Support for Structured Mining in Knowledge Representation and Reasoning

Matthias van der Hallen - GOA (20/10/15)

Structured Mining

  • Lack of genericity w.r.t. different concepts of matching:
    • Graph mining, sequence mining, itemset mining
    • All have different concepts of when a pattern matches
      but main ideas stay the same
  • Tasks higher up the polynomial hierarchy

Challenges:

  • Lack of genericity w.r.t. different concepts of matching:
    • Graph mining, sequence mining, itemset mining
    • All have different concepts of when a pattern matches
      but main ideas stay the same

Lack of genericity

Our solution: Templates

Templates are Second Order Definitions

{
    homomorphism(F,G1,G2) ← (∀x : (∃y (G1(x,y) ∧ G1(y,x)) ⇔ ∃y : y=F(x)) ∧
                             (∀x,y : G1(x,y) ⇒ G2(F(x),F(y))).
}

Interesting templates might be: 

  • Homomorphism, isomorphism
  • Matches (Args: Pattern, Example, Matching)
  • Canonical

Templates

  • allow for higher modularity

  • library building constructs

I. Dasseville, M. van der Hallen, G. Janssens, M. Denecker: Semantics of templates in a compositional framework for building logics, Theory and Practice of Logic Programming 15  

Work published in paper:

High computational Complexity

We need quantification over predicates and functions:

Eg. homomorphism, isomorphism

∃F : homomorphism(F,G1,G2).

We study Quantifier Elimination and Oracles to allow these theories with higher computational complexity.

Quantifier Elimination

Reducing the number of quantifiers makes for smaller groundings

Skolemization: Replace existential quantifiers by (Skolem) constants:

  • Unnested ∃: introduce skolem constant

  • Nested ∃: introduce functions

∃a : P(a).
∀a : ∃b : Q(a,b).
P(S).
∀a : Q(a,S(a)).

Extensions & other techniques exist:

  • Rewrite ∀ quantifiers to ​∃

  • Rewritings such as in Presburger Arithmetic (Cooper1972)

Oracles

Propagation technique using nested solvers to allow quantification over predicates:
 

  • Ensure quantifications are existential

  • Encapsulate quantification body as theory

  • Solve this Existential SO theory using
    subsolver as oracle

  • Require it to pass/fail

Questions?

Matthias van der Hallen

matthias.vanderhallen@cs.kuleuven.be

Celestijnenlaan 200A, Leuven

+32 16 37 39 84

Oracles

  • Efficiently share resources between different (sub)solver processes

  • Evaluate eager / lazy activation

  • How to represent information learnt from subsolver conflicts

Upcoming work:

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