Answer-set programming

ASP
Rules and Ontologies
Debugging in ASP

Answer-Set Programming
Basics, Combinations with Ontologies, and Debugging

Jörg Pührer

Knowledge Based Systems Group, Institute for Information Systems
Vienna University of Technology

Partially supported by the European Commission under
FP7 Project ONTORULE, Marie Curie Actions Project NET2,
and the Austrian Science Fund (FWF) P21698

Jörg Pührer Answer-Set Programming 1 / 71

ASP Paradigm
Rules and Ontologies Syntax and Semantics
Debugging in ASP Applications

Context
We deal with answer-set programming (ASP)
Programming paradigm


ASP Paradigm

Context
Term coined by Lifschitz (1999)
Proposed by others people (Marek and Truszczyński, Niemelä)
at about the same time


ASP Paradigm

Context
Fruitful approach for declarative problem solving


ASP Paradigm

Context
Fruitful approach for declarative problem solving

Problem

?

Solution


ASP Paradigm

Declarative vs. Procedural Programming

In procedural programming, a problem is solved by writing a
program that corresponds to a recipe.
General Questions:
What steps do I need to take to solve the problem?
How do I solve it?


ASP Paradigm

Declarative vs. Procedural Programming

In procedural programming, a problem is solved by writing a
program that corresponds to a recipe.
General Questions:
What steps do I need to take to solve the problem?
How do I solve it?

Problem
Step 1
Program Recipe ...

Step n
Solution


ASP Paradigm

Declarative vs. Procedural Programming (ctd.)
In declarative programming, a problem is solved by writing a
program that corresponds to a description of the problem.
General Question:
What is the problem?


ASP Paradigm

Declarative vs. Procedural Programming (ctd.)
In declarative programming, a problem is solved by writing a
program that corresponds to a description of the problem.
General Question:
What is the problem?

Problem
Property 1
Program Description ...

Property n
Solution

Control ﬂow is not provided in the program.
No algorithm is given for solving the problem.


ASP Paradigm

ASP Big Picture
In Answer Set Programming (ASP), the problem is encoded in
terms of a logic theory - typically using logic programming rules.
Answer-Set Solver computes dedicated models of the theory.
They are called answer sets.


ASP Paradigm

ASP Big Picture
In Answer Set Programming (ASP), the problem is encoded in
terms of a logic theory - typically using logic programming rules.
Answer-Set Solver computes dedicated models of the theory.
They are called answer sets.
Correspond one-to-one to the (multiple) solutions of the
problem.
Problem
H1 ← B 1
Answer-Set Program ...

Hn ← B n
Solver

Answer Set 1 ... Answer Set m

Solution 1 ... Solution m


ASP Paradigm

Answer Sets
Multiple Answer Sets:
Non-monotonic reasoning (use of default negation)

Ability to express non-determinism (guessing)

Sometimes useful: Preference relations between answer sets
Most used semantics:
Stable model semantics ( Answer-Set semantics)

Diﬀerent modes of reasoning:
Model generation: compute 1, n, or all answer sets
Cautious reasoning: query holds in all answer sets
Brave reasoning: there is an answer set for which query holds


ASP Paradigm

Syntax

We deal with disjunctive logic programs (DLPs) under the
answer-set semantics, which are sets of rules of form:
a1 ∨ · · · ∨ al ← al+1 , . . . , am , not am+1 , . . . , not an
it must hold that n ≥ 1
all ai are literals over some ﬁxed function-free ﬁrst-order
language L
“not” denotes default negation (negation as failure)
a (strong) literal is either an atom or an atom preceded by
strong negation, −.


ASP Paradigm

Syntax (ctd.)
We deal with rules of form:
a1 ∨ · · · ∨ al ← al+1 , . . . , am , not am+1 , . . . , not an

We distinguish the following parts of rule r of the form above:
the head H(r ) = {a1 , · · · , al },
the body B(r ) = {al +1 , . . . , am , not am+1 , . . . , not an },
the positive body B+ (r ) = {al +1 , . . . , am }, and
the negative body B− (r ) = {am+1 , . . . , an }.

A rule r is
a fact if B(r ) = 0,
/
a constraint if H(r ) = 0,
/
normal if |H(r )| ≤ 1,
positive if B− (r ) = 0, or
/
Horn if it is both, normal and positive.

ASP Paradigm

Semantics

The answer-set semantics for disjunctive logic programs was
proposed by Gelfond and Lifschitz (1991).

Deﬁnition based on ground programs:
Rules do not contain variables

Semantics for a program P with variables based on the
program’s grounding gr (P):
Replace each non-ground rule r by all ground rules that can be
obtained from r by substituting variables by constants in the
Herbrand Universe of the program.


ASP Paradigm

Semantics (ctd.)

An interpretation is a consistent set I of ground literals
a is true iﬀ a ∈ I.


ASP Paradigm

Semantics (ctd.)

An interpretation is a consistent set I of ground literals
a is true iff a ∈ I.

For a ground rule r ,
I |= B(r ) iff B+ (r ) ⊆ I and B− (r ) ∩ I = 0;
/

I |= H(r ) iff H(r ) ∩ I = 0;
/

I |= r iff (I |= B(r ) implies I |= H(r ))

I is a model of a ground program P iff I |= r for all r ∈ P.


ASP Paradigm

Semantics (ctd.)

The reduct of a ground program P w.r.t. I is the program

P I := {H(r ) ← B+ (r ) | r ∈ P and I ∩ B− (r ) = 0}
/


ASP Paradigm

Semantics (ctd.)

The reduct of a ground program P w.r.t. I is the program

P I := {H(r ) ← B+ (r ) | r ∈ P and I ∩ B− (r ) = 0}
/

I is an answer set of a program P if it is the minimal model of
gr (P)I

By AS(P), we denote the set of all answer sets of P.


ASP Paradigm

Semantics (ctd.)

Many alternative deﬁnitions
By means of propositional formulas (Lin and Zhao, 2004)

In terms of equilibrium logic and logic of here-and-there
(Pearce, 1996)
Using the FLP-reduct PFLP instead of P I (Faber, Leone, and
I
Pfeifer, 2004):
I is an answer set of a program P if it is a minimal model of
the rules r ∈ gr (Π) such that I |= B(r ).
Allows for intuitive handling of recursive aggregates.

...


ASP Paradigm

Small example

P ={ a(X ) ←not e(X ), d(X ),
e(X ) ←not a(X ), d(X ),
c(X , Y ) ←a(X ), a(Y ), not −b(X ),
d(1) ←,
d(2) ←}.


ASP Paradigm

Small example

P ={ a(X ) ←not e(X ), d(X ),
e(X ) ←not a(X ), d(X ),
c(X , Y ) ←a(X ), a(Y ), not −b(X ),
d(1) ←,
d(2) ←}.

The grounding of P is given by
gr (P) = { a(1) ←not e(1), d(1), a(2) ←not e(2), d(2),
e(1) ←not a(1), d(1), e(2) ←not a(2), d(2),
c(1, 1) ←a(1), a(1), not −b(1), c(2, 2) ←a(2), a(2), not −b(2),
c(1, 2) ←a(1), a(2), not −b(1), c(2, 1) ←a(2), a(1), not −b(2),
d(1) ←, d(2) ←}.


ASP Paradigm

Small example (ctd.)
P ={ a(X ) ←not e(X ), d(X ),
e(X ) ←not a(X ), d(X ),
c(X , Y ) ←a(X ), a(Y ), not −b(X ),
d(1) ←,
d(2) ←}.

Consider the interpretation I1 = {d(1), d(2)}

gr (P)I1 = { a(1) ←not e(1), d(1),
FLP a(2) ←not e(2), d(2),
e(1) ←not a(1), d(1), e(2) ←not a(2), d(2),
(
((( (
(((
((((( −b(1),
←
c(1, 1) a(1), a(1), not ←
((((( −b(2),
c(2, 2) a(2), a(2), not
(
((( (
(((
((((( −b(1),
←
c(1, 2) a(1), a(2), not ((((( −b(2),
←
c(2, 1) a(2), a(1), not
d(1) ←, d(2) ←}.

I1 is not a model of gr (P)I1 .
FLP

ASP Paradigm

P ={ a(X ) ←not e(X ), d(X ),
e(X ) ←not a(X ), d(X ),
c(X , Y ) ←a(X ), a(Y ), not −b(X ),
d(1) ←,
d(2) ←}.

Consider the interpretation I2 = {d(1), d(2), e(1), a(2), e(2), c(2, 2)}

gr (P)I2 = { not e(1), d(1),
( (
FLP a(1) ←((((( ←((((
a(2) not e(2), d(2),
(
(
e(1) ←not a(1), d(1), e(2) ←(((((
not a(2), d(2),
(
(((
←((((( −b(1),
c(1, 1) a(1), a(1), not c(2, 2) ←a(2), a(2), not −b(2),

(
((( (
(((
←((((( −b(1),
c(1, 2) a(1), a(2), not ←
((((( −b(2),
c(2, 1) a(2), a(1), not

d(1) ←, d(2) ←}.

I2 is a model of gr (P)I2
FLP .

ASP Paradigm

P ={ a(X ) ←not e(X ), d(X ),
e(X ) ←not a(X ), d(X ),
c(X , Y ) ←a(X ), a(Y ), not −b(X ),
d(1) ←,
d(2) ←}.

Consider the interpretation I2 = {d(1), d(2), e(1), a(2), e(2), c(2, 2)}

gr (P)I2 = { not e(1), d(1),
( (
FLP a(1) ←((((( ←((((
a(2) not e(2), d(2),
(
(
e(1) ←not a(1), d(1), e(2) ←(((((
not a(2), d(2),
(
(((
←((((( −b(1),
c(1, 1) a(1), a(1), not c(2, 2) ←a(2), a(2), not −b(2),

(
((( (
(((
←((((( −b(1),
c(1, 2) a(1), a(2), not ←
((((( −b(2),
c(2, 1) a(2), a(1), not

d(1) ←, d(2) ←}.

I2 is a model of gr (P)I2 but not minimal.
FLP

ASP Paradigm

P ={ a(X ) ←not e(X ), d(X ),
e(X ) ←not a(X ), d(X ),
c(X , Y ) ←a(X ), a(Y ), not −b(X ),
d(1) ←,
d(2) ←}.

Consider the interpretation I3 = {d(1), d(2), e(1), a(2), c(2, 2)}

gr (P)I3 = { not e(1), d(1),
(
FLP a(1) ←((((( a(2) ←not e(2), d(2),
(
e(1) ←not a(1), d(1), e(2) ←(((((
not a(2), d(2),
(
(((
←((((( −b(1),
c(1, 1) a(1), a(1), not c(2, 2) ←a(2), a(2), not −b(2),

(
((( (
(((
((((( −b(1),
←
c(1, 2) a(1), a(2), not ((((( −b(2),
←
c(2, 1) a(2), a(1), not
d(1) ←, d(2) ←}.


ASP Paradigm

P ={ a(X ) ←not e(X ), d(X ),
e(X ) ←not a(X ), d(X ),
c(X , Y ) ←a(X ), a(Y ), not −b(X ),
d(1) ←,
d(2) ←}.

Consider the interpretation I3 = {d(1), d(2), e(1), a(2), c(2, 2)}

gr (P)I3 = { not e(1), d(1),
(
FLP a(1) ←((((( a(2) ←not e(2), d(2),
(
e(1) ←not a(1), d(1), e(2) ←(((((
not a(2), d(2),
(
(((
←((((( −b(1),
c(1, 1) a(1), a(1), not c(2, 2) ←a(2), a(2), not −b(2),

(
((( (
(((
((((( −b(1),
←
c(1, 2) a(1), a(2), not ((((( −b(2),
←
c(2, 1) a(2), a(1), not
d(1) ←, d(2) ←}.

I3 is a minimal model of gr (P)I3 , hence I3 is an answer set of P.
FLP

ASP Paradigm


P ={ a(X ) ←not e(X ), d(X ),
e(X ) ←not a(X ), d(X ),
c(X , Y ) ←a(X ), a(Y ), not −b(X ),
d(1) ←,
d(2) ←}.

P has the following answer sets:
{d(1), d(2), a(1), a(2), c(1, 1), c(1, 2), c(2, 1), c(2, 2)},
{d(1), d(2), a(1), e(2), c(1, 1)},
{d(1), d(2), e(1), a(2), c(2, 2)},
{d(1), d(2), e(1), e(2)}


ASP Paradigm

Uniform Problem Encoding

Typically, answer-set programs are uniform problem encodings.
The program itself encodes the general problem,
whereas individual problem instances represented by facts can
be joined with the program on demand.

Problem Instance General Problem
F1 ← H1 ← B 1
... Set of Facts ∪ Answer-Set Program ...

Fl ← Hn ← B n
Solver

Answer Set 1 ... Answer Set m

Solution 1 ... Solution m


ASP Paradigm

Graph Colouring
Uniform problem encoding:
P1 = {another _col(V , C ) ←vertex (V ), col(C ), col(D),
col_of (V , D), C = D,
col_of (V , C ) ←vertex (V ), col(C ),
not another _col(V , C ),
←vertex (U), vertex (V ), edge(U, V ),
col_of (U, C ), col_of (V , C )}.
Input instance as set of facts:
F1 = {col(red), col(green), col(blue),
vertex (a), vertex (b), vertex (c), vertex (d), vertex (e),
edge(a, b), edge(a, c), edge(a, d),
edge(b, e), edge(c, d), edge(d, e)}.


ASP Paradigm

Graph Colouring (ctd.)

The answer sets of P1 ∪ F1 correspond to the solutions of the
problem instance.

{col_of (a, red), col_of (b, blue), col_of (c, blue), col_of (d, green), col_of (e, red), . . . }


ASP Paradigm

Use
Complexity captured by uniform problem encodings:
normal logic programs: NP-hard problems
disjunctive logic programs: ΣP -hard problems
2

Applications of ASP
configuration, computer-aided verification, software testing,
health care, bio informatics, planning, music composition,
database repair, Semantic-Web reasoning, . . . .

Strengths:
Concise representation, elaboration tolerance,
non-determinism, use of recursion
Efficient solvers:
Clasp (University of Potsdam)
DLV (University of Calabria, Vienna University of Technology)


ASP Paradigm

ASP Language Extensions

Many extensions have been proposed, partly motivated by
applications
Some are syntactic sugar, other strictly add expressiveness
Incomplete list:
optimisation: minimize/maximize statements, weak constraints
aggregates, cardinality constraints
templates (for macros)
function symbols (undecidability)
Frame Logic syntax
KR frontends (diagnosis, planning,...) in DLV
higher-order features (variables as predicates)
external atoms


ASP ONTORULE
Rules and Ontologies Ontologies and Rules
Debugging in ASP DL-Programs

ONTORULE ONTOR UL E

Project ONTORULE Ontologies meet Business Rules

Main Objectives:
Acquisition of ontologies and rules from the most appropriate
sources, including natural language documents
their separate management and maintenance
their transparent operationalisation in IT applications.
Consortium:
IBM France (Coordinator), Ontoprise, PNA, Fundación CTIC,
Audi, ArcelorMittal,
Free University of Bolzano, Vienna University of Technology,
Université Paris 13
Partially funded by the European Union’s 7th Framework
Programme (ICT-231875).

ASP ONTORULE

ONTORULE (ctd.) ONTOR UL E

Ontologies meet Business Rules
Project ONTORULE

Business Documents

Acquisition (e.g., NLP)

Business Vocabulary
Business Rules

Production Rules
OWL Ontology Rules
Logical Rules


ASP ONTORULE

ONTOR UL E
ONTORULE (ctd.)

Tasks led by TUWIEN: Ontologies meet Business Rules

in WP2 - Business Rules and Ontologies Ownership and
Management:
Consistency maintenance.
techniques to detect when the creation or modiﬁcation of
rules and/or the ontology aﬀects the overall consistency
focus on detecting inconsistencies across the rule/ontology
boundary
techniques for repairing detected inconsistencies
testing and debugging across the rule/ontology boundary
in WP3 - Execution and Inference:
Theoretical foundations for combining Logic Programs and
Ontologies
Complexity, expressiveness, and optimization of combinations
of LP and Ontologies

ASP ONTORULE

Formalisms combining Ontologies and Rules
Hybrid formalisms - Ontologies / Logical rules.
loosely coupled approaches
DL-programs
F-Logic#

tightly coupled approaches
SWRL and restrictions (DLP, Carin, DL-safe rules, Desription
Logic Rules)
R-hybrid KBs
F-hybrid KBs

Embedding approaches
hybrid MKNF KBs
Quantiﬁed Equilibrium Logic (QEL)


ASP ONTORULE

Ontologies and Rules in the Semantic Web
Why a “Semantic” Web?

Undoubtedly, the World Wide Web is one of the most signiﬁcant
technical innovations of the last decades,
The WWW will be (and already is) strongly impacting, and
changing, our living, culture, and society.


ASP ONTORULE



The WWW has been designed for human users, not for machines


ASP ONTORULE




To process the data and information out on the Web, semantic
annotation and description is needed.


ASP ONTORULE




To process the data and information out on the Web, semantic
annotation and description is needed.

The Semantic Web is the vision of such an enriched, future
generation Web

Logic and logic-based formalisms (should/might) play an important
role in this endeavor.

ASP ONTORULE

Building the Semantic Web (T. Berners-Lee, 04/2005)

RDF is the data model of the Semantic Web
RDF Schema semantically extends RDF by simple taxonomies and
hierarchies
OWL is a W3C standard, which builds on Description Logics
Rule languages: Rule Interchange Format (RIF) WG of W3C

ASP ONTORULE

ASP vs. Ontologies

ASP and OWL/DLs have related yet different underlying
settings

At the heart, the difference is between LP and Classical logic

Main Differences:

Closed vs. Open World Assumption (CWA/OWA)
Negation as failure vs. classical negation
Strong negation vs. classical negation
Unique names, equality


ASP ONTORULE

LP / Classical Logic: CWA vs. OWA

LP aims at building a single model, by closing the world
Reiter’s CWA:
If T |= A, then conclude ¬A, for ground atom A

FO logic / description logics keep the world open

In the Semantic Web, this is often reasonable

However, taking the agnostic stance of OWA may be not helpful for
drawing rational conclusions under incomplete information


ASP ONTORULE

LP / Classical Logic: NAF vs. Classical Negation

P: wine(X ) ← whiteWine(X ).
nonWhite(X ) ← not whiteWine(X ).
wine(myDrink).

T: ∀X . (whiteWine(X ) ⊃ wine(X )) ∧
∀X . (¬whiteWine(X ) ⊃ nonWhite(X )) ∧
wine(myDrink).


ASP ONTORULE


wine(myDrink).

wine(myDrink).

Query nonWhite(myDrink)?


ASP ONTORULE


wine(myDrink).

wine(myDrink).


Conclude nonWhite(myDrink) from P.


ASP ONTORULE


wine(myDrink).

wine(myDrink).


Conclude nonWhite(myDrink) from P.
Do not conclude nonWhite(myDrink) from T .


ASP ONTORULE

LP / Classical Logic: Strong vs. Classical Negation

−wine(myDrink).
T: ∀X . (WhiteWine(X ) ⊃ Wine(X )) ∧
¬Wine(myDrink).


ASP ONTORULE


−wine(myDrink).
¬Wine(myDrink).

Conclude ¬WhiteWine(myDrink) from T ;


ASP ONTORULE


−wine(myDrink).
¬Wine(myDrink).

Conclude ¬WhiteWine(myDrink) from T ;
Do not conclude −whiteWine(myDrink) from P
Note: no contraposition in LP!

−whiteWine(X ) ← −wine(X ).
is not equivalent to
wine(X ) ← whiteWine(X ).


ASP ONTORULE

LP / Classical Logic: Unique Names, Equality
In LP, usally we have Unique Names Assumption (UNA):
Syntactically different ground terms are different objects.
Thus, usually only Herbrand interpretations are considered in LP
Ontology languages like OWL don’t make UNA, and allow to link
objects (owl:sameAs)
OWL considers also non-Herbrand interpretations
Further, related problems with existential quantifiers:

T: ∀X ∃Y . (Wine(X ) ⊃ hasColor (X , Y ))

(in DL Syntax, Wine ∃hasColor )

Simple skolemisation does not work in general


ASP ONTORULE

DL-programs

Recently, much eﬀort has been put into the study of
combinations of rules and ontologies in the context of the
Semantic Web.

DL-programs have been proposed as a fruitful approach for
combining logic programs (LP) under the answer-set
semantics and description logic (DL) ontologies [Eiter et al.
2004,2008].

The interface beetween LP and DL is based on loose coupling.


ASP ONTORULE

Loose Coupling
The two components do not rely on a shared language or
common logical models.
Instead, they interact via dedicated atoms, called DL-atoms,
in the LP.
A DL-atom can be seen as a query to the ontology.
Bi-directional ﬂow of information:
before the query is evaluated, predicates from the LP can be
temporarily imported to the DL.
this way, the DL-reasoner takes information into account that
has been derived by rules in the LP.

ASP Solver ? DL Engine


ASP ONTORULE

Syntax
A DL-atom a(t) has the form
DL[S1 op 1 p1 , . . . , Sm op m pm ; Q](t), where
m ≥ 0,
each Si is either a DL concept or a role predicate
op i ∈ { , ∪, ∩},
− −

pi is a unary, resp. binary, LP predicate symbol, and
Q(t) is a DL-query
We call γ = S1 op 1 p1 , . . . , Sm op m pm the input signature.
Intuitively,
op i = increases Si by the extension of pi ,
op i = ∪ increases ¬Si by the extension of pi , and
−

op i = ∩ constrains Si to pi .
−


ASP ONTORULE

Syntax (ctd.)
A DL-rule r is a normal rule of form:
a ← b1 , . . . , bk , not bk+1 , . . . , not bm , m ≥ k ≥ 0, where
a is a strong literal, and
b1 , . . . , bm are strong literals or DL-atoms.
“not” denotes default negation (negation as failure).
H(r ) = a, B+ (r ) = {b1 , . . . , bk }, B− (r ) = {bk +1 , . . . , bm },
B(r ) = {b1 , . . . , bk , not bk +1 , . . . , not bm }

A DL-program KB = (T , P) consists of a DL ontology T and
a ﬁnite set P of DL-rules.
The signatures of T and P have the same set F of (0-ary)
function symbols.
The grounding gr (P) of P is the set of all ground rules that
can be obtained by replacing variables of rules in P by
constants occurring in F .

ASP ONTORULE

Semantics

Let KB = (T , P) be a DL-program.

An interpretation for KB is a consistent set of strong literals
from gr (P).

A ground strong literal a is true under interpretation I
(I |=T a) iﬀ a ∈ I.


ASP ONTORULE

Semantics (ctd.)
A ground DL-atom a = DL[S1 op 1 p1 , . . . , Sm op m pm ; Q](c) is
true under DL T (I |=T a) iff
T ∪ τI (a) |= Q(c), where
m
the extension of a under I is τI (a) = i =1 Ai (I) such that
Ai (I) = {Si (e) | pi (e) ∈ I}, for op i = ;
Ai (I) = {¬Si (e) | pi (e) ∈ I}, for op i = ∪;
−

Ai (I) = {¬Si (e) | pi (e) ∈ I}, for op i = ∩.
/ −

I |=T B+ (r ) iff I |=T a for all a ∈ B+ (r ).
I |=T B− (r ) iff I |=T a for all a ∈ B− (r ).
I |=T B(r ) iff I |=T B+ (r ) and I |=T B− (r ).
I |=T r iff I |=T H(r ) whenever I |=T B(r ).
I is a model of KB (I |= KB) iff I |=T r for all r ∈ gr (P).


ASP ONTORULE

Semantics (ctd.)
Answer sets deﬁned analogously to answer-set programs using
the FLP-reduct

remember: introduced for an intuitive handling of
nonmonotone aggregates in ASP (Faber, Leone, Pfeifer 2004).
The FLP-reduct KB I FLP of DL-program KB = (T , P) relative
to I is the set of rules r ∈ gr (P) such that I |=T B(r ).
Definition
An interpretation I is an answer set of KB if it is a minimal model
of KB I .
FLP
The set of all answer sets of KB is denoted by AS(KB).

Originally, two different semantics have been defined (Eiter,
Lukasiewicz,Schindlauer, and Tompits 2004):
Strong Answer-Set semantics
Weak Answer-Set semantics

ASP ONTORULE

Example

A computer store obtains hardware from several vendors.
DL knowledge base Tex contains information about the
product range provided by each vendor.
For some parts, a shop may already be contracted as supplier.
Tex T-Box
≥ 1 provides shop; ∀provides.part;

Tex A-Box
provides(s1 , cpu); provides(s1 , case); provides(s2 , cpu);
provides(s3 , harddisk); provides(s3 , case);
supplies(s3 , case);


ASP ONTORULE

Example (ctd.)

DL-program KB ex = (Tex , Pex ) not-deterministically chooses a
vendor for each needed part.
LP part Pex :
needed(X ) ←DL[; part](X );
alreadyContracted(P) ←DL[; supplies](S, P), needed(P);
offer (S, P) ←DL[; provides](S, P), needed(P),
not alreadyContracted(P);
chosen(S, P) ←offer (S, P), not notChosen(S, P);
notChosen(S, P) ←offer (S, P), not chosen(S, P);
supplier (S, P) ←DL[supplies chosen; supplies](S, P), needed(P);
anySupplied(P) ←supplier (S, P);
←not needed(P), not anySupplied(P).


ASP ONTORULE

Example (ctd.)
KB ex has two answer sets, I1 and I2
both containing the same atoms of predicates needed, offer ,
alreadyContracted, and anySupplied:
I = {needed(cpu), needed(harddisk), needed(case), alreadyContracted(case),
offer (s1 , cpu), offer (s2 , cpu), offer (s3 , harddisk),
anySupplied(cpu), anySupplied(harddisk), anySupplied(case)}

The remaining atoms of I1 are given by
I1 I = {chosen(s1 , cpu), chosen(s3 , harddisk), notChosen(s2 , cpu),
supplier (s1 , cpu), supplier (s3 , harddisk), supplier (s3 , case)}

The remaining atoms of I2 are given by
I2 I = {chosen(s2 , cpu), chosen(s3 , harddisk), notChosen(s1 , cpu),
supplier (s2 , cpu), supplier (s3 , harddisk), supplier (s3 , case)}


ASP ONTORULE

Complexity of DL-programs

Theorem
Given and a dl-program KB = (T , P) with P in SHIF (D), deciding
whether KB has an answer set is complete for NEXP in the
general case, and complete for EXP when KB is positive or
stratiﬁed.


ASP ONTORULE

Tractable DL-programs
Tractable Reasoning with DL-programs (Heymans, Eiter, and
Xiao 2010)
Datalog-rewritable Description Logics
DL-program is translated to a “pure” logic program
Approximation of answer-set semantics: well-founded
semantics
new DL deﬁned: LDL+
OWL2 fragments:
OWL 2 RL: fully datalog rewritable (strict subset of LDL+ )
OWL 2 EL: partly d. r. (neg. and existential qu. in axiom
r.h.s.)
OWL 2 QL: partly d. r. (neg. and existential qu. in axiom
r.h.s.)


ASP ONTORULE

HEX-programs

DL-programs where extended to HEX-programs (Eiter, Ianni,
Schindlauer, and Tompits 2005)
Nonmonotonic logic programs admitting higher-order atoms as
well as external atoms
Higher-order: Variables may appear in predicate position (does
not change expressibility)
External atoms extend DL-atoms: use of arbitrary sources
instead of queries to DL
Implemented in the DLVHEX system

SPARQL-queries can be translated to HEX-programs using
external atoms for RDF-import (Polleres 2007)


ASP Context
Rules and Ontologies Framework
Debugging in ASP Application

Debugging in ASP
Problem: What if an answer-set program contains a bug?
Syntax is simple
Every syntactically correct program has well-deﬁned semantics

Resulting answer sets diﬀer from expected answer sets
Often the answer sets change drastically when a bug was
introduced.

Work on debugging in ASP has started recently:
Oetsch, Pührer, and Tompits (2010),
Pontelli and Son (2009),
Gebser et al. (2008),
Mikitiuk, Mosely, and Truszczyński (2007),
Brain et al. (2007),
Syrjänen (2006),
Brain and De Vos (2005)

ASP Context

Debugging in ASP (ctd.)
Research has so far focused on declarative methods.
Two main motivations:
1. debugging systems should be independent of any particular
solving tool or solving method.
2. it is commonly argued that procedural-style debugging would
ruin the declarative nature of answer-set programs.
¯ destroys the view of a program as a description of a problem
¯ where no rule takes precedence
¯ and each of the rules is equally important as the others.


ASP Context

Debugging in ASP (ctd.)
Research has so far focused on declarative methods.
Two main motivations:
1. debugging systems should be independent of any particular
solving tool or solving method.
2. it is commonly argued that procedural-style debugging would
ruin the declarative nature of answer-set programs.
¯ destroys the view of a program as a description of a problem
¯ where no rule takes precedence
¯ and each of the rules is equally important as the others.

Following a procedural debugging approach the program could
be seen as parameters for the solving algorithm.
Could compel the user to a programming style adjusted to the
algorithm rather than clarity of representation.


ASP Context

Declarative Debugging in ASP
Most proposed approaches for debugging ASP follow the
scheme:
1. Take a declarative characterisation of the answer-set semantics.
2. Use it to explain an unexpected semantics of the given
program P.


ASP Context

scheme:
program P.

For example, answer sets can be characterised as
interpretations I such that
(i) all some rules in P are classically satisﬁed by I and
(ii) I contains no loop of P that is unfounded by P w.r.t. I.


ASP Context

scheme:
program P.

For example, answer sets can be characterised as
interpretations I such that
(i) all some rules in P are classically satisﬁed by I and
(ii) I contains no loop of P that is unfounded by P w.r.t. I.

For instance, a declarative debugging system could state:
I is not an answer set because rule r3 is not
classically satisﬁed by I!


ASP Context

Declarative Debugging in ASP (ctd.)

Moreover, we could also receive answers like:


ASP Context


I is not an answer set because rules r2 , r3 , r5 , r6 , r7 ,
r9 , r10 , r11 , r14 , r15 , r16 , r17 , r18 , r19 , r20 , r21 , r22 ,
r23 , r24 , r25 , r26 , r27 , r28 , r29 , r30 , r31 , r32 , r33 , r34 ,
r35 , r36 , r37 , r38 , r39 , r40 , r41 , r42 , r43 , r44 , r45 , r46 ,
r47 , r48 , r49 , r50 , r51 , r52 , r53 , r54 , r55 , r56 , r57 , r58 ,
r59 , r61 are not classically satisﬁed by I!


ASP Context


r9 , r10 , r11 , r14 , r15 , r16 , r17 , r18 , r19 , r20 , r21 , r22 ,
r23 , r24 , r25 , r26 , r27 , r28 , r29 , r30 , r31 , r32 , r33 , r34 ,
r35 , r36 , r37 , r38 , r39 , r40 , r41 , r42 , r43 , r44 , r45 , r46 ,
r47 , r48 , r49 , r50 , r51 , r52 , r53 , r54 , r55 , r56 , r57 , r58 ,
Problem:
¯ whole program is typically considered at once
¯ hard to identify the most useful information about a bug


We need means to focus debugging information


ASP Context

The User’s Intuition

How would a user “manually” verify whether an interpretation
is an answer set?


ASP Context


is an answer set?
Probably, by inspecting how the program’s rules step-by-step
built up the interpretation.


ASP Context


is an answer set?
Most likely, following an intuitive order according to the
semantics of the rules.


But what about the declarative view?


No rule takes precedence?


No rule takes precedence?
Each of the rules is equally important as the others?


ASP Context

The User’s Intuition (ctd.)

There are “natural” orders in which to consider rules.
For example, orders imposed by
stratiﬁcation,
Guess Check components,
importance of subprograms w.r.t. the problem domain, or
the order in which the program is written.


We want to make use of these orders for debugging.


ASP Context

Idea
aim at a debugging approach that allows the user to
build an answer set he or she has in mind by
stepwise adding further applicable rules.

emulation of a bottom-up generation of answer sets
considered interpretation grows monotonically
user serves as an oracle, choosing rules considered to be
supporting
bugs can be found when computation deviates from
expectations

Needed: A computation model for ASP that
is simple and intuitive
does not require a ﬁx order in which to consider the rules
concentrates on computation steps interesting to the user

ASP Context

States
Deﬁnition
Let S be a set of ground (normal) rules. Then, the interpretation
induced by S is given by IS = r ∈S H(r ).

Deﬁnition
A set S of ground rules is self-supporting if IS |= B(r ), for all r ∈ S,
and stable if IS ∈ AS(S). A state of a normal program P is a set
S ⊆ gr (P) of ground rules which is self-supporting and stable.

Intuitively,
S contains the already considered ground rules (source code
level) and
IS contains the already derived literals (output level).
States serve as breakpoints.

ASP Context

Computations
Definition
For a state S of a program P and a set S ⊆ gr (P) of ground rules,
S is a successor of S in P, symbolically S P S , if S = S ∪ {r },
for some rule r ∈ gr (P) S with
(i) IS |= B(r ),
(ii) H(r ) = 0, and
/
(iii) H(r ) ∩ (B− (r ) ∪ −
r ∈S B (r )∪ a∈IS a) = 0.
¯ /

The successor relation suffices to “step” from one state to another,
i.e., S is always self-supporting and stable, and hence a state of P.
Definition
A computation for a program P is a finite sequence C = S0 , . . . , Sn
of states such that, for all 0 ≤ i n, Si P Si+1 .


ASP Context

Computations (ctd.)
Deﬁnition
A computation C = S0 , . . . , Sn for P
has failed at Step i if there is no answer set I of P such that
every rule r in Si is active under I (i.e., I |= B(r ));
is stuck if there is no successor of Sn in P but there is a rule
r ∈ gr (P) Sn that is active under ISn ;
is complete if, for each rule r ∈ gr (P) that is active under ISn ,
we have r ∈ Sn .
Intuitive meanings
failure: computation reached a point where no answer set of
the program can be reached
stuck: last state activated rules deriving literals that are
inconsistent with previously chosen active rules
complete: there are no more unconsidered active rules

ASP Context

Computations (ctd.)
The following result guarantees the soundness of our stepping
framework.
Theorem
Let P be a program and C = S0 , . . . , Sn a complete computation for
P. Then, ISn is an answer set of P.
The computation model is also complete in the sense that
stepping, starting from an arbitrary state S0 of P as breakpoint,
can reach every answer set I ⊇ IS0 of P, where every rule in S0 is
active under I.
Theorem
Let I ∈ AS(P) be an answer set of program P and S0 a state of P
such that IS0 ⊆ I and every rule in S0 is active under I. Then, there
is a complete computation C = S0 , . . . , Sn with
Sn = {r ∈ gr (P) | I |= B(r )} and ISn = I.

ASP Context

Maze Example

The problem of Maze Generation:
Consider an n × m grid where each cell is either a wall or
empty and there are two dedicated empty cells located at the
border,the maze’s entrance and its exit.
The maze grid has to satisfy the following conditions:
(i) border cells are walls (exception: entrance and exit)
(ii) there is a path from the entrance to every empty cell
(iii) for two diagonally adjacent walls, one of their common
neighbours is a wall
(iv) no 2 × 2 blocks of empty cells or walls
(v) no wall can be completely surrounded by empty cells


ASP Context

Maze Example (ctd.)

Input of the problem: facts that specify
the size of the maze,
positions of the entrance and the exit, and
for an arbitrary subset of cells whether they are walls or empty.
Example input:

F = {row (1), row (2), row (3), row (4), row (5), col(1), col(2), col(3),
col(4), col(5), entrance(1, 2), exit(5, 4), wall(3, 3), empty (3, 4)}

Output: completion of the input to a valid maze structure.

ASP Context

Maze Example (ctd.)
First step: identify border cells.
Our initial program is P0 = F ∪ PBdr , where PBdr is given by
maxcol(X ) ← col(X ), not col(X + 1),
maxrow (Y ) ← row (Y ), not row (Y + 1),
border (1, Y ) ← col(1), row (Y ),
border (X , Y ) ← row (Y ), maxcol(X ),
border (X , 1) ← col(X ), row (1),
border (X , Y ) ← col(X ), maxrow (Y ).

A set of facts is always a state.
Consider F as breakpoint.
We start stepping from F by choosing, e.g., the ground rule
r = maxcol(5) ← col(5), not col(6),
that is active under F , as next rule to add.
We obtain the computation C = F , F ∪ {r }.

ASP Context

Maze Example (ctd.)

It is not feasible to step through all rules in the (exponential)
grounding of a program.

Start at breakpoints “close” to an interesting debugging
situation.

We need means to obtain such interesting states.


We need means to obtain such interesting states.

Proposition
For programs P and P ⊆ P ∪ gr (P) such that I ∈ AS(P ), the set
of rules in gr (P ) that are active under I is a state of P.


ASP Context

Maze Example (ctd.)
maxcol(X ) ← col(X ), not col(X + 1),
maxrow (Y ) ← row (Y ), not row (Y + 1),
border (1, Y ) ← col(1), row (Y ),
border (X , Y ) ← row (Y ), maxcol(X ),
border (X , 1) ← col(X ), row (1),
border (X , Y ) ← col(X ), maxrow (Y ).

We want to step through rules deriving the border /2 atoms.
Calculate an answer set of P0 ⊆ P0 that is given by
P0 = F ∪ {maxcol(X ) ← col(X ), not col(X + 1),
maxrow (Y ) ← row (Y ), not row (Y + 1)}.
Answer set given by I0 = F ∪ {maxcol(5), maxrow (5)}
The desired breakpoint S0 : the rules in gr (P0 ) active under I0 :
P0 = F ∪ {maxcol(5) ← col(5), not col(6),
maxrow (5) ← row (5), not row (6)


ASP Context

Maze Example (ctd.)

During stepping one might want to jump several steps forward.
We extend the maze program P0 to P1 by adding the rule
rbw = wall(X , Y ) ← border (X , Y ), not entrance(X , Y ), not exit(X , Y )
Stepping from S1 that already contains all border atoms:


ASP Context

Maze Example (ctd.)

S2 = S1 ∪ {wall(1, 1) ← border (1, 1), not entrance(1, 1), not exit(1, 1)}


ASP Context

Maze Example (ctd.)



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