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

• 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).``

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:

GOA

By Matthias van der Hallen

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