=Paper=
{{Paper
|id=Vol-2445/paper_5
|storemode=property
|title=InfOCF-Lib: A Java Library for OCF-based Conditional Inference
|pdfUrl=https://ceur-ws.org/Vol-2445/paper_5.pdf
|volume=Vol-2445
|authors=Steven Kutsch
|dblpUrl=https://dblp.org/rec/conf/ki/Kutsch19
}}
==InfOCF-Lib: A Java Library for OCF-based Conditional Inference==
https://ceur-ws.org/Vol-2445/paper_5.pdf
Copyright c 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
InfOCF-Lib: A Java Library for OCF-based
Conditional Inference
Steven Kutsch
Dept. of Computer Science, FernUniversität in Hagen, Hagen, Germany
steven.kutsch@fernuni-hagen.de
Abstract. Conditionals of the form “If A, then usually B” are often
used to define nonmonotonic inference relations. Typically, a finite set of
conditionals R, called a knowledge base, is inductively completed to an
inference relation containing all the explicit and implicit beliefs of an in-
telligent agent. One way of representing such a complete epistemic state
is by a ranking function, assigning degrees of disbelief to propositional
interpretations. Each ranking function defines a nonmonotonic inference
relation and each set of ranking functions defines several inference re-
lations by employing different modes of inference. We propose the Java
library InfOCF-Lib that is capable of reading a knowledge base from a
file, calculating various sets of ranking functions proposed in the litera-
ture, and answering queries of the form “Does A entail B in the context
of the knowledge base R using the set of ranking functions M?”.
Keywords: conditional logic, conditional knowledge base, ranking func-
tion, nonmonotonic inference, Java
1 Introduction
Rules with possible exceptions, i.e. statements of the form “if A, then usually
B”, play a prominent role in the area of knowledge representation and reason�
ing. Several semantics have been proposed for defining nonmonotonic inference
relations over sets of such rules. Examples are Lewis’ system of spheres [17],
conditional objects evaluated using Boolean intervals [9], or possibility measures
[8, 10].
Here, we will consider Spohn’s ranking functions [20] that assign degrees of
disbelief to propositional interpretations. C-Representations [13, 14] are a subset
of all ranking functions for a knowledge base, which can conveniently be calcu�
lated by solving a constraint satisfaction problem dependent on the knowledge
base. A first Prolog implementation of this approach was introduced in [4].
In [3], the tool InfOCF was introduced, that allows the user to load knowledge
bases, calculate admissible ranking functions (in particular c-representations and
system Z [19]), and perform inference using these sets of ranking functions.
System P entailment [1] was also implemented. For calculating c-representations,
InfOCF interfaced with a Prolog component akin to the one introduced in [4].
47
While InfOCF supports many tasks when working with conditional knowledge
bases, the need for new features or variants of old features arose frequently, as
new use cases became relevant in the active research of conditionals and inference
relations defined by them. Since it is impossible to foresee every possible use case
for a tool such as InfOCF, and because the architecture of InfOCF did not support
a practical API, we developed the library InfOCF-Lib, that allows the user to
quickly write personal tools for any use case, using the core functionalities of
InfOCF. Thus, InfOCF-Lib provides a redesign of InfOCF that is more easily
extensible and provides a convenient API. InfOCF-Lib can be accessed online1
and comes with a detailed user manual.
Other implementations for conditional inference have been proposed. Z-log
[18] offers inference based on system Z. The Java library for conditional logic of
The Tweety Project2 offers some limited capabilities for calculating c-represen�
tations, i.e. calculation of a single c-representation for a given knowledge base,
as well as System Z inference. Since the implementation of rank based inference
in The Tweety Project, several new theoretical approaches have been proposed,
such as different notions of minimal c-representations [2] and inference using
sets of ranking functions [5, 7]. These new approaches have been implemented in
InfOCF-Lib.
2 Background
2.1 Conditional Logic and OCFs
Let L be a propositional language over a finite set ⌃ of atoms a1 , . . . , an . The
formulas of L will be denoted by letters A, B, C, . . . . We write AB for A ^ B and
A for ¬A. We identify the set of all complete conjunctions over ⌃ with the set
⌦ of possible worlds over L. For ! 2 ⌦, ! |= A means that A 2 L holds in !.
A conditional (B|A) formalises a statement of the form “if A then usually
B” and establishes a plausible connection between the antecedent A and the
consequence B. It partitions the set of worlds ⌦ in three parts: those worlds
satisfying AB, thus verifying the conditional, those worlds satisfying AB, thus
falsifying the conditional, and those worlds not fulfilling the premise A and to
which the conditional may not be applied to at all. This allows us to associate to
(B|A) a generalised indicator function (B|A) going back to [11] (where u stands
for unknown or indeterminate):
8
<1 if ! |= AB
(B|A) (!) = 0 if ! |= AB (1)
:
u if ! |= A
InfOCF-Lib uses Spohn’s ranking functions (also called ordinal conditional
functions, OCFs) [20]. A ranking function is a function : ⌦ ! N expressing
1
www.fernuni-hagen.de/wbs/data/InfOCF-Lib.zip
2
tweetyproject.org
48
degrees of plausibility of possible worlds where a higher degree denotes “less
plausible” or “more surprising”. At least one world must be regarded as being
normal; therefore, (!) = 0 for at least one ! 2 ⌦. Each such can be taken as
the representation of a full epistemic state of an agent, and it uniquely extends
to a function (also denoted by ) mapping sentences to N [ {1} by:
(
min{(!) | ! |= A} if A is satisfiable
(A) = (2)
1 otherwise
A ranking function accepts a conditional (B|A) (denoted by |= (B|A))
if the verification of the conditional is less surprising than its falsification, i.e.,
if (AB) < (AB). Every ranking function gives rise to a nonmonotonic
inference relation |⇠ defined as
A |⇠ B iff A ⌘ ? or (AB) < (AB) (3)
A non-empty finite set of conditionals R is called a knowledge base. An OCF
accepts R if accepts all conditionals in R, and R is consistent if an OCF
accepting R exists [12]. To test a conditional knowledge base for consistency, an
algorithm using the notion of tolerance has been proposed.
Definition 1 (Tolerance [12]). Let R = {(B1 |A1 ), . . . , (Bn |An )} be a knowl-
edge base and r = (B|A) a conditional. The knowledge base R tolerates the
conditional r if there is an interpretation ! such that
^
! |= AB ^ (Ai _ Bi ) (4)
16i6n
The algorithm OrderedPartition (Listing 1.1) calculates the inclusion maximal
ordered partition Rp = (R0 , . . . , Rk ) such that for every 0 6 i 6 k, every
Sk
r 2 Ri is tolerated by j=i Rj . Since OrderedPartition returns NULL if R is
inconsistent, it can be used as a consistency test [12].
2.2 Inference Modes for Sets of Ranking Functions
Typically, there is more then one valid viewpoint when representing a domain
that involves uncertainty. We can model reasoning with multiple viewpoints
(ranking functions) by performing skeptical, weakly skeptical [2] or credulous
inference over a set of ranking functions M .
Definition 2. Let R be a knowledge base, M a set of ranking functions accepting
R, and A and B be formulas.
sk,M
B is a skeptical inference over M from A, denoted by A |⇠R B, if for all
2 M it holds that A |⇠ B.
cr,M
B is a credulous inference over M from A, denoted by A |⇠R B, if there is
a 2 M such that A |⇠ B holds.
ws,M
B is a weakly skeptical inference over M from A, denoted by A |⇠R B, if
there is a 2 M such that A |⇠ B holds and there is no 2 M such that A |⇠ B
holds.
49
Listing 1.1. Algorithm to test for consistency of R (cf. [12]).
PROCEDURE : OrderedPartition
INPUT : Knowledge base R={(B1 |A1 ),...,(Bn |An )}
OUTPUT : Ordered partition (R0 , R1 , . . . , Rk ) if R is consistent ,
NULL otherwise
INT i :=0;
WHILE (R 6= ;) DO
Ri := { (B|A) 2 R | R tolerates (B|A) };
IF (Ri 6= ;)
THEN
R:=R \ Ri ;
i := i +1;
ELSE
RETURN NULL ; // R is inconsistent
RETURN Rp = (R0 , . . . , Rk );
2.3 C-Representations
In this section we will discuss c-representations [13, 14] that assign an individual
impact ⌘i to each conditional (Bi |Ai ) and generate the world ranks as a sum of
impacts of falsified conditionals:
Definition 3 (c-representation [13, 14]). A c-representation of a knowledge
base R = {(B1 |A1 ), . . . , (Bn |An )} is an OCF constructed from non-negative
integer impacts ⌘i 2 N0 assigned to each conditional (Bi |Ai ) such that accepts
R and is given by:
X
(!) = ⌘i (5)
16i6n
!|=Ai B i
C-representations can conveniently be specified using a constraint satisfaction
problem (for detailed explanations, see [13, 14, 5]):
Definition 4 (CR(R) [4]). Let R = {(B1 |A1 ), . . . , (Bn |An )}. The constraint
satisfaction problem for c-representations of R, denoted by CR(R), on the con-
straint variables ⌘1 , . . . , ⌘n ranging over N is given by the conjunction of the
constraints, for all i 2 {1, . . . , n}:
X X
⌘i > min ⌘j min ⌘j (6)
!|=Ai Bi !|=Ai Bi
j6=i j6=i
!|=Aj Bj !|=Aj Bj
A solution of CR(R) is a vector #»
⌘ = (⌘1 , . . . , ⌘n ) of n natural numbers. With
Sol (CR(R)) we denote the set of solutions of CR(R). For #» ⌘ 2 Sol (CR(R)) and
as in Equation (5), is the OCF induced by #» ⌘ , denoted by #»⌘.
50
Since there are in general infinitely many solutions of CR(R), three notions
of minimality of impact vectors have been proposed [2].
Definition 5 (sum-, cw- and ind-minimality [2]). Let R be a knowledge
base, and #» ⌘ 2 Sol (CR(R)). Then
⌘ , #»
X X
⌘ 4+ #»
#» ⌘ 0 iff ⌘i 6 ⌘0 i (7)
16i6n 16i6n
#» ⌘ 4+ #»
⌘ is sum-minimal iff #» ⌘ 0 for all #»
⌘ 0 2 Sol (CR(R)). We write #»
⌘ #»0 iff
+ ⌘
⌘ 4 ⌘ and ⌘ 64 ⌘ .
#»
+
#»0 #» 0 #»
+
⌘ 4cw #»
#» ⌘0 iff ⌘i 6 ⌘ 0 i for all i 2 {1, . . . , n} (8)
and ⌘i < ⌘i0 for some i 2 {1, . . . , n}
⌘ is cw-minimal (or Pareto-minimal) iff there is no vector #»
#» ⌘ 0 2 Sol (CR(R))
such that ⌘ 4cw ⌘ and ⌘ 64cw ⌘ .
#»0 #» #» #» 0
⌘ 4O #»
#» ⌘0 iff ⌘ (!) 6 #»
#» ⌘ 0 (!) for all ! 2 ⌦ (9)
⌘ is ind-minimal iff there is no vector #»
#» ⌘ 0 4O #»
⌘ 0 2 Sol (CR(R)) such that #» ⌘
and ⌘ 64 ⌘ .
#»
O
#»0
Since none of the three order relations is total, there are in general multiple
sum-, cw-, and ind-minimal c-representations for a given knowledge base. We can
therefor perform skeptical, credulous and weakly skeptical inference over four sets
of c-representations: all solutions of CR(R), or only sum-, cw-, or ind-minimal
solutions.
3 Architecture
In this section we will detail the architecture of InfOCF-Lib. Since the imple�
mentation of the answering of queries using finte sets of ranking functions is
straightforward (loop over ranking functions and check rank of verification and
falsification), we will not go into detail about the concrete implementation of
these aspects of the library.
The representation of formulas and elements of classical propositional logic is
also straightforward and is detailed in the manual that comes with InfOCF-Lib.
Here we will detail the architecture of the non classical elements of the library.
Figure 1 shows the package conditionalLogic containing the two classes
Conditional and ConditionalKnowledgeBase. Conditionals are composed of
two propositional formulas. The evaluation of conditionals (B|A) is realised in
the method conditionalIndicatorFunction that evaluates the conditional in
an interpretation ! according to the generalised indicator function (B|A) (!)
(cf. (1)). Knowledge bases are collections of conditionals together with a name
and a signature. Since InfOCF-Lib uses Jasper3 to interface with a SICStus
3
sicstus.sics.se/sicstus/docs/4.1.0/html/sicstus/lib_002djasper.html
51
propositionalLogic
conditionalLogic
Conditional
antecedent : PropositionalFormula
consequent : PropositionalFormula
conditionalIndicatorFunction(PropositionalInterpretation w)
: ConditionalIndicatorValue
0..*
0..*
ConditionalKnowledgeBase
name : String
signature : PropositionalSignature
toPrologString() : String
orderedPartition() : List>
isConsistent() : Boolean
Fig. 1. The package conditionalLogic containing the classes Conditional and
ConditionalKnowledgeBase.
Prolog component, knowledge bases need to be translatable to Prolog programs.
The method orderedPartition calculates the ordered partition according to
Algorithm 1.1. The consistency test realised in the method isConsistent tries to
construct the order partition and returns true or false depending of the existence
of the ordered partition.
Figure 2 shows the essential components of InfOCF-Libs representation of
ranking functions and sets of ranking functions. As defined in Section 2.1, ranking
functions assign integers to every interpretation in a propositional universe (set
of all interpretations for a fixed signature). The interpretations for a signature
are ordered according to [6]. This allows the correct association with the ranks.
C-Representations (cf. Section 2.3) are ranking functions that assign the rank
of interpretations due to falsified conditionals and their impacts. They therefore
always need to know the knowledge base R that they are accepting, together
with the impacts, which are obtained from solving CR(R) (see Definition 4).
The solving of this constraint problem is implemented in Prolog [3].
In order to represent inference using sets of ranking functions, as defined in
Section 2.2, a RankingFunctionSet represents a set of ranking functions that ac�
cept a common knowledge base. The ranking functions in a RankingFunctionSet
52
ocf
RankingFunction
signature : PropositionalSignature
universe : PropositionalUniverse
ranks : ArrayList
getInterpretationRank(PropositionalInterpretation w) : Integer
getFormulaRank(PropositionalFormula w) : Integer
0..*
RankingFunctionSet
0..*
acceptedKB : ConditionalKnowledgeBase
CRepresentation
context : ConditionalKnowledgeBase
impacts : ArrayList
Fig. 2. The package ocf containing the classes RankingFunction, CRepresentations
and RankingFunctionSet.
also share a common ModelSetKind. The Enum ModelSetKind identifies the fol�
lowing sets of ranking functions:
– all c-representations
– ind-minimal c-representations
– cw-minimal c-representations
– sum-minimal c-representations
– system Z [19]
– all ranking functions
While the construction of a RankingFunctionSet using a ModelSetKind rep�
resenting a set of c-representations always result in a call to the Prolog compo�
nent for solving CR(R), the RankingFunctionSet for ModelSetKind represent�
ing system Z or all ranking functions are calculated internally. If the construc�
tor of RankingFunctionSet is called with the ModelSetKind representing all
ranking functions, no ranking functions are actually calculated. The resulting
RankingFunctionSet simply serves as a placeholder object for defining system
P inference, which can be characterised as skeptical inference over all ranking
functions [16].
Objects of the class RankedInferenceRelation (shown in Figure 3) are con�
structed from
– a set of ranking functions, and
53
inference
RankedInferenceRelation
context : ConditionalKnowledgeBase
rankingModels : RankingFunctionSet
mode : InferenceMode
contains(Conditional c) : Boolean
Fig. 3. The class RankedInferenceRelation representing inference relations defined
over sets of ranking functions.
– a mode of inference (skeptical, weakly skeptical or credulous) (cf. Def. 2).
The context for the inference relation is clear, since the set of ranking func�
tions accepts a common knowledge base. A ranked inference relation represents a
set of conditionals, namely all conditionals entailed from the knowledge base by
performing inference over the ranking function set with the specified mode of in�
ference. Therefore, the primary method in the class RankedInferenceRelation
is contains, that checks whether the supplied conditional is contained in the
inference relation, by performing inference.
If a RankedInferenceRelation is constructed with a RankingFunctionSet
with the ModelSetKind for all ranking functions, inference is realised by negating
the query conditional r and checking for the consistency of R [ {r} [9].
4 Examples
To give an impression of how InfOCF-Lib works, we illustrate the usage with
two examples. The first loads a knowledge base from a file, calculates the set
of ind-minimal c-representations for the knowledge base and answers a set of
queries, also loaded from a file. In the second example, InfOCF-Lib is used to
load a set of knowledge bases from a file and check every knowledge base for
consistency and print the name of every inconsistent knowledge base to the
console.
4.1 Loading a Knowledge Base and Answering Queries with
Respect to Minimal C-Representations
The knowledge base we want to load is stored in the file birds.cl with the
following content:
signature
b,p,f
54
conditionals
birds001{
(f | b), // birds usually fly
(b | p), // penguins are birds
(!f| p) // penguins don’t fly
}
In this file, formulas are build up from propositional variables introduced
under the keyword signature. The junctors and (comma), or (semicolon) and
not (exclamation point) can be used.
To load a file containing a knowledge base, first the content of the file needs
to be read into a string using Java’s standard functionalities for reading files. We
call this string birdsString. This string can then be parsed using the functions
InfOCF-Lib provides.
String birdsString = new String ( Files . readAllBytes ( Paths . get (
" birds . cl " ) ) ) ;
List < ConditionalKnowledgeBase > kbs =
p a r s e C o n d i t i o n a l K n o w l e d g e B a s e ( birdsString ) ;
C o nd i t io n a lK n o wl e d ge B a se birds = kbs . get (0) ;
The method parseConditionalKnowledgeBase returns a list of knowledge
bases, since .cl-files can contain multiple knowledge bases. We will see the use
of these lists in the next example. Here we just select the first and only knowl�
edge base in the file. To answer some queries in the context of this knowledge
base, we need to define a RankedInferenceRelation. Inference relations are de�
fined using sets of ranking functions that accept a common knowledge base. To
generate the set of ind-minimal c-representations accepting the knowledge base
birds, we simply pass the knowledge base and the parameter specifying ind-min�
imal c-representations to the constructor of the class RankingFunctionSet. The
skeptical inference relation over the ind-minimal c-representations accepting the
knowledge base birds can then be defined.
RankingFunctionSet in dM inC Rep Ra nki ng Mod el =
new RankingFunctionSet ( birds , ModelSetKind . CREP_IND ) ;
R an ke dI n fe re nc e Re la t io n indMinCRepInference =
new Ra nk ed I nf er en c eR el at i on ( indMinCRepRankingModel ,
InferenceMode . SKEPTICAL ) ;
In this example, we want to answer the queries in the file birdsQueries.cl.
The file contains a comma separated list of conditionals:
(b | p),
(f | p),
(f | p,b)
We can load this list of conditionals in the same way we loaded the knowl�
edge base by reading the content into a string and parsing that string with
InfOCF-Lib:
55
String queriesString = new String ( Files . readAllBytes ( Paths .
get ( " birdsQueries . cl " ) ) ) ;
List < Conditional > queries = p a r s e C o n d it i o n a l Q u e r y Li s t (
queriesString ) ;
We can now iterate over our queries and answer each query using our
RankedInferenceRelation:
for ( Conditional query : queries ) {
System . out . print ( query . toString () + " : " ) ;
if ( indMinCRepInference . contains ( query ) ) {
System . out . println ( " 1 " ) ;
} else {
System . out . println ( " 0 " ) ;
}
}
The code above produces the following output:
( b | p ) : 1
( f | p ) : 0
( f | (b,p) ) : 0
4.2 Checking a Set of Knowledge Bases for Consistency
Here we again load knowledge bases from a file. The file birds2.cl contains
multiple knowledge bases:
signature
b,p,f
conditionals
birds001{
(f | b), // birds usually fly
(b | p), // penguins are birds
(!f| p) // penguins don’t fly
}
birds002{
(f | b), // birds usually fly
(!f | b) // birds usually don’t fly
}
As mentioned above, we will check every knowledge base in this file for con�
sistency, and output the names of the inconsistent knowledge bases.
String birdsString = new String ( Files . readAllBytes ( Paths . get (
" birds2 . cl " ) ) ) ;
List < ConditionalKnowledgeBase > kbs =
p a r s e C o n d i t i o n a l K n o w l e d g e B a s e ( birdsString ) ;
56
Iterating over the list of knowledge bases read from the file, we can call the
method isConsistent for every knowledge base and identify the inconsistent
knowledge bases:
for ( Co n d it i o na l K no w l ed g e Ba s e kb : kbs ) {
if (! kb . isConsistent () )
System . out . println ( kb . getName () + " is inconsistent " ) ;
}
producing the output
birds002 is inconsistent
5 Conclusions and Future Work
In internal use, InfOCF-Lib has proven to be a useful and effective library for
working with conditionals and ranking functions. New use cases (e.g. [15]) are
quickly realised and experiments using conditional knowledge bases can be de�
signed and conducted easily.
As InfOCF-Lib is still in development, many theoretical efficiency benefits
are not fully implemented jet. A thorough evaluation of the efficiency of the
implementation is part of our future work. We are currently also working on a
web based interface for InfOCF-Lib, that offers convenient access to the most
basic features.
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