Extending an Isabelle Formalisation of CDCL to Optimising CDCL - - PowerPoint PPT Presentation

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Extending an Isabelle Formalisation of CDCL to Optimising CDCL - - PowerPoint PPT Presentation

Oct 17, 2022 307 likes •594 views

Extending an Isabelle Formalisation of CDCL to Optimising CDCL Mathias Fleury joint work with Christoph Weidenbach Matryoshka+Veridis 2019 Lets find a model with minimal weight 10 4 20 13 Optimal partial model: Optimal

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SLIDE 1

Extending an Isabelle Formalisation


  • f CDCL to Optimising CDCL

Mathias Fleury joint work with Christoph Weidenbach

Matryoshka+Veridis 2019

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SLIDE 2

Let’s find a model with minimal weight

∨

10 20 Optimal partial model: Optimal total model: 4 13

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How reliable is the theory?

Conference version Journal version

Branch and Bound for Boolean Optimization and
 the Generation of Optimality Certificates


Javier Larrosa, Robert Nieuwenhuis, Albert Oliveras, and Enric Rodríguez-Carbonell (SAT 2009)

A Framework for Certified Boolean Branch-and-Bound Optimization

Javier Larrosa, Robert Nieuwenhuis, Albert Oliveras, and Enric Rodríguez-Carbonell (JAR 2011)

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SLIDE 4

How reliable is the theory?

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10 20

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10 1

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Let’s optimise our problem

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Let’s optimise our problem

OCDCL = CDCL + identify better models
 + conflicts based on weights

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Let’s optimise our problem

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10 20

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10 1

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Optimal model 11

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How lazy do you like your
 formalisation?

OCDCLW = CDCL + improve + conflict rules Christoph’s view: copy-paste of proofs My first idea: reuse CDCL proofs OCDCL = CDCL + improve +
 {-M. cost M ≥ min_cost}

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How lazy do you like your
 formalisation?

My first idea: reuse CDCL proofs OCDCL = CDCL + improve +
 {-M. cost M ≥ min_cost} My second idea: CDCLbnb = CDCL + improve +
 T(min_cost)

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reuse CDCL proofs

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SLIDE 10

How lazy do you like your
 formalisation?

CDCLbnb = CDCL + improve +
 T(min_cost) OCDCL = CDCLbnb where
 T(min_cost) = {-M. cost M ≥ min_cost} OCDCLW = OCDCL + restrictions

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SLIDE 11

Reuse!

CDCLbnb CDCL

by removing the additional component,
 is a special case of definitions and invariants

CDCL

properties

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can be seen as a fragment of

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SLIDE 12

Reuse!

C ∨ L ∈ N ⟹ M ⊨as ¬C ⟹ undefined_lit M L ⟹
 (M, N) ⇒CDCL (L # M, N)

in Isabelle

Propagate rule

C ∨ L ∈ N ⟹ M ⊨as ¬C ⟹ undefined_lit M L ⟹
 (M, N, O) ⇒CDCLbnb (L # M, N, O)

in Isabelle

Propagate rule

  • btained for free, thanks to abstraction over the state!


also invariants and theorems can be reused

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Reuse!

CDCLbnb CDCL

by removing the additional component,
 is a special case of definitions and invariants

CDCL

properties

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can be seen as a fragment of

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SLIDE 14

Translate to reuse

C ∨ L ∈ N+T(min_cost) ⟹ M ⊨as ¬C ⟹
 undefined_lit M L ⟹
 (M, N + T(min_cost), O) ⇒CDCL 


(L # M, N + T(min_cost), O)

in Isabelle

Propagate rule

C ∨ L ∈ N ⟹ M ⊨as ¬C ⟹ undefined_lit M L ⟹
 (M, N, O) ⇒CDCLbnb (L # M, N, O)

in Isabelle

Propagate rule

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Reuse!

CDCLbnb CDCL

by removing the additional component,
 is a special case of can be seen as a fragment of definitions and invariants

CDCL

properties

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CDCLbnb = CDCL + improve +
 T(min_cost)

ignore the additional
 component

Inherited: Definitions (for free)

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Reuse in practise!

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CDCLbnb = CDCL + improve +
 T(min_cost)

no strategy
 but terminating well-founded
 for most applications

Termination (for free) Inherited:

Reuse in practise!

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Definitions (for free)

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CDCLbnb = CDCL + improve +
 T(min_cost)

strategy strategy invariants
 holds improve can
 always be applied


  • n total models

Ends with an unsat set (nearly for free) Inherited: Termination (for free)

⟹

Definitions (for free)

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Reuse in practise!

CDCLbnb does not know anything about what is optimised!

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OCDCL = CDCLbnb where
 T(min_cost) = {-M. cost M ≥ min_cost} If I is a total model of N with cost < min_cost,
 then I is a model of N+T(min_cost)

Why does it work?

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Lemma

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If I is a total model of N with cost < min_cost,
 then I is a model of N+T(min_cost)

Why does it work?

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Lemma

Fails for partial models!

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How lazy do you like your
 formalisation?

OCDCLW = OCDCL + restrictions

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make sure that the rules on paper
 and in Isabelle are the same

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Another application:
 Dead features

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Can every option be true?

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How lazy do you like your
 formalisation?

CDCLcmW = CDCL + improve + conflict rules Christoph’s view: copy-paste of proofs My idea: CDCLcm = CDCLbnb where
 T(models_founds) = {-M. there is a model with
 more trues in models_founds}

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How lazy do you like your
 formalisation?

Lines of codes (kloc) CDCLbnb 2,0 OCDCL 1,8 OCDCLW 1,0 Partial Encoding 1,2 MaxSAT 0,4

can solve (a lot of boilerplate)

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Concrete outcome

  • CDCL with branch and bound
  • Via an encoding, also partial optimal models

Conclusion

Methodology

  • Locales, locales, locales
  • Be lazy!

Future work

  • CDCL(T)

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OCDCL = CDCLbnb where
 T(min_cost) = {-M. cost M ≥ min_cost} OCDCL = CDCLbnb where
 T(min_cost) = {-D. {M. cost M ≥ min_cost} ⊧ D}

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Conclusion: How about CDCL(T)?

CDCLbnb where
 T = {clauses entailed theory} But isn’t CDCL(T) exactly: Not exactly, because the wrong conflict
 clause (negation of the trail) is used

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Translate to reuse

C ∨ L ∈ N + T(min_cost) ⟹ M ⊨as ¬C ⟹
 undefined_lit M L ⟹
 (M, N + T(min_cost), O) ⇒CDCLbnb 


(L # M, N + T(min_cost), O)

in Isabelle

Propagate rule

Theory propagation

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