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:
GOA
By Matthias van der Hallen
GOA
- 1,091
