VSIDS/PS and the CDCL search increase weights in conflict clauses

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Background Message Passing Modifying the PMPF Conclusions

Magnetic Agitation (1)

Originally, the MP framework defines ω(c, v) =

• l∈c\{v,¯

v}

δ(l, c) there is no way to influence an “overall direction” ignores external magnetic flields in spin glasses Introducing a direction preference. . . as a parameter Φ ∈ [−1.0, +1.0] (magnetic field parameter) ω(c, v, Φ) = (1 − |Φ|)  

• l∈c\{v,¯

v}

δ(l, c, Φ)   + |Φ| + sgn(v, c) · Φ 2 Φ = 0 results in no modification Φ > 0 will increase warnings send from positive literals Φ < 0 will increase warnings send from negative literals

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Background Message Passing Modifying the PMPF Conclusions

Magnetic Agitation (2)

What can we do with the Φ value? In theory we can study the effects of Φ → −1 or Φ + 1 on an MP algorithm’s behavior

using Φ → −1 forces the MP algorithm to simulate

the zero-first value ordering β(v) = −1

using Φ → 1 forces the MP algorithm to simulate

the one-first value ordering β(v) = +1

In practice we can gradually interpolate between zero-first/one-first harness a-priori knowledge about “0-1 ratios” in satisfying assignments

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Background Message Passing Modifying the PMPF Conclusions

Thermal Agitation (1)

Originally, the MP framework defines ω(c, v) =

• l∈c\{v,¯

v}

δ(l, c) clauses constitute a “link” between variables in the formula information can “travel unhindered” through the clauses ignores thermal agitation in spin glasses Introducing a dampening effect. . . as a parameter Υ ∈ [0, 1] (temparature parameter) ω(c, v, Φ) = (1 − Υ)  

• l∈c\{v,¯

v}

δ(l, c, Υ)   Υ > 0 smaller warnings (variables become more independent) Υ = 1 results in zero-warnings (variables are completely independent)

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Background Message Passing Modifying the PMPF Conclusions

Thermal Agitation (2)

What can we do with the Υ value? In theory we can use Υ to define a new measure for variables: heat resistance increase Υ gradually and monitor the biases biases will decay at high temperature, but not necessarily equally fast large biases for large temperature values constitute a high heat restistance suggests more stable assignments In practice we can use Υ to control how much computational time is spend

higher temperature values result in faster convergence during the cycles computing biases at various temperature levels can sugges the “stability” of assignments

allows a trade-off between computational time and bias quality

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Background Message Passing Modifying the PMPF Conclusions

Putting it all together

Replacing ω(c, v) =

• l∈c\{v,¯

v}

δ(l, c) in the original product-based MP framework with ω(ci, v, Ψ, Φ, Υ) = (1 − Υ)   (1 − |Φ|)  

• l∈ci\{v,v}

ψi,l

• δ(l, ci, Ψ, Φ, Υ)

  + |Φ| + sgn(v, c) · Φ 2    results in the extended product-based MP framework (ePMPF).

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Background Message Passing Modifying the PMPF Conclusions

Conclusions

Modifying the ω equation in the product-based MP framework allows us to influence the behavior of the MP algorithm (and the biases it computes) by harnessing additional concepts from physics

site agitation with arbitrary coupling strength, Ψ

gradually interpolate between conflict avoiding/seeking behavior realize feedback from the search to the MP algorithm

magnetic agitation, Φ

gradually interpolate between zero-first/one-first direction preferences harness a-priori knowledge about 0-1 ratios in solutions tuneable parameter

thermal agitation, Υ

allows to define a heat resistance measure for variables control how much computational effort is spend by MP algorithm tuneable parameter

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Background Message Passing Modifying the PMPF Conclusions

Thanks you for your attention!

An Ising Model inspired Extension of the Product-based MP Framework for SAT Oliver Gableske

• liver@gableske.net

Thank you for your attention.

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