Resolving query translation ambiguity using a decaying co-occurrence model and syntactic dependence relations Bilingual dictionaries have been commonly used for query translation in cross-language information retrieval (CLIR). However, we are faced with the problem of translation selection. Several recent studies suggested the utilization of term cooccurrences in this selection. This paper presents two extensions to improve them. First, we extend the basic co-occurrence model by adding a decaying factor that decreases the mutual information when the distance between the terms increases. Second, we incorporate a triple translation model, in which syntactic dependence relations (represented as triples) are integrated. Our evaluation on translation accuracy shows that translating triples as units is more precise than a word-by-word translation. Our CLIR experiments show that the addition of the decaying factor leads to substantial improvements of the basic co-occurrence model; and the triple translation model brings further improvements. Bilingual dictionaries have been commonly used for query translation in CLIR, but we are always faced with the problem of translation ambiguity, i.e. several translation words are provided for a source word by the dictionary, but only part of them are appropriate.. Recently, several studies [2, 4, 7, 11, 14, 19] have been proposed the use of co-occurrence information to deal with this problem. In those studies, usually a mutual information value between terms is estimated according to their co-occurrences within a predefined window. The translation word that has the highest mutual information score with the other translation words is selected. While these approaches have successfully improved the quality of query translation, there are still two main drawbacks. First, the standard calculation of mutual information does not take into account the distance between terms; whereas our intuition is that closer terms have stronger relationships. Second, the syntactic dependency between terms is not considered. This paper presents two methods to improve the basic cooccurrence approaches. First, we incorporate a decaying factor that decreases the mutual information when the distance between the terms increases. Second, we present a statistical triple translation model, in which syntactic dependence relations (represented as triples) are integrated.. Our evaluation results show that the triple translation is more precise than the word-by-word translation with the co-occurrence model. The CLIR experiments on TREC collections show that the decaying co-occurrence method performs better than the basic cooccurrence method, and the triple translation model brings additional improvements.. The remainder of this paper is organized as follows: Section 2 provides a brief description on the related work. Sections 3 and 4 describe our co-occurrence approach and the triple translation model. Section 5 evaluates the proposed approaches. Section 6 presents our conclusion. This paper proposed two statistical models for dealing with the problem of query translation ambiguity. We focused on the method that exploits linguistic dependency information represented as triples. The translations using triples showed three main benefits: a more precise translation; an extension of the coverage of the bilingual dictionary; and the possibility to train the model using unrelated bilingual corpora. Our experiments of CLIR showed that the triple translation has a positive impact on the query translation, and results in significant improvements of CLIR performance over the co-occurrence method. We also presented a revised version of the co-occurrence model. It differs from previous ones in that it includes a distance component that decays the mutual information between terms when the distance between them increases. Our experiments showed that the decaying co-occurrence model performs better than the standard co-occurrence model, and brings significant improvements over the simple dictionary approaches in CLIR.. An important problem we currently have with the triple translation is the robustness of the parser. A certain portion of incorrect triples are extracted, especially those with low frequencies. Many other triples cannot be extracted because the parser fails to parse the sentence completely. In order to increase the impact of triple translation, the robustness of the parser has to be improved. In fact, as we only need to extract triples, a partial parser may be more suitable. This alternative will be investigated in the future.

O B B O O O O B B O O O O O O O O O O O B B O O O O B O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O B B O O O O B O O O B I O O B O O O O O O O O O B B O O O O O O O O O O O B O B B O O O O O O O O B O O O B O B O O O O O O O O B O O O O O O O B B O O O B B O O O O O O O O O O B B O B O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B I O O O O O O O O O O O O O O B O O O O O B I O O O O B O O O O O O O O O O O O B B O O O O O O O O O O O B I O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O B I O O B O O O O O O O O O O B B O O O O O O O O O O O B O O O O O B O O B O O O B O O B B O B O O O O O O O O B O O O O O O O O O O O B B O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O B B O O O O O O O O O O O O B O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O B O O O O O O O B O O O O B O O O O O O B B O O O O O O B O O O B O O O O O O O O O B B O O O O O O O O O O B O O O O B I O O O O B O O O O O O O O O B B O O O O O B B O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Fast fully dynamic landmark-based estimation of shortest path distances in very large graphs Computing the shortest path between a pair of vertices in a graph is a fundamental primitive in graph algorithmics. Classical exact methods for this problem do not scale up to contemporary, rapidly evolving social networks with hundreds of millions of users and billions of connections. A number of approximate methods have been proposed, including several landmark-based methods that have been shown to scale up to very large graphs with acceptable accuracy. This paper presents two improvements to existing landmarkbased shortest path estimation methods. The first improvement relates to the use of shortest-path trees (SPTs). Together with appropriate short-cutting heuristics, the use of SPTs allows to achieve higher accuracy with acceptable time and memory overhead. Furthermore, SPTs can be maintained incrementally under edge insertions and deletions, which allows for a fully-dynamic algorithm. The second improvement is a new landmark selection strategy that seeks to maximize the coverage of all shortest paths by the selected landmarks. The improved method is evaluated on the DBLP, Orkut, Twitter and Skype social networks. Finding the shortest path between a given pair of vertices in a graph is a fundamental primitive in graph algorithmics. For example, in the context of social network analysis, this primitive is useful for socially-sensitive search [18, 20] , whereby a user of a social network searches for an individual by name and expects to see the results ranked in the order of their shortest path distance to him. Similarly, a user may wish to know what chain of contacts allows him to reach another user in the network.. There exists a large body of methods to address this problem [21] . Existing methods can be broadly classified into exact and approximate. Exact methods, such as those based on Dijkstra's traversal, are prohibitively slow for performing online queries on graphs with hundreds of millions of vertices, which is a typical size for a contemporary social network. Among the approximate methods, a family of scalable algorithms for this problem are the so-called landmark-based (or sketch-based ) approaches [4, 5, 7, 9, 18] . In this family of techniques, a fixed set of landmark nodes is selected and distances or actual shortest paths are precomputed from each vertex to some or all of the landmarks. Knowledge of the distances to the landmarks, together with the triangle inequality, typically allows one to compute approximate distance between any two vertices in O(k) time, where k is the number of landmarks, and O(kn) space, where n is the number of vertices in the network. Those estimates can then be used as-is, or exploited further as a component of a graph traversal or routing strategy in order to obtain an exact shortest path [7, 10] .. Although landmark-based algorithms do not provide strong theoretical guarantees on approximation quality [11] , they have been shown to perform well in practice, scaling up to graphs with millions or even billions of edges with acceptable accuracy and response times of under one second per query [5, 9, 16] .. In this paper we describe two improvements to existing landmark-based estimation methods for undirected graphs.. These improvements allow us to achieve higher accuracy than existing methods with millisecond-execution times on a graph with over three billion edges.. The first improvement derives from the use of shortest path trees (SPTs) to maintain the paths between each landmark and every vertex in the graph. Based on this data structure, we present three novel strategies for computing an approximate shortest path between any pair of nodes. Moreover, the use of SPTs makes the proposed method suitable for continuously evolving graphs, since practically efficient algorithms exist for incremental SPT update [3, 6] . For completeness' sake we explicitly present a version of the incremental update algorithm from [6] , simplified to unweighted graphs.. The second improvement relates to the selection of landmarks. Specifically, we propose a greedy approach to select those landmarks that provide the best coverage of all shortest paths in a random sample of vertex pairs.. We evaluate the proposed method on four large social network, the largest of them (Skype) consisting of circa 500 million users and 3 billion connections. The evaluation demonstrates improved precision of distance estimates gained by both the landmark selection technique and the use of SPTs.. The rest of the paper is structured as follows. Section 2 introduces the notation used to present the algorithms, which are themselves given in Section 3. Section 4 summarizes the experimental evaluation results. Finally, Section 5 discusses related work, while Section 6 provides concluding remarks. This paper described and evaluated two improvements to existing approaches for landmark-based estimation of shortest paths. These improvements strike a tradeoff between accuracy, query execution time and disk usage for precomputed data. With respect to previous related work, we achieve notable accuracy improvements while maintaining the response time per query within a few milliseconds -even for a graph with billions of edges -and a space consumption comparable to previous state of the art methods. An exclusive property of the proposed methods is the support for dynamic updates.. Several extensions to the proposed method are possible.. Generalization to directed weighted graphs. We have presented the algorithms for the case of an undirected unweighted graph. The generalization to weighted graphs is obtained by replacing the BFS in the SPT precomputation phase and in the Landmark-BFS algorithm with a Dijkstra traversal. The generalization to directed graphs requires computing two shortest path trees for each landmark -the first one holding distances to the landmark, and the second one with distances from the landmark. The algorithms then need to be updated slightly to use both trees appropriately (e.g. lines 4 and 5 of Algorithm 3 will refer to two different trees rather than one).. Exact distance estimation. Being a distance approximation scheme, a landmark-based algorithm can be used as a heuristic in the (unidirectional or bidirectional) A* search, as described in [7, 10] . In particular, this ability to efficiently estimate exact shortest paths allows us to take larger samples for best coverage landmark selection. Note that due to the incremental update capabilities of our approach, the result is a fast fully-dynamic exact shortest path algorithm.. Evolutionary landmark selection. In the presented approach, all landmarks are selected ex ante and the selection of landmarks is never revised. A further improvement might be obtained by using information collected during the processing of queries in order to add or remove landmarks. Each time a query is answered, we can identify which vertices are used in the shortest path. Based on this information we could promote certain vertices, which lie on shortest paths frequently, to become landmarks, or we could drop landmarks that are infrequently used. The task of finding a shortest path between an arbitrary pair of vertices is known as a point-to-point shortest path (PPSP) problem. The naïve solution is to run the Dijkstra's traversal from the source to the destination vertex on every query. An improvement to this can be obtained by running a bi-directional search [15] . A further improvement is obtained by exploiting the A* search with landmark-based heuristics [7, 8, 10] .. Sometimes it makes sense to precompute all pairwise distances. The corresponding algorithms are typically referred to as all pairs shortest paths (APSP). Although a variety of elegant techniques have been proposed for this task, such as dynamic programming, path-forming and Boolean matrix multiplications, most known exact APSP algorithms are about O(n 3 ) in time, with a few subcubic solutions for cer- tain types of graphs [21] . This is not significantly better than simply performing a separate SSSP run for each vertex. For many practical applications, finding out approximate distances between vertices can be sufficient. Careful indexing of the graph can provide a way to later provide approximate answers to arbitrary distance queries in constant or logarithmic time. Thorup and Zwick [17] coin the term approximate distance oracles for such algorithms. They present a method which requires O(kn 1+1/k ) space and O(kmn 1/k ) expected time for preprocessing and allows constant O(k) query time. The idea of their method is to compute a sequence of nested sets of randomly picked landmarks. The closest landmark from each set is located and stored for each vertex together with the distance to it. A distance query d(s, t) can then be answered by finding a landmark u which is one of the closest to both s and t and using the approximation given by Equation (2) . The authors prove that their algorithm provides distance estimates which can be wrong up to a factor of 2k − 1.. An interesting analogy of our method to this algorithm can be observed. Indeed, let u be a single landmark. Consider the collection of nested sets A0 ⊆ A1 · · · ⊆ AD, where Ai = {v ∈ V , d(u, v) ≤ i}, and D is the diameter of the graph. Then the complete shortest path πs,u corresponds to the bunch concept from Thorup's algorithm, and the process of finding the LCA is similar to the distance estimation technique used in that paper. However, the strategy of selecting the nested sets Ai is completely different. Besides, the analogy does not hold as soon as more than one landmark is used in our algorithm.. In general, the idea of landmark-based estimation has been widely studied in previous works (see, e.g., [4, 5, 7, 9, 16, 18] ). A closely related work is that of Potamias et al. [16] , where the Landmarks-Basic algorithm is evaluated under different landmark selection strategies. The major innovation of our work is the use of SPTs instead just keeping the distance from each landmark to every vertex.. The algorithms Landmarks-LCA, Landmarks-SC and Landmarks-BFS are also similar to those suggested by Gubichev et al. [9] . However, the algorithms in [9] use different sets of landmarks for each vertex and thus store complete shortest paths to each landmark at each vertex. This results in higher memory requirements (O(Dkn) instead of O(kn) for storing only a single pointer per vertex-landmark pair in our approach) and prevents the possibility of performing incremental updates. The execution times reported in [9] are considerably slower when compared to ours -more than 4 seconds on a graph with 10 times less edges and 100 times less vertices than the Skype graph.. An important aspect of landmark-based methods is the way landmarks are selected -a careful selection strategy can have a significant positive effect. Strategies, which rely on selecting landmarks with high degree, betweenness-and closeness centrality as well as ensuring proper dispersion of landmarks over the graph and its parts, have been suggested, with highest degree and closeness centrality being shown to typically yield highest accuracy in [16] . In our work, we proposed and evaluated an additional method that can compete with the highest degree approach, outperforming it slightly on some datasets. The combined effect of our improvements (best coverage landmark selection with Landmarks-BFS algorithm) for the large Skype network is a 40% smaller error. To evaluate the performance of our methods we first select a random sample of 500 vertex pairs from each graph and precompute the actual shortest path distance for every pair. We then apply the distance approximation techniques to those vertex pairs and measure their Approximation error, Time, and Disk space.. Approximation error is computed as ( − )/ , where is the approximation and is the actual distance. Time refers to the average time required to compute a distance approximation for a pair of vertices. Finally, the Disk space corresponds to the amount of data stored for each landmark.. We run our measurements on a server with 32× Quadcore AMD Opteron 64bit 2.2GHz processors, 256G of RAM, accessing IBM DS 3400 FC SAN disk array, running Red Hat Enterprise Linux 5 operating system. We used C++ for algorithm implementation.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Expectation

Maximization

over

Binary

Decision

Diagrams

for

Probabilistic

Logic

Programs

Recently

much

work

in

Machine

Learning

has

concentrated

on

using

expressive

representation

languages

that

combine

aspects

of

logic

and

probability.

A

whole

field

has

emerged,

called

Statistical

Relational

Learning,

rich

of

successful

applications

in

a

variety

of

domains.

In

this

paper

we

present

a

Machine

Learning

technique

targeted

to

Probabilistic

Logic

Programs,

a

family

of

formalisms

where

uncertainty

is

represented

using

Logic

Programming

tools.

Among

various

proposals

for

Probabilistic

Logic

Programming,

the

one

based

on

the

distribution

semantics

is

gaining

popularity

and

is

the

basis

for

languages

such

as

ICL,

PRISM,

ProbLog

and

Logic

Programs

with

Annotated

Disjunctions.

This

paper

proposes

a

technique

for

learning

parameters

of

these

languages.

Since

their

equivalent

Bayesian

networks

contain

hidden

variables,

an

Expectation

Maximization

(EM)

algorithm

is

adopted.

In

order

to

speed

the

computation

up,

expectations

are

computed

directly

on

the

Binary

Decision

Diagrams

that

are

built

for

inference.

The

resulting

system,

called

EMBLEM

for

"EM

over

Bdds

for

probabilistic

Logic

programs

Efficient

Mining",

has

been

applied

to

a

number

of

datasets

and

showed

good

performances

both

in

terms

of

speed

and

memory

usage.

In

particular

its

speed

allows

the

execution

of

a

high

number

of

restarts,

resulting

in

good

quality

of

the

solutions.

Machine

Learning

has

seen

the

development

of

the

field

of

Statistical

Relational

Learning,

where

logical-statistical

languages

are

used

in

order

to

effectively

learn

in

complex

domains

involving

relations

and

uncertainty.

These

techniques

have

been

successfully

applied

in

social

networks

analysis,

entity

recognition,

collective

classification

and

information

extraction,

to

name

a

few..

Similarly,

in

the

field

of

Logic

Programming,

a

large

number

of

works

have

started

to

appear

that

combine

logic

and

probability.

Among

these,

many

share

a

common

approach

to

defining

the

semantics

of

the

proposed

languages:

the

distribution

semantics

It

underlies

for

example

Probabilistic

Logic

Programs

Probabilistic

Horn

Abduction

(PHA)

PRISM

Independent

Choice

Logic

(ICL)

pD

Logic

Programs

with

Annotated

Disjunctions

(LPADs)

ProbLog

and

CP-logic

The

approach

is

particularly

appealing

for

its

intuitiveness

and

because

efficient

inference

algorithms

have

started

to

appear

Most

of

these

techniques

use

Binary

Decision

Diagrams

(BDD)

for

inference:

explanations

for

the

query

are

found

and

the

probability

of

the

query

is

computed

by

building

a

BDD..

In

this

paper

we

present

the

EMBLEM

system

for

"EM

over

Bdds

for

probabilistic

Logic

programs

Efficient

Mining"

that

learns

parameters

of

probabilistic

logic

programs

under

the

distribution

semantics

by

using

an

Expectation

Maximization

(EM)

algorithm.

The

system

exploits

the

fact

that

the

translation

of

these

programs

into

graphical

models

generates

models

with

hidden

variables

and

therefore

an

EM

approach

is

necessary.

Its

main

characteristic

is

that

it

computes

the

values

of

expectations

using

BDDs.

EMBLEM

is

developed

for

the

language

of

LPADs

and

tested

on

the

IMDB

Cora

and

UW-CSE

datasets

and

compared

with

RIB

LeProbLog

Alchemy

and

CEM,

an

implementation

of

EM

based

on

the

cplint

interpreter

The

paper

is

organized

as

follows.

Section

presents

Probabilistic

Logic

Programming,

concentrating

on

LPADs.

Section

describes

EMBLEM

together

with

an

example

of

its

execution.

Section

discusses

related

work.

In

Section

the

results

of

the

experiments

performed

are

presented.

Finally

Section

concludes

the

paper.

Our

work

has

close

connection

with

various

other

works.

proposed

an

EM

algorithm

for

learning

the

parameters

of

Boolean

random

variables

given

observations

of

a

Boolean

function

over

them,

represented

by

a

BDD.

EMBLEM

is

an

application

of

that

algorithm

to

probabilistic

logic

programs.

Independently

also

proposed

an

EM

algorithm

which

computes

expectations

over

decision

diagrams.

The

algorithm

learns

parameters

for

the

CPT-L

language,

a

simple

probabilistic

logic

language

for

describing

sequences

of

relational

states,

that

is

less

expressive

than

LPADs.

applies

the

algorithm

of

to

the

problem

of

computing

the

probabilistic

parameters

of

abductive

explanations.

recently

presented

the

CoPrEM

algorithm

that

performs

EM

for

the

ProbLog

language.

We

differ

from

this

work

in

the

construction

of

BDDs:

while

they

build

a

BDD

for

an

interpretation

that

represents

the

application

of

the

whole

theory

to

the

interpretation,

we

focus

on

a

target

predicate,

the

one

for

which

we

want

to

obtain

good

predictions,

and

we

build

BDDs

starting

from

atoms

for

the

target

predicate.

Moreover,

while

we

compute

the

contributions

of

deleted

paths

with

the

ς

table,

CoPrEM

treats

missing

nodes

as

if

they

were

there

and

updates

the

counts

accordingly..

Other

approaches

for

learning

probabilistic

logic

programs

can

be

classified

into

three

categories:

those

that

employ

constraint

techniques,

those

that

use

EM

and

those

that

adopt

gradient

descent.

In

the

first

class

learn

a

subclass

of

ground

programs

by

first

finding

a

large

set

of

clauses

satisfying

certain

constraints

and

then

applying

mixed

integer

linear

programming

to

identify

a

subset

of

the

clauses

that

form

a

solution..

Among

the

approaches

that

use

EM,

first

proposed

to

use

the

EM

algorithm

to

induce

parameters

of

ground

LPADs

and

the

Structural

EM

algorithm

to

induce

ground

LPAD

structures.

Their

EM

algorithm

however

works

on

the

underlying

Bayesian

network..

RIB

performs

parameter

learning

using

the

information

bottleneck

approach,

which

is

an

extension

of

EM

targeted

especially

towards

hidden

variables.

However,

it

works

best

when

interpretations

have

the

same

Herbrand

base,

which

is

not

always

the

case..

The

PRISM

system

is

one

of

the

first

learning

algorithms

based

on

EM.

It

exploits

Logic

Programming

techniques

for

computing

expectations

but

imposes

restrictions

on

the

language..

In

the

authors

use

EM

to

learn

the

structure

of

first-order

rules

with

associated

probabilistic

uncertainty

parameters.

Their

approach

involves

generating

the

underlying

graphical

model

using

a

Knowledge-Based

Model

Construction

approach.

EM

is

then

applied

on

the

graphical

model..

Among

the

works

that

use

a

gradient

descent

technique,

LeProbLog

starts

from

a

set

of

queries

annotated

with

a

probability

and

from

a

ProbLog

program.

It

tries

to

find

the

values

of

the

parameters

of

the

program

that

minimize

the

mean

squared

error

of

the

probabilities

of

the

queries.

LeProbLog

uses

the

Binary

Decision

Diagrams

that

represent

the

queries

to

compute

the

gradient..

Alchemy

is

a

state

of

the

art

Statistical

Relational

Learning

system

that

offers

various

tools

for

inference,

weight

learning

and

structure

learning

of

Markov

Logic

Networks

(MLNs).

discusses

how

to

perform

weight

learning

by

applying

gradient

descent

of

the

conditional

likelihood

of

queries

for

target

predicates.

MLNs

significantly

differ

from

the

languages

under

the

distribution

semantics

since

they

extend

first-order

logic

by

attaching

weights

to

logical

formulas,

reflecting

"how

strong"

they

are.

MLNs

allow

the

use

of

logical

formulas

without

syntactic

restrictions,

but

do

not

allow

to

exploit

logic

programming

techniques..

Experiments

EMBLEM

has

been

tested

over

three

real

world

datasets:

IMDB

UW-CSE

and

Cora

We

implemented

EMBLEM

in

Yap

Prolog

and

we

compared

it

with

RIB

CEM,

an

implementation

of

EM

based

on

the

cplint

inference

library

LeProblog

and

Alchemy

All

experiments

were

performed

on

Linux

machines

with

an

Intel

Core

Duo

MHz)

processor

and

GB

of

RAM..

To

compare

our

results

with

LeProbLog

we

exploited

the

translation

of

LPADs

into

ProbLog

proposed

in

in

which

a

disjunctive

clause

with

k

head

atoms

and

vector

of

variables

X

is

modeled

with

k

ProbLog

clauses

and

k

probabilistic

facts

with

variables

X..

To

compare

our

results

with

Alchemy

we

exploited

the

translation

between

LPADs

and

MLN

used

in

and

inspired

by

the

translation

between

ProbLog

and

MLNs

proposed

in

An

MLN

clause

is

translated

into

an

LPAD

clause

in

which

the

head

atoms

of

the

LPAD

clause

are

the

atom

plus

the

positive

literals

of

the

MLN

clause

while

the

body

atoms

are

the

negative

literals..

For

the

probabilistic

logic

programming

systems

(EMBLEM,

RIB,

CEM

and

LeProbLog)

we

consider

various

options.

The

first

consists

in

choosing

between

associating

a

distinct

random

variable

to

each

grounding

of

a

probabilistic

clause

or

a

single

random

variable

to

a

non-ground

probabilistic

clause

expressing

whether

the

clause

is

used

or

not.

The

latter

case

makes

the

problem

easier,

as

stated

previously.

The

second

option

is

concerned

with

putting

a

limit

on

the

depth

of

derivations

as

done

in

thus

eliminating

explanations

associated

to

derivations

exceeding

the

depth

limit.

This

is

necessary

for

problems

that

contain

cyclic

clauses,

such

as

transitive

closure

clauses.

The

third

option

involves

setting

the

number

of

restarts

for

EM

based

algorithms..

All

experiments

for

probabilistic

logic

programming

systems

have

been

performed

using

open-world

predicates,

meaning

that,

when

resolving

a

literal,

both

facts

in

the

database

and

rules

are

used

to

prove

it..

All

datasets

are

partitioned

into

five

mega-examples,

so

a

five-fold

cross-validation

approach

has

been

adopted

in

the

experiments:

of

the

five

mega-examples,

a

single

example

is

retained

for

testing,

and

the

remaining

four

are

used

as

training

data.

The

datasets

are

described

in

Table

in

terms

of

target

predicates,

number

of

different

constants,

number

of

different

predicates

and

number

of

tuples

(ground

atoms)

in

the

interpretations.

As

part

of

the

test,

we

drew

a

Precision-Recall

curve

and

a

Receiver

Operating

Characteristics

curve,

and

computed

the

Area

Under

the

Curve

(AUCPR

and

AUCROC

respectively)

using

the

methods

reported

in

IMDB

regards

movies,

actors,

directors

and

movie

genres.

Each

mega-example

contains

all

the

information

regarding

four

movies.

We

defined

different

LPADs,

two

for

predicting

the

target

predicate

and

two

for

predicting

We

had

one

positive

example

for

each

fact

that

is

true

in

the

data,

while

we

sampled

from

the

complete

set

of

false

facts

three

times

the

number

of

true

instances

in

order

to

generate

negative

examples..

For

predicting

we

used

the

same

LPAD

of

where

p

is

a

tunable

parameter.

We

ran

EMBLEM

on

it

with

the

following

settings:

no

depth

bound,

random

variables

associated

to

instantiations

of

the

clauses

and

a

number

of

restarts

chosen

to

match

the

execution

time

of

EMBLEM

with

that

of

the

fastest

other

algorithm.

The

queries

that

LeProbLog

takes

as

input

are

obtained

by

annotating

with

each

positive

example

for

and

with

each

negative

example.

We

ran

LeProbLog

for

a

maximum

of

iterations

or

until

the

difference

in

Mean

Squared

Error

(MSE)

between

two

iterations

got

smaller

than

−5

For

Alchemy

we

used

the

preconditioned

rescaled

conjugate

gradient

discriminative

algorithm

and

we

specified

as

the

only

non-evidence

predicate..

A

second

LPAD

has

been

created

to

evaluate

the

performance

of

the

algorithms

when

some

atoms

are

unseen:

The

and

predicates

are

unseen

in

the

data.

Alchemy

was

run

with

the

−withEM

option

that

turns

on

EM

learning.

The

other

parameters

for

Alchemy

and

for

the

other

algorithms

are

set

as

before.

Table

and

show

respectively

the

AUCPR

and

AUCROC

averaged

over

the

five

folds

for

EMBLEM,

RIB,

LeProbLog,

CEM

and

Alchemy.

Results

for

the

two

programs

are

shown

respectively

in

the

IMDB-SP

and

IMDBu-SP

rows

(where

u

stands

for

unseen).

Table

shows

the

learning

times

in

hours..

For

predicting

we

used

the

LPAD:

To

test

the

behaviour

when

unseen

predicates

are

present,

we

transformed

the

program

for

as

we

did

for

thus

introducing

the

unseen

predicates

samemovie

and

samemovie

We

ran

EMBLEM

on

them

with

no

depth

bound,

one

variable

for

each

instantiation

of

a

rule

and

one

random

restart.

As

regards

LeProbLog

and

Alchemy,

we

ran

them

with

the

same

settings

as

IMDB-SP

and

IMDBu-SP,

by

replacing

sameperson

with

samemovie.

Table

and

show

respectively

the

AUCPR

and

AUCROC

averaged

over

the

five

folds.

Results

for

the

two

LPADs

are

shown

respectively

in

the

IMDB-SM

and

IMDBu-SM

rows.

RIB

in

this

case

obtained

a

memory

error

(indicated

with

"me"),

due

to

the

exhaustion

of

the

available

stack

space

during

the

execution

of

the

algorithm..

The

Cora

database

contains

citations

to

computer

science

research

papers.

For

each

citation

we

know

the

title,

the

authors,

the

venue

and

the

words

that

appear

in

them.

The

task

is

to

determine

which

citations

are

referring

to

the

same

paper,

by

predicting

the

predicate

The

database

contains

facts

for

the

predicates

sametitle(tit1,tit2),samevenue(ven1,ven2),haswordtitle(title,word)

haswordauthor(author,word)

and

haswordvenue(venue,word)..

From

for

a

total

of

rules.

In

this

case

we

had

to

run

EMBLEM

with

a

depth

bound

equal

to

two

and

a

single

variable

for

each

non-ground

rule;

the

number

of

restarts

was

one.

As

for

LeProbLog,

we

separately

learned

the

four

predicates

because

learning

the

whole

theory

at

once

would

give

a

lack

of

memory

error.

We

annotated

with

each

positive

example

for

and

with

the

negative

examples

for

the

same

predicates.

We

ran

it

for

a

maximum

of

iterations

or

until

the

difference

in

MSE

between

two

iterations

got

smaller

than

−5

For

Alchemy

we

used

the

preconditioned

rescaled

conjugate

gradient

discriminative

training

algorithm

and

we

specified

the

four

predicates

as

the

non-evidence

predicates.

Table

and

show

respectively,

in

the

Cora

and

CoraT

(Cora

transitive)

rows,

the

average

AUCPR

and

AUCROC.

On

CoraT,

CEM

and

Alchemy

gave

a

memory

error,

for

memory

exhaustion

and

a

segmentation

fault

(during

the

use

of

learnwts

command)

respectively,

while

RIB

was

not

applicable

because

it

was

not

possible

to

split

the

input

examples

into

smaller

independent

interpretations

as

required

by

RIB..

The

UW-CSE

dataset

contains

information

about

the

computer

science

department

of

the

University

of

Washington.

It

contains

different

predicates,

such

as

and

so

on.

The

predicates

are

typed,

where

possible

types

are

person,

course,

publication,

etc.

Each

mega-example

contains

facts

for

a

particular

area

of

the

CS

department:

artificial

intelligence,

graphics,

programming

languages,

systems

and

theory.

The

goal

here

is

to

predict

the

predicate,

namely

the

fact

that

a

person

is

advised

by

another

person:

this

was

our

target

predicate..

The

theory

used

was

obtained

from

the

MLN

of

It

contains

rules,

such

as

for

instance:.

advisedby(S,

P)

:p

:-courselevel(C,level_500),taughtby(C,P,Q),

ta(C,

S,

Q).

tempadvisedby(S,

P)

:p

taughtby(C,

P,

Q),

ta(C,

S,

Q).

professor(P)

:p

:-courselevel(C,level_500),taughtby(C,P,Q)..

We

ran

EMBLEM

on

it

with

a

single

variable

for

each

non-ground

rule,

a

depth

bound

of

two

and

one

random

restart.

The

negative

examples

have

been

generated

by

considering

all

couple

of

persons

(a,b)

where

a

and

b

appear

in

an

fact

in

the

data

and

by

adding

a

negative

example

advisedby(a,b)

if

it

is

not

in

the

data..

The

annotated

queries

that

LeProbLog

takes

as

input

have

been

created

by

annotating

with

each

positive

example

for

and

with

each

negative

example.

We

ran

LeProbLog

for

a

maximum

of

iterations

or

until

the

difference

in

MSE

between

two

iterations

got

smaller

than

−5

and

we

used

a

single

variable

for

each

nonground

rule.

For

Alchemy,

we

used

the

preconditioned

rescaled

conjugate

gradient

discriminative

training

algorithm

to

learn

weights,

by

specifying

as

the

only

non-evidence

predicate.

RIB

was

non

applicable

to

this

dataset

because

it

does

not

allow

to

have

variables

for

non-ground

rules.

Table

and

show

respectively

the

AUCPR

and

AUCROC

averaged

over

the

five

departments

for

all

the

algorithms.

Table

and

show

the

p-value

of

a

paired

two-tailed

t-test

at

the

significance

level

of

the

difference

respectively

in

AUCPR

and

AUCROC

between

EMBLEM

and

RIB/LeProbLog/CEM/Alchemy

(significant

differences

in

bold).

Table

Results

of

the

experiments

on

all

datasets

in

terms

of

Area

Under

the

PR

Curve.

IMDBu

refers

to

the

IMDB

dataset

with

the

theory

containing

unseen

predicates.

CoraT

refers

to

the

theory

containing

transitive

rules.

Numbers

in

parenthesis

followed

by

r

mean

the

number

of

random

restarts

(when

different

from

one)

to

reach

the

area

specified.

"me"

means

memory

error

during

learning.

"no"

means

that

the

algorithm

was

not

applicable.

AUCPR

is

the

area

under

the

precision-recall

curve

averaged

over

the

five

folds.

R

is

RIB,

L

is

LeProbLog,

C

is

CEM,

A

is

Alchemy.

Table

Results

of

the

experiments

on

all

datasets

in

terms

of

Area

Under

the

ROC

Curve.

IMDBu

refers

to

the

IMDB

dataset

with

the

theory

containing

unseen

predicates.

CoraT

refers

to

the

theory

containing

transitive

rules.

Numbers

in

parenthesis

followed

by

r

mean

the

number

of

random

restarts

(when

different

from

one)

to

reach

the

area

specified.

"me"

means

memory

error

during

learning.

"no"

means

that

the

algorithm

was

not

applicable.

AUCROC

is

the

area

under

the

Receiver

Operating

Characteristics

curve

averaged

over

the

five

folds.

R

is

RIB,

L

is

LeProbLog,

C

is

CEM,

A

is

Alchemy.

Table

Results

of

t-test

on

all

datasets,

relative

to

AUCPR.

p

is

the

p-value

of

a

paired

two-tailed

t-test

(significant

differences

in

AUCPR

at

the

level

in

bold)

between

EMBLEM

and

all

the

others.

R

is

RIB,

L

is

LeProbLog,

C

is

CEM,

A

is

Alchemy.

From

the

results

we

can

observe

that

over

IMDB

EMBLEM

has

comparable

performances

with

CEM

for

IMDB-SP,

with

similar

execution

time.

On

IMDBu-SP

it

has

better

performances

than

all

other

systems

(see

AUCPR),

with

a

learning

time

equal

to

the

fastest

other

algorithm.

On

IMDB-SM

it

reaches

the

highest

area

value

in

less

time

(only

one

restart

is

needed).

On

IMDBu-SM

it

still

reaches

the

highest

area

with

one

restart

but

with

a

longer

execution

time..

Over

Cora

it

has

comparable

performances

with

the

best

other

system

CEM

but

in

significant

lower

time

and

over

CoraT

is

one

of

the

few

systems

to

be

able

to

complete

learning,

with

better

performances

in

terms

of

area

(especially

AUCPR)

and

time..

Over

UW-CSE

it

has

better

performances

with

respect

to

all

the

algorithms.

A

difference

in

the

learning

times

between

EMBLEM

and

the

other

systems,

in

favour

of

the

latter,

can

be

found

with

the

IMDBu-SM

and

UW-CSE

datasets,

in

which

EMBLEM

takes

a

few

hours,

but

it

must

be

noted

that

in

both

cases

there

is

also

a

significant

gap

in

the

area

values:

EMBLEM

on

IMDBu-SM

reaches

the

highest

possible

area

and

on

UW-CSE

obtains

a

significantly

higher

AUCPR

with

respect

to

the

other

algorithms..

Among

the

probabilistic-logic

systems,

the

closest

to

EMBLEM

are

RIB

and

LeProblog.

RIB

is,

on

one

hand,

based

on

an

efficient

algorithm,

shown

to

be

superior

to

EM

for

learning

parameters

of

Bayesian

networks

with

hidden

variables

because

it

can

avoid

some

local

maxima,

but

on

the

other

hand,

it

requires

a

different

"format"

for

input

examples

with

respect

to

EMBLEM,

which

makes

it

unapplicable

on

one

dataset,

and

is

less

performing

in

the

presence

of

partially

hidden

variables.

LeProblog

is

the

only

system

able

to

complete

learning

for

all

datasets

as

EMBLEM,

with

good

performances

in

almost

all

cases,

but

it

takes

longer

execution

times

(except

for

IMDBu-SM

and

UW-CSE)..

Looking

at

the

overall

results,

EMBLEM

achieves

higher

or

equal

AUCPR

and

AUCROC

with

respect

to

all

other

systems,

except

on

IMDBu-SP

where

LeProbLog

achieves

a

non-statistically

significant

higher

AUCROC.

In

the

other

cases

the

differences

between

EMBLEM

and

the

other

systems

are

statistically

significant

in

out

of

cases.

B I O B I I O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O B I I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O B I I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Modelling defeasible and prioritized support in bipolar argumentation Abstract Cayrol and Lagasquie-Schiex introduce bipolar argumentation frameworks by introducing a second relation on the arguments for representing the support among them. The main drawback of their approach is that they cannot encode defeasible support, for instance they cannot model an attack towards a support relation. In this paper, we introduce a way to model defeasible support in bipolar argumentation frameworks. We use the methodology of meta-argumentation in which Dung's theory is used to reason about itself. Dung's well-known admissibility semantics can be used on this meta-argumentation framework to compute the acceptable arguments, and all properties of Dung's classical theory are preserved. Moreover, we show how different contexts can lead to the alternative strengthening of the support relation over the attack relation, and converse. Finally, we present two applications of our methodology for modeling support, the case of arguments provided with an internal structure and the case of abstract dialectical frameworks. Support is a controversial, but central issue in classical approaches to argumentation theory. Toulmin [27] proposes a model where support is a relation between data and claims having that data sustains the claim, and the beliefs in Pollock's [23] OSCAR 1 are justified if and only if they are supported by an argument that is undefeated. In Dung's theory of abstract argumentation [15] , support is not explicit, and only one kind of interaction between arguments is considered, the attack relation.. Three approaches have been proposed to model support in Dung's theory: by Dung's notion of defence, by instantiating abstract arguments and by adding support relations among arguments. The following example illustrates several kinds of interactions between arguments in an informal argument exchange during a degree committee meeting. The related argumentation framework is visualized in Fig. 1 . In this paper, a meta-argumentation framework for representing bipolar argumentation is presented. We start from the drawbacks of Cayrol and Lagasquie-Schiex's meta-argumentation approach and we propose an alternative way of modelling support using meta-argumentation. We do not take a stance towards the usefulness of the support relation among arguments, though we show that if one would introduce it, it can be done without extending Dung's theory.. We prevent the drawbacks of Cayrol and Lagasquie-Schiex's approach by considering deductive support where given a ⇒ b it holds that: if a is acceptable then b is acceptable too and if b is not acceptable then a is not acceptable either. Differently from them, we avoid secondary attacks which may introduce inconsistencies, but we introduce mediated attacks which help us in avoiding the "loss of admissibility in Dung's sense".. We extend deductive support to defeasible support by allowing second-order attacks not only on attack relations but also on support relations. Given a ⇒ b and a second-order attack on this support relation c → (a ⇒ b ), we have that the semantics of deductive support does not hold anymore. In Cayrol and Lagasquie-Schiex [13] , no attacks on the support relations are introduced, and it has been shown that this absence leads to undesirable extensions, e.g., in the swimming pool example.. Moreover, different contexts allow to distinguish various kinds of priorities on support and attack. On the one hand, when the support relation is stronger, then the supporting argument can attack either the argument attacking the argument it supports or the attack relation itself. On the other hand, when the attack relation is stronger, then the attacking argument attacks the supporting argument, i.e., with a mediated attack or an attack against the support.. We introduce a methodology which makes it possible to define various kinds of support in a relatively easy way, without the need to introduce additional machinery. Since there are various kinds of support, it is preferable not to extend argumentation frameworks, but to instantiate them. We show, for instance, how to represent the evidence in support of the conclusion of structured arguments and the representation of arcs which are neither attacking nor supporting, as proposed in abstract dialectical frameworks [11] .

B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A risk management capability model for use in medical device companies Medical device software is a risky business. Failure of the software can have potentially catastrophic effects, leading to injury of patients or even death. It is therefore no surprise that regulators throughout the world are penalising medical device manufacturers that do not demonstrate that sufficient attention is devoted to the areas of hazard analysis and risk management (RM) throughout the software lifecycle. If a medical device company fails to comply with the regulations of a given country, in effect they surrender their legal right to market their device in that country. With so much at stake, it is in everybody's best interest that the medical device manufacturer gets it right. However, with so many different standards, regulatory guidance papers and industry guides on RM, the task of collating this information into a usable model is itself daunting. This paper seeks to extract the important concepts from a number of industry accepted standards and guides, and present them as a generic usable model for the medical device software industry. Software quality may be defined and measured in many ways. For medical device companies, one way to define and measure software quality is in terms of the risk of software-containing devices in exposing patients, operators, bystanders, service personnel or the environment to hazards. Although medical devices are developed to increase the well-being of patients, on occasion, they fail to operate properly, or are misused in ways that are associated with injuries and death [6] . According to the Institute of Medicine report 'To Err is Human' [13] , between 4400 to 9800 people die throughout the world in hospital from preventative medical errors. The report also says that more people die every year as a result of medical errors than from motor vehicle accidents, breast cancer or AIDS.. Medical device companies are responsible for ensuring that they take adequate precautions to produce safe and effective software that does not pose a severe hazard should a software-related failure occur. Therefore, medical device companies who market within the USA must ensure that they comply with the regulations outlined by the Food and Drug Administration (FDA), for medical devices. They must also be able to produce sufficient evidence to support this compliance. The FDA "is responsible for protecting the public health by assuring the safety, efficiency, and security of…medical devices…" [11] . To this end, the Center for Devices and Radiological Health (CDRH) has published guidance papers for industry and medical device staff which include risk-based activities to be performed during software validation [7] , premarket submission [8] and when using off-the-shelf (OTS) software in a medical device [9] . Although the CDRH guidance documents provide information on which software activities should be performed, including risk based activities, they do not enforce any specific method for performing these activities.. Within the medical device industry, two predominant standards have emerged for performing risk-based activities: ANSI/AAMI/ISO 14971 (ISO 14971) [3] and ANSI/AAMI SW68 (SW68) [4] . However, several authors have asserted that both ISO 14971 and SW68 require that RM activities include software, without providing any detail on how it should be done [1, 2] . In 2005, a Technical Information Report (TIR32) was published by the Association for the Advancement of Medical Instrumentation (AAMI) to rectify this situation [2] . Likewise, the International Society for Pharmaceutical Engineering (ISPE), published a guide, GAMP 4 [12] , which included guidance for performing validation of software systems and risk-based activities.. This paper outlines a proposed Risk Management Capability Model (RMCM) for use by medical device companies to assist them in producing safe and effective software. It seeks to extract, interpret and combine the disparate knowledge within the medical device industry and associated standards. It does so in the context of the following regulations: ISO 14971, SW68, TIR32 and GAMP 4 and CDRH (FDA specific) guidance documents. With respect to the specific goals and practices of the RM process area, it is clear that following the medical device regulations will only partially meet the goals of this CMMI ® process area, with only specific goal 1 being fully satisfied. As might reasonably be expected, there is little guidance provided within the medical device regulations for the advanced practices of RM. Since failure to perform any specific practice implies failure to meet the specific goal, with respect to CMMI ® , it is clear, that the goals of RM cannot be obtained by satisfying the medical device regulations during software development. However, the RMCM shows that 35 specific sub-practices would have to be performed in order to satisfy medical device regulations and that only an additional 6 sub-practices would be required in order to satisfy all the CMMI ® level 1 (or RMCM level 1) requirements. But is the opposite true, can meeting the CMMI ® goals for RM successfully meet the medical device software regulations? Meeting the goals of the RM process area by performing the CMMI ® specific practices would not meet the requirements of the medical device software regulations in this area as an additional 17 medical device specific sub-practices had to be added to meet the objectives of RMCM. We are currently progressing the development of the RMCM. Next, we plan is to trial this model within a number of Irish medical device companies. Our vision is to provide a complete SPI framework that is specifically tailored to the needs of the medical device industry. This SPI framework will consist of a capability model for each process area that is relevant to the development of software within the medical device software industry (RMCM will be one of these capability models). RMCM will support medical device companies in pursuing a continuous SPI path that will produce more efficient software development and safer medical devices in compliance with regulatory requirements.

O B I O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Resultants for unmixed bivariate polynomial systems produced using the Dixon formulation A necessary and sufficient condition on the support of a generic unmixed bivariate polynomial system is identified such that for polynomial systems with such support, the Dixon resultant formulation produces their resultants. It is shown that Sylvester-type matrices, called Dixon dialytic matrices, can also be obtained for such polynomial systems. These results are shown to be a generalization of related results reported by Chionh as well as Zhang and Goldman. For a support not satisfying the above condition, the degree of the extraneous factor in the projection operator computed by the Dixon formulation is calculated by analyzing how much the support deviates from a related rectangular support satisfying the condition. This degree estimate is an upper bound on the degree of the extraneous factor in the case when the polynomial system with a given unmixed support is generic. The concept of a support hull interior point of a support is introduced. A generic inclusion of terms corresponding to support hull interior points in an unmixed polynomial system is shown not to affect the degree of the projection operator computed by the Dixon construction.It is shown that the proposed construction for Dixon dialytic matrices works especially well for mixed bivariate systems. "Good" Sylvester-type matrices can be constructed by solving an optimization problem on their supports (by translating supports so that they have maximal overlap). The determinant of such a matrix gives a projection operator with a low degree extraneous factor, i.e. the degree of the extraneous factor in a projection operator computed from the original polynomial system is at least as much as the degree of the extraneous factor in a projection operator computed from the corresponding "translated" polynomial system. The results are illustrated on a variety of examples. Resultant computations of bivariate polynomial systems using the Dixon formulation are investigated. A necessary and sufficient condition on bivariate supports (defining the terms in a polynomial system) is identified such that for a generic unmixed polynomial system with such a support, 1 its resultant can be computed exactly using constructions based on the Dixon resultant formulation. Such bivariate supports are shown to include Chionh's supports (Chionh, 2001) as well as Zhang and Goldman's corner-cut supports (Zhang, 2000; Zhang and Goldman, 2000) for which they proved that Dixon matrices as well as Sylvester-type dialytic matrices can be constructed whose determinant is the resultant. The results in this paper are shown to be related to and more general than those in Chionh (2001) , Zhang and Goldman (2000) and Zhang (2000) . A method for constructing Sylvester-type matrices using the Dixon formulation is given; such matrices are called Dixon dialytic matrices. Earlier in Chtcherba and Kapur (2000a) , Chtcherba and Kapur had shown that the support of a bivariate unmixed polynomial system not including an orderable simplex is a necessary and sufficient condition for the determinant of the associated Dixon matrix being an exact resultant (without any extraneous factors). 2 For a bivariate polynomial system whose support is not corner-cut, the proposed construction estimates the degree of the extraneous factor in a projection operator computed from a Dixon matrix. This estimate serves as an upper bound in the case when the polynomial system is generic.. The algorithm for constructing dialytic matrices based on the Dixon resultant formulation works in general for a polynomial system from which more than two variables need to be eliminated (even when the polynomial system is not necessarily unmixed). These dialytic matrices can be used to extract (in most cases) the resultants as determinants of their maximal minors. The approach generalizes a related method for constructing dialytic matrices from the Dixon resultant formulation discussed in Chtcherba and Kapur (2000b) . Beside being a generalization, the approach has the advantage of generating Sylvester-like dialytic matrices whose determinants are resultants even in cases where the earlier method of Chtcherba and Kapur produces an extraneous factor.. It is also shown that for the bivariate case, the proposed construction produces dialytic matrices with resultants as their determinants even in some mixed systems. The supports of the polynomial system are translated so that they have an nonempty intersection and then a term in the nonempty intersection of translated supports is used for constructing the dialytic matrix. This is formulated as an optimization problem that minimizes the size of the Dixon dialytic matrix. The approach is compared with other approaches, and is shown to be more efficient and to work better on many examples of practical interest.. The results reported in this paper also apply to scaled supports (Kapur and Saxena, 1997) . That is, a corner-cut support remains corner-cut even if it is scaled; similarly, if a support is not corner-cut, then any scaling of it will not produce a corner-cut support either.. 1 A polynomial system is unmixed if each of the polynomials in the system has exactly the same set of terms (with nonzero coefficients).. 2 A simplex corresponding to three terms The concept of corner-cut support does not generalize easily from bivariate systems to systems with more than two variables. For instance, the determinant of a Dixon dialytic matrix for a generic unmixed trivariate system with support in which box-like corners are removed from a box support 3 is shown to be not necessarily the resultant (the support in Fig. 1 is one such example). Nevertheless, preliminary results show that with a proper generalization, the results hold when more than two variables are eliminated. For more details, the reader can refer to Chtcherba and Kapur (2003) where a corner-cut support is generalized for the multivariate case; the Dixon based formulations compute the resultant exactly of a generic unmixed polynomial system with such a support. The concept of an almost corner-cut support is also introduced there; it is shown that the Dixon based formulations compute the resultant exactly for a generic unmixed polynomial with an almost corner-cut support provided that new variables are introduced in an appropriate order as constrained by the definition of an almost corner-cut support. However, unlike in the bivariate case, it cannot be shown even for the trivariate case that a support being cornercut is necessary for Dixon based formulations to compute the resultant exactly of a generic unmixed system. See also the conclusion section for more discussion on generalization to the multivariate case. Section 2 defines supports of a polynomial and a polynomial system, and reviews the BKK bound for toric roots and toric resultants of a polynomial system. Section 3 reviews the Dixon formulation of resultants, where the Dixon matrix and the Dixon polynomial of a given polynomial system are introduced. Section 3.1 analyzes the support of the Dixon polynomial in terms of the support of the polynomial system. It is shown that the support of the Dixon polynomial (which determines the size of the Dixon matrix) can be expressed as a union of the support of the Dixon polynomials of polynomial systems corresponding to simplexes (a simplex support has three distinct points).. Section 4 is a detailed analysis of the support of the Dixon polynomial in relation to the support of unmixed polynomial systems. For a simplex support, the support of its Dixon polynomial is precisely characterized in terms of the projection sum expressed in terms of the coordinates of the simplex. Points inside the convex hull of the support of a polynomial system are classified into two categories: (i) support hull interior points such that when terms corresponding to these support points are included in the polynomial system, the support of the Dixon polynomial and, hence, the size of the Dixon matrix do not change, (ii) other support points such that the corresponding terms when included in the polynomial system contribute to the extraneous factors in the projection operator computed from the associated Dixon matrix. Using these concepts, the notion of the support hull of a support is defined which includes, along with the support, all its support hull interior points. Using the support hull of the support of an unmixed polynomial system, the support of its Dixon polynomial is precisely characterized using the projection sum of the support.. The concept of a support complement characterizing how different a given support is from a bidegree support (in the case of bivariate systems) is introduced; this support complement can be partitioned into four corners. It is shown that for a given polynomial system, the support of its Dixon polynomial can be shown to be a rectangle (constructed from the bounding bidegree system) from which the four-corner support complement determined from the support of the polynomial system is removed. Thus the size of the Dixon matrix (which is the cardinality of the support of the Dixon polynomial) can be precisely determined on the basis of the size of the corners. It is also proved that if a term corresponding to a support hull interior point of a given support is generically included, the modified unmixed polynomial system will lead to the Dixon matrix of the same size as obtained from the original unmixed polynomial system.. We thus provide a geometric characterization of support points which do not contribute to the extraneous factor in the computation of a projection operator.. Section 5 has one of the main results of the paper. It is shown that if the support of the polynomial system after inclusion of its support hull interior points is a rectangle with four rectangular corners removed, then the size of the Dixon matrix is the same as the BKK bound; this implies that for generic unmixed polynomial systems with such supports, the Dixon formulation computes the resultant exactly. In contrast, Chionh (2001) proved that the Dixon formulation computes the exact resultant for generic unmixed polynomial systems whose support is a rectangle with four rectangular corners removed.. Section 6 proves a result about the degree of the extraneous factor in the projection operator computed using the Dixon resultant formulation for generic unmixed polynomial systems whose supports do not satisfy the above-stated condition.. In Section 7, these results are extended to Dixon dialytic matrices, which are Sylvestertype matrices but constructed using the Dixon formulation. Zhang and Goldman's results about corner-cut supports are shown to be a special case of our results discussed in Sections 5 and 7. It is shown that an obvious generalization of corner-cut supports does not work even for trivariate polynomial systems. This is followed by Section 8 discussing examples of unmixed systems and a comparison of different methods, including Canny and Emiris's subdivision algorithm (Canny and Emiris, 2000) , Zhang and Goldman's method (Zhang and Goldman, 2000) , D'Andrea and Emiris's method (D'Andrea and Emiris, 2001) as well as Khetan's method (Khetan, 2003) . The Dixon dialytic matrix method turns out to have many advantages over other methods for computing resultants.. Section 9 considers mixed polynomial systems. A heuristic for generating "good" Dixon dialytic matrices whose determinants are projection operators having extraneous factors of minimal degree is discussed. This heuristic utilizes terms common in the supports of the mixed polynomial system for generating the Dixon dialytic matrix. Supports are translated to maximize overlap among them. Determining how much supports ought to be translated as well as the term to be selected for generating the Dixon dialytic matrix can be formulated as an optimization problem, minimizing the support of the Dixon polynomial. An example illustrating this idea is discussed in detail. Section 10 compares our results experimentally with other approaches on examples of mixed polynomial systems.. Section 11 discusses issues for further investigation as well as possible generalization of these results to multivariate polynomial systems. We have identified a necessary and sufficient condition on the support of an unmixed bivariate polynomial system such that the methods based on the Dixon resultant formulation can compute its resultant exactly. When this cannot be done, the degree of the projection operator can be predicted, from which the degree of the extraneous factor appearing in it can be computed. Knowing the degree of the extraneous factor in a projection operator is helpful in identifying the resultant in the projection operator.. A method for computing the Dixon dialytic matrices based on the Dixon formulation is proposed. Unlike the Dixon matrices, the Dixon dialytic matrices are of Sylvester type in the sense that matrix entries are either 0 or coefficients of terms in the polynomial systems. These results are thus strict generalizations of the related results reported in Chionh (2001) , Zhang and Goldman (2000) and Zhang (2000) .. For mixed bivariate systems, heuristics are developed for translating supports and selecting a monomial for computing the Dixon dialytic matrices so that projection operators computed from these matrices are either resultants or, besides the resultants, they have extraneous factors of low degrees.. The above results still do not lead to precise identification of the extraneous factor, as is known in the case of eliminating a single variable. This issue will be investigated in future.. For a nongeneric polynomial system for which the upper bound on the number of toric roots is still the BKK bound, the degree of the extraneous factor in a projection operator cannot be estimated. It appears that the discrepancy between the BKK bound and the size of the Dixon matrix is due to the difference between the volume of the Newton polytope and the size of the corresponding support. Experimental evidence suggests that the coefficients of terms in polynomials also play a role in determining the support of the Dixon polynomial; this is also reflected in the formula for the Dixon polynomial based on the Cauchy-Binet formula. We are interested in analyzing whether the genericity requirements for obtaining the BKK bound are sufficient to preclude any role that the coefficients of terms in a polynomial system play in determining the support of the Dixon polynomial and, hence, the size of Dixon matrix.. The focus of this paper has been on bivariate systems. We have succeeded in generalizing these results, particularly the concepts of a support interior point, support hull, and corner-cut support in an arbitrary dimension. As illustrated above, it can be shown that the determinant of the Dixon dialytic matrix is not exactly the resultant for a trivariate generic unmixed system even if its support is corner-cut. 12 However, a necessary and sufficient condition on supports based on the exclusion of support interior points has been developed. As stated in the introduction, a straightforward generalization of a cornercut support to the arbitrary dimension does not work. An appropriate generalization of a multivariate corner-cut support is defined in Chtcherba and Kapur (2003) and it is shown that the Dixon based formulations compute the resultant exactly for generic unmixed polynomial systems with such supports. If an order can be imposed on the new variables introduced in the Dixon formulation, then the concept of an almost corner-cut support is introduced such that the Dixon based formulations compute the resultant exactly for generic unmixed polynomial systems with such supports; it is however crucial that Dixon matrices are constructed by substituting new variables in the particular order constrained by the almost corner-cut support.. There are many other aspects of the above Dixon based formulations which need further investigation, the most important ones being:. • finding a necessary and sufficient condition on the support of a generic unmixed polynomial system for which the Dixon based formulations compute the resultant exactly; • identifying the extraneous factor otherwise;. • performing a similar analysis for the mixed case, and, most importantly, • analyzing the dependence of the extraneous factor on the nongenericity of coefficients of terms in a polynomial system. As stated in the introduction, the results reported here are a generalization of results in Zhang (2000) , Chionh (2001) and Zhang and Goldman (2000) which characterize conditions on unmixed supports under which the determinant of the corresponding Dixon matrix is the resultant. A necessary and sufficient condition on the support of a generic unmixed polynomial system is given in the paper for which the determinants of the corresponding Dixon matrix as well as the Dixon dialytic matrix are precisely the result. A detailed comparison with the results in Zhang (2000) , Chionh (2001) and Zhang and Goldman (2000) is given in Sections 5 and 7.. In Sections 8 and 10, a comparison of different approaches including Canny and Emiris's subdivision algorithm (Canny and Emiris, 2000) , Zhang and Goldman's method (Zhang and Goldman, 2000) , D'Andrea and Emiris's method (D'Andrea and Emiris, 2002), as well as Khetan's method (Khetan, 2003) , with the Dixon dialytic method is given on a class of examples both with mixed and unmixed supports. The Dixon dialytic matrix method is shown to have many advantages over other methods for computing resultants, especially when a polynomial system is mixed.. During the review of this paper, we were made aware of Khetan (2002) in which a method for constructing a hybrid matrix of a generic unmixed bivariate polynomial system is given such that the determinant of such a matrix is the resultant of the polynomial system. Most of the rows in a hybrid matrix in this method are expressed in terms of brackets and are similar to the entries in a Dixon matrix obtained from the Dixon resultant formulation; some of its rows are however of Sylvester type, i.e., the entries are the coefficients of the polynomials in an original polynomial system. The size of Khetan's matrix is 4 A − B + 4, where A is the area of the convex hull of the support and B is the number of boundary points in the support. Whereas the Dixon construction yields smaller matrices in nongeneric cases (thus implicitly exploiting the specific coefficients of the terms in a given polynomial system), the hybrid matrices in Khetan (2003) appear to be the same irrespective of whether the coefficients are generic or specialized.. In Arries and Senoussi (2001) , a method for solving the implicitization problem for a surface defined using a rational parametrization without base points is proposed based on a construction from Salmon's book (Salmon, 1885) . The method works for four homogeneous polynomials in three variables; this problem can be reformulated in terms of three polynomials over two affine variables. This method also constructs hybrid matrices in which about half of the rows consist of entries which are the coefficients of the terms in a polynomial system and the remaining half (approximately) consist of entries similar to those in the corresponding Dixon matrix. A hybrid matrix constructed by the method is exact when all polynomials are all of the same degree and contain all the possible terms, and the parametrization of the surface is faithful; in the case where the parametrization is not faithful, the determinant of the hybrid matrix is then some power of the resultant of the polynomial system. However, the method cannot handle nongeneric cases, particularly when there are base points.

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