MARKOV LOGIC NASSLLI 2010 Mathias Niepert

MAP INFERENCE IN MARKOV LOGIC NETWORKS We’ve tried Alchemy (MaxWalkSAT) with poor results  Better results with integer linear programming (ILP)  ILP performs exact inference  Works very well on the problems we are concerned with  Originated in the field of operations research 

LINEAR PROGRAMMING A linear programming problem is the problem of maximizing (or minimizing) a linear function subject to a finite number of linear constraints  Standard form of linear programming: 

n

maximize

c x j

j 1 n

subject to

a x j 1

ij

xj

j

j

 bi

(i



( j  1, 2, ...,

0

 1, 2, ..., m) n)

INTEGER LINEAR PROGRAMMING An integer linear programming problem is the problem of maximizing (or minimizing) a linear function subject to a finite number of linear constraints  Difference to LP: Variables only allowed to have integer values 

n

maximize

c x j

j 1

subject to

n

a x j 1

ij

xj

j

j

 bi

(i



( j  1, 2, ...,

0

 1, 2, ..., m) n)

x j  {...,1,0,1,...} 4

MAP INFERENCE 1.5 x Smokes( x )  Cancer ( x ) 1.1 x, y Friends ( x, y )  Smokes( x )  Smokes( y ) 

Two constants: Anna (A) and Bob (B) Evidence: Friends(A,B), Friends(B,A), Smokes(B) :Smokes(A) _ Cancer(A) 1.5 :Smokes(B) _ Cancer(B) 1.5 :Smokes(A) _ Cancer(B) 1.5 :Smokes(B) _ Cancer(A) 1.5 :Friends(A,B) _ :Smokes(A) _ Smokes(B) 0.55 :Friends(A,B) _ :Smokes(B) _ Smokes(A) 0.55 …

MAP INFERENCE - EXAMPLE :Smokes(A) _ Cancer(A) 1.5

Introduce one variable for each ground atom: sa , ca Introduce one variable for each formula: xj Add the following three constraints:  -sa + -xj · -1  ca - xj · 0  xj + sa - ca · 1 Add 1,5xj to the objective function

n

maximize

c x j 1

subject to

j

j

j

 bi

n

a x j 1

ij

x j  {0,1}

(i  1, 2, ..., m)

ONTOLOGY MATCHING Ontology O1

Ontology O2

Person

People Author

Author

CommitteeMember

Reviewer

PCMember

Document

Doc

reviews Paper Review

reviews

writes

Paper

writes

MARKOV LOGIC & ONTOLOGY MATCHING Markov logic supports hard and soft constraints  Ontology alignment involves both types of constraints 



Hard constraints To ensure 1-1 and functional alignment  To mitigate incoherence in the merged ontology 



Soft Constraints “A-priori” confidence for correspondences of concepts and roles based on lexical similarity measures  “Stability constraints”: Alignment should use structural information of the two ontologies during the alignment process 

DESCRIPTION LOGICS 

Logic-based knowledge representation formalisms Descendants of semantic networks and KL-ONE  Describe domain in terms of concepts (classes), roles (properties, relationships) and individuals (instances) 



Typical Properties of a DL Formal model theoretic semantics  Decidable fragments of FOL  Closely related to Propositional Modal & Dynamic Logics  Availability of inference algorithms  Decision procedures for key problems (satisfiability, subsumption, etc)  Highly optimized implemented systems 

DESCRIPTION LOGIC BASICS 

Concept names are equivalent to unary predicates 



In general, concepts are equivalent to formulas with one free variable

Role names are equivalent to binary predicates 

In general, roles are equivalent to formulas with two free variables

Individual names are equivalent to constants  Operators restricted so that: 

Language is decidable and, if possible, of low complexity  No need for explicit use of variables 





Restricted form of 9 and 8

Features such as counting can be succinctly expressed

DL SYSTEM ARCHITECTURE

Man ´ Human u Male Happy-Father ´ Man u 9 has-child Woman v :Man

Abox (data) John : Happy-Father hJohn, Maryi : has-child John: 6 1 has-child

(Horrocks 2005)

Interface

Tbox (schema)

Inference System

Knowledge Base

SYNTAX OF DESCRIPTIONS (ALC) A description C or D can be: A > ? C C1 u D1 C1 t D1 R.C R.C

an atomic concept (top) the universal concept (bottom) the null concept a negated concept the intersection of concepts the union of two concepts (restriction) (existential quantification)

* C atomic concept

* *

DL SEMANTICS  Semantics  An 

defined by interpretations

interpretation I = (DI, ¢I), where

DI is the domain (a non-empty set of individuals)

 ¢I

is an interpretation function that maps:



Concept (class) name A to subsets AI of DI



Role (property) name R to binary relation RI over DI



Individual name i to iI element of DI

DL SEMANTICS (CONT.) Interpretation function ¢I extends to concept (and role) expressions

DL KNOWLEDGE BASE 

A DL Knowledge base K is a pair hT ,Ai where T is a set of “terminological” axioms (the Tbox)  A is a set of “assertional” axioms (the Abox) 



Tbox axioms are of the form:

C v D, C ´ D, R v S, R ´ S and R+ v R where C, D concepts, R, S roles, and R+ set of transitive roles



Abox axioms are of the form: x:D, hx,yi:R where x,y are individual names, D a concept and R a role

DL KNOWLEDGE BASE Knowledge Base Tbox (schema) Man ´ Human u Male Happy-Father ´ Man u 9 has-child Woman v :Man has-child.Female isEmployedBy.Farmer

Abox (data) John : Happy-Father hJohn, Maryi : has-child John: 6 1 has-child

DL KNOWLEDGE BASE SEMANTICS 

An interpretation I satisfies (models) a Tbox axiom A (I ² A): I ² C v D iff CI µ DI I ² R v S iff RI µ SI





I satisfies a Tbox T (I ² T ) if and only if I satisfies every axiom A in T An interpretation I satisfies an Abox axiom A (I ² A) I ² x:D iff xI 2 DI





I ² C ´ D iff CI = DI I ² R ´ S iff RI = SI

I ² hx,yi:R iff (xI,yI) 2 RI

I satisfies an Abox A (I ² A) if and only if I satisfies every axiom A in A I satisfies an KB K (I ² K) if and only if I satisfies both T and A

ONTOLOGY MATCHING Ontology O1

Ontology O2

Person

People Author CommitteeMember

PCMember

Document

Author

< Author, Author, =, 0.97 > < Paper, Paper, =, 0.94 > < reviews, reviews, =, 0.91 > < writes, writes, =, 0.7 > < Person, People, =, 0.8 > < Document, Doc, =, 0.7 > < Reviewer, Review, ≤, 0.6 > …

Reviewer

reviews

Doc

writes reviews Paper Review

Paper

writes

ONTOLOGY MATCHING WITH ML 

We can compute confidence values for matching correspondences (equivalence) < Author, Author, =, 0.97 >

< Paper, Paper, =, 0.94 > < reviews, reviews, =, 0.91 > < writes, writes, =, 0.7 > < Person, People, =, 0.8 > < Document, Doc, =, 0.7 > < Reviewer, Review, ≤, 0.6 > …

We used the Levensthein distance between the labels of properties and concepts  More sophisticated approaches possible  The goal is a 1-1 and functional alignment 

ONTOLOGY MATCHING AND ML Typed predicates cmap and pmap modeling the sought-after matching correspondences  Incorporating confidence values sX,Y 



Constraining the alignment to be 1-1 and functional (cardinality constraints)

ONTOLOGY MATCHING AND ML We want the merged ontology to be “as coherent as possible” (no unsatisfiable concepts)  First, we need to introduce predicates to the ML formulation  Predicates for subsumption, disjointness, domain restrictions, range restrictions, …  Used the tableau reasoner Pellet for this preprocessing step 

COHERENCE CONSTRAINTS 

Hard (!) constraints (formulas with “infinite weight”) that ensure that incoherences cannot occur in the merged ontology

Incoherence!

EXAMPLE

EXAMPLE – GROUND MARKOV LOGIC NETWORK

EXAMPLE – INTEGER LINEAR PROGRAM

STABILITY CONSTRAINTS Stability formulas (soft!) to propagate evidence derived from structural properties  Example: 

Formula reduces the probability of alignments that map concept X to Y and X’ to Y’ if X’ subsumes X but Y’ does not subsume Y  Would introduce new structural knowledge 

MANUALLY SET WEIGHTS

We set the weights for the first group to −0.5 and the weights for the second group to −0.25  Subsumption axioms between concepts are specified by ontology engineers more often than domain and range restriction of properties (Ding and Finin 2006).  A pair of two correct correspondences will less often violate constraints of the first type than constraints of the second type 

LEARNING WEIGHTS An Online learner roughly works as follows: 1. set number of epochs to 0. 2. for each instance (xj, yj) of the training corpus do • run inference to calculate y ^ = arg maxy s(xj ; y) • 3.

4.

update current weights wj by comparing y^ to yj

If number of epochs is larger than some predefined value go to 4, otherwise increase number of epochs and go to 2. Return last solution

LEARNING WEIGHTS Run inference to calculate y^ = arg maxy s(xj ; y) ^ to yj  Update current weights wi by comparing y ^ )]  wi ← wi + η [counti(yj) – counti(y  Voted perceptron rule 

RESULTS

RESULTS – MANUAL VS. LEARNED

THANKS

Thank you for attending the course!

Markov Logic

MAP INFERENCE IN MARKOV LOGIC. NETWORKS. ○ We've tried Alchemy (MaxWalkSAT) with poor results. ○ Better results with integer linear programming.
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