Answer Set Programming Tutorial: Modeling Problems Declaratively

An Answer Set Programming Tutorial

Minh Dao-Tran

Institut für Informationsysteme, TU Wien

Supported by Austrian Science Fund (FWF) project P20841, and the Marie Curie action IRSES
under Grant No. 2476

Based on slides from Thomas Eiter, Giovambattista Ianni, Thomas Krennwallner in the Reasoning Web Summer School 2009
and slides from Torsten Schaub

DERI, July 2011

An Answer Set Programming Tutorial

Outline

1. Introduction
2. Horn Logic Programming
2.1 Positive Logic Programs
2.2 Minimal Model Semantics
3. Stable Logic Programming
4. Extensions
4.1 Disjunction
4.2 Integrity Constraints
5. Answer Set for the Semantic Web
5.1 DL-Programs
5.2 HEX-Programs
6. Future Directions of ASP

Minh Dao-Tran DERI, July 2011 1/33

An Answer Set Programming Tutorial 1. Introduction

Introduction

Answer Set Programming (ASP) is a recent problem solving
approach

ASP has its roots in:
• (logic-based) knowledge representation and reasoning
• (deductive) databases

• constraint solving, SAT solving

• logic programming (with negation)

ASP allows for solving all search problems in NP (and NPNP ) in a
uniform way



Knapsack Problem



Sudoku

6 1 4 5
8 3 5 6
2 1
8 4 7 6
6 3
7 9 1 4
5 2
7 2 6 9
4 5 8 7

Task:
Fill in the grid so that every row, every column, and every 3x3 box
contains the digits 1 through 9.


Wanted!

A general-purpose, declarative approach for modeling and solving
these and many other problems.

Declarative:
• “What is the problem?”

instead of
• “How to solve the problem?”



Wanted!

A general-purpose, declarative approach for modeling and solving
these and many other problems.

Declarative:
• “What is the problem?”

instead of
• “How to solve the problem?”

Proposal:
Answer Set Programming (ASP) paradigm!


An Answer Set Programming Tutorial 2. Horn Logic Programming 2.1 Positive Logic Programs

Positive Logic Programs
Definition (Positive Logic Program)
A positive logic program P is a finite set of clauses (rules) in the form

a ← b1 , . . . , bm

where a, b1 , . . . , bm are atoms of a first-order language L.
a is the head of the rule
b1 , . . . , bm is the body of the rule.
If m = 0, the rule is a fact (written shortly a)



Positive Logic Programs
Definition (Positive Logic Program)
A positive logic program P is a finite set of clauses (rules) in the form

a ← b1 , . . . , bm

where a, b1 , . . . , bm are atoms of a first-order language L.
a is the head of the rule
b1 , . . . , bm is the body of the rule.
If m = 0, the rule is a fact (written shortly a)

Example
prof (supervisor(X)) ← phd_student(X).
poor(X) ← phd_student(X).
phd_student(Y) ← phd_student(X), couple(X, Y).


Herbrand Semantics by an Example
A logic program P:
prof (supervisor(X)) ← phd_student(X). phd_student(peter).
poor(X) ← phd_student(X). couple(peter, jane).

Constant symbols: peter, jane; Function symbols: supervisor.



A logic program P:


Herbrand Universe
peter, supervisor(peter), supervisor(supervisor(peter)), . . . ,
HU(P) =
jane, supervisor(jane), supervisor(supervisor(jane)), . . .



A logic program P:


Herbrand Universe
HU(P) =

Herbrand Base
 
 phd_student(peter), phd_student(jane), phd_student(supervisor(peter)), . . . ,
 

prof (peter), prof (jane), prof (supervisor(peter)), prof (supervisor(jane)), . . . ,
 
HB(P) =
 poor(peter), poor(jane), poor(supervisor(peter)), poor(supervisor(jane)), . . . ,
 

couple(peter, jane), couple(peter, supervisor(peter)), couple(jane, jane), . . .
 



A logic program P:


Herbrand Universe
HU(P) =

Herbrand Base
 
 phd_student(peter), phd_student(jane), phd_student(supervisor(peter)), . . . ,
 

prof (peter), prof (jane), prof (supervisor(peter)), prof (supervisor(jane)), . . . ,
 
HB(P) =
 poor(peter), poor(jane), poor(supervisor(peter)), poor(supervisor(jane)), . . . ,
 

couple(peter, jane), couple(peter, supervisor(peter)), couple(jane, jane), . . .
 

Herbrand Interpretation
I1 = ∅ I2 = HB(P)
I3 = {phd_student(peter), phd_student(jane), couple(peter, jane), prof (supervisor(peter))}


Grounding

Grounding a rule

r = poor(X) ← phd_student(X).

A ground instance of r is obtained by replacing its variables by terms from HU P .
 

 poor(peter) ← phd_student(peter). 

poor(jane) ← phd_student(jane).

 

grnd(r) = poor(supervisor(peter)) ← phd_student(supervisor(peter)).
 

 .
.


.
 

Grounding a program

grnd(P) = r∈P grnd(r)



Herbrand Models

Deﬁnition (Model, satisfaction)
An interpretation I is a (Herbrand) model of a
a ground (variable-free) rule r = a ← b1 , . . . , bm , if either
{b1 , . . . , bm } I or a ∈ I ; (I |= r)

a rule r, if I |= r for every r ∈ grnd(r); (I |= r)

a program P, if I |= r for every rule r in P. (I |= P)



Example (Program P)

Which of the following interpretations are models of P?
I1 = ∅
I2 = HB(P)
phd_student(peter), couple(peter, jane), phd_student(jane),
I3 =
poor(peter), poor(jane), prof (supervisor(peter))



Example (Program P)

I1 = ∅ no
I2 = HB(P)
I3 =



Example (Program P)

I1 = ∅ no
I2 = HB(P) yes
I3 =



Example (Program P)

I1 = ∅ no
I2 = HB(P) yes
I3 = no


An Answer Set Programming Tutorial 2. Horn Logic Programming 2.2 Minimal Model Semantics

Minimal Model Semantics

Prefer models with true-part as small as possible.

Deﬁnition
A model I of P is minimal, if there exists no model J of P such that J ⊂ I .




Prefer models with true-part as small as possible.

Deﬁnition

Theorem
Every logic program P has a single minimal model (called the least
model), denoted LM(P).



Computation
The minimal model can be computed via ﬁxpoint iteration.

Example

0
TP = ∅



Computation

Example

0
TP = ∅
1
TP = {phd_student(peter), couple(peter, jane)}



Computation

Example

0
TP = ∅
1
2 phd_student(peter), couple(peter, jane), phd_student(jane),
TP =
poor(peter), prof (supervisor(peter))



Computation

Example

0
TP = ∅
1
TP =
 
 phd_student(peter), couple(peter, jane), phd_student(jane), 
3
TP = poor(peter), prof (supervisor(peter)), poor(jane),
prof (supervisor(jane))
 



Computation

Example

0
TP = ∅
1
TP =
 
 phd_student(peter), couple(peter, jane), phd_student(jane), 
3
TP = poor(peter), prof (supervisor(peter)), poor(jane),
prof (supervisor(jane))
 
4 3
TP = TP


An Answer Set Programming Tutorial 3. Stable Logic Programming

Negation in Logic Programs
Why negation?

Natural linguistic concept
Facilitates convenient, declarative descriptions (deﬁnitions)
E.g., "Men who are not husbands are singles.”



Negation in Logic Programs
Why negation?

Natural linguistic concept
Facilitates convenient, declarative descriptions (definitions)
E.g., "Men who are not husbands are singles.”

Definition
A normal logic program is a set of rules of the form

a ← b1 , . . . , bm , not c1 , . . . , not cn (n, m ≥ 0) (1)

where a and all bi , cj are atoms in a first-order language L.
not is called “negation as failure”, “default negation”, or “weak negation”

Things get more complex!


Stable model semantics

First, for variable-free (ground) programs P
Treat “not ” specially
Intuitively, literals not a are a source of “contradiction” or “unstability”.





Example (Dilbert, program P2 )
man(dilbert). (f1 )
single(dilbert) ← man(dilbert), not husband(dilbert). (r1 )
husband(dilbert) ← man(dilbert), not single(dilbert). (r2 )





man(dilbert). (f1 )
M = {man(dilbert)}, get {man(dilbert), single(dilbert), husband(dilbert)}





man(dilbert). (f1 )

M = {man(dilbert), single(dilbert), husband(dilbert)}, get {man(dilbert)}.



Stable Models

Deﬁnition (Gelfond-Lifschitz Reduct PM 1988)
The GL-reduct (simply reduct) of a ground program P w.r.t. an
interpretation M , denoted PM , is the program obtained from P by
1 removing rules with not a in the body for each a ∈ M ; and
2 removing literals not a from all other rules.



Stable Models


Intuition:
M makes an assumption about what is true and what is false.
The reduct PM incorporates this assumptions.
As a “not ”-free program, PM derives positive facts, given by LM(PM ).
If this coincides with M , then the assumption of M is “stable”.



Stable Models


Intuition:
M makes an assumption about what is true and what is false.
The reduct PM incorporates this assumptions.
As a “not ”-free program, PM derives positive facts, given by LM(PM ).
If this coincides with M , then the assumption of M is “stable”.
Deﬁnition (stable model)
An interpretation M of P is a stable model of P, if M = LM(PM ).


Example (P2 cont’d)

man(dilbert). (f1 )

Candidate interpretations:
M1 = {man(dilbert), single(dilbert)},
M2 = {man(dilbert), husband(dilbert)},
M3 = {man(dilbert), single(dilbert), husband(dilbert)},
M4 = {man(dilbert)}




man(dilbert). (f1 )

Candidate interpretations:
M1 = {man(dilbert), single(dilbert)},
M2 = {man(dilbert), husband(dilbert)},
M3 = {man(dilbert), single(dilbert), husband(dilbert)},
M4 = {man(dilbert)}

M1 and M2 are stable models.




man(dilbert). (f1 )

M1 = {man(dilbert), single(dilbert)}:




man(dilbert). (f1 )

reduct PM1 :
2
man(dilbert).
single(dilbert) ← man(dilbert).

The least model of PM1 is {man(dilbert), single(dilbert)} = M1 .
2




man(dilbert). (f1 )

reduct PM1 :
2
man(dilbert).

The least model of PM1 is {man(dilbert), single(dilbert)} = M1 .
2

M2 = {man(dilbert), husband(dilbert)}: by symmetry of husband and
single, also M2 is stable.



man(dilbert). (f1 )

M3 = {man(dilbert), single(dilbert), husband(dilbert)}:



man(dilbert). (f1 )

PM3 is
2
man(dilbert).

LM(PM3 ) = {man(dilbert)} = M3 .
2



man(dilbert). (f1 )

PM3 is
2
man(dilbert).

2

M4 = {man(dilbert)}:



man(dilbert). (f1 )

PM3 is
2
man(dilbert).

2

M4 = {man(dilbert)}:
PM4 is
2
man(dilbert).
husband(dilbert) ← man(dilbert).

LM(PM4 ) = {man(dilbert), single(dilbert), husband(dilbert)} = M4 .
2


Programs with Variables

As for positive programs, view a program rule as a shorthand for all
its ground instances.
Recall: grnd(P) is the grounding of program P.

Deﬁnition (stable model, general case)
An interpretation M of P is a stable model of P, if M is a stable model of
grnd(P).


An Answer Set Programming Tutorial 4. Extensions

Extensions

Many extensions exist, partly motivated by applications
Some are syntactic sugar, other strictly add expressiveness
Incomplete list:
• disjunction
• integrity constraints
• strong negation
• nested expressions
• cardinality constraints (Smodels)
• optimization: weight constraints, minimize (Smodels);
weak constraints (DLV)
• aggregates (Smodels, DLV)
• templates (for macros), external functions (DLVHEX)
• Frame Logic syntax (for Semantic Web)
• preferences: e.g., PLP, LOPDs
• KR frontends (diagnosis, inheritance, planning,...) in DLV
Comprehensive survey: [Niemelä (ed.), 2005]

An Answer Set Programming Tutorial 4. Extensions 4.1 Disjunction

Disjunction
The use of disjunction is natural to express indeﬁnite knowledge.



Disjunction

Example

female(X) ∨ male(X) ← person(X).

single(dilbert) ∨ husband(dilbert) ← man(dilbert).



Disjunction

Example



Disjunction is natural for expressing a “guess” and to create
non-determinism



Disjunction

Example



Disjunction is natural for expressing a “guess” and to create
non-determinism

Example

ok(C) ∨ notok(C) ← component(C).


An Answer Set Programming Tutorial 4. Extensions 4.2 Integrity Constraints

Integrity Constraints

Adding
fail ← b1 , . . . , bm , not c1 , . . . , not cn , not fail.
to P “kills” all stable models of P that
• contain b1 , . . . , bm , and
• do not contain c1 , . . . , cn

This is convenient to eliminate scenarios which does not satisfy
integrity constraints.



Integrity Constraints

Adding
fail ← b1 , . . . , bm , not c1 , . . . , not cn , not fail.
to P “kills” all stable models of P that
• contain b1 , . . . , bm , and
• do not contain c1 , . . . , cn

This is convenient to eliminate scenarios which does not satisfy
integrity constraints.

Short:

Integrity Constraint

← b1 , . . . , bm , not c1 , . . . , not cn .




man(dilbert). (f1 )
single(dilbert) ∨ husband(dilbert) ← man(dilbert). (r1 )
← husband(X), not wedding_ring(X). (c1 )

The constraint c1 eliminates models in which there is no evidence for
a husband having a wedding ring.
Single stable model: M1 = {man(dilbert), single(dilbert)}


An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web

Answer Set Solvers

DLV1 http://www.dbai.tuwien.ac.at/proj/dlv/
Smodels2 http://www.tcs.hut.fi/Software/smodels/
GnT http://www.tcs.hut.fi/Software/gnt/
Cmodels http://www.cs.utexas.edu/users/tag/cmodels/
ASSAT http://assat.cs.ust.hk/
NoMore(++) http://www.cs.uni-potsdam.de/~linke/nomore/
Platypus http://www.cs.uni-potsdam.de/platypus/
clasp http://www.cs.uni-potsdam.de/clasp/
XASP http://xsb.sourceforge.net/, distributed with XSB
aspps http://www.cs.engr.uky.edu/ai/aspps/
ccalc http://www.cs.utexas.edu/users/tag/cc/

1
+ many extensions, e.g., DLVEX, DLVHEX, DLVDB , DLT, DLV-Complex
2
+ Smodels_cc

An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web

Answer Set Programming Systems for Debian/Ubuntu

Easy to install:
• $ sudo add-apt-repository ppa:tkren/asp
• $ sudo apt-get update
• $ sudo apt-get install clasp gringo potassco-guide
dlv-installer

With the next version of Ubuntu in October 2011, clasp and gringo
will be built in

More details at
http://www.kr.tuwien.ac.at/staff/tkren/deb.html


An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web 5.1 DL-Programs

DL-Programs
A clean approach to combine description logics and logic programs
under answer set semantics

A dl-program consists of two parts:
a logic program P,
a description logic knowledge base L
L is accessible from P by means of dl-atoms.

ASP Solver ? DL Engine



The famous tweety example
 
 Flier ¬NonFlier, Penguin Bird, 
L= Penguin NonFlier, —————–
Bird Flier
Bird(tweety), Penguin(joe)
 



 
Bird Flier
 

flies(X) ← DL[Flier flies; Bird](X),
P=
not DL[Flier flies; ¬Flier](X).



 
Bird Flier
 

P=
The dl-atom device

Can specify a query to L:

DL[Bird](X)

DL[Bird](tweety) true for y s.t. L |= Bird(tweety).



 
Bird Flier
 

P=
The dl-atom device

Can specify a query to L:

DL[Bird](X)

DL[Bird](tweety) true for y s.t. L |= Bird(tweety).
Can push knowledge to L before querying:

DL[Flier ﬂies; ¬Flier](X).


An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web 5.2 HEX-Programs

HEX-programs

Extends dl-programs from one-to-one coupling to many-one.
• Outer Knowledge sources are not constrained to DL knowledge
bases only.



HEX-programs

bases only.
P can interface multiple external sources of knowledge of any sort
via so called external atoms



HEX-programs

bases only.
P can interface multiple external sources of knowledge of any sort
via so called external atoms
P has higher order atoms



An example

subRelation(brotherOf , relativeOf ).
brotherOf (john, al).
relativeOf (john, joe).
brotherOf (al, mick).
invites(john, X) ∨ skip(X) ← X = john, &reach[relativeOf , john](X).
R(X, Y) ← subRelation(P, R), P(X, Y).
someInvited ← invites(john, X).
← not someInvited.
← &degs[invites](Min, Max), Max > 2.

Example
Input: Some data about John’s neighborhood
Output: Possible picks for persons John might want to invite, according to some
constraints


Higher order atoms


The device of higher order atoms

Predicate names can be variables
Constants can appear both as terms values or as predicate values



Higher order atoms



Allows (comfortable) meta-reasoning




Higher order atoms



Allows (comfortable) meta-reasoning

subRelation(brotherOf , relativeOf ). ⇒ relativeOf (X, Y) ← brotherOf (X, Y).



External atoms

&reach[relativeOf , john](X) (2)
&degs[invites](Min, Max) (3)

The device of external atoms

Each external predicate is tied to a corresponding evaluation
function
E.g. &reach corresponds to f&reach .

For a given interpretation I , I |= &reach[relativeOf , john](x) if
f&reach (I, relativeOf , john) = 1


An Answer Set Programming Tutorial 6. Future Directions of ASP

Current state-of-the-art
Semantics
Introduction of Function Symbols [Syrjänen, 2001],[Bonatti,
2004],[Calimeri et al., 2008],[Šimkus and Eiter, 2007],[Eiter and Šimkus,
2009]
Modularity [Dao-Tran et al., 2009],[Janhunen et al., 2007],[Oikarinen and
Janhunen, 2008],[Tari et al., 2005],[Balduccini, 2007],[Baral et al.,
2006],[Calimeri and Ianni, 2006],[Polleres et al., 2006],[Analyti et al., 2008]
Study of equivalence [Lifschitz et al., 2001],[Gelfond, 2008],[Eiter et al.,
2007],[Eiter et al., 2004],[Woltran, 2008]



Current state-of-the-art
Semantics
Introduction of Function Symbols [Syrjänen, 2001],[Bonatti,
2004],[Calimeri et al., 2008],[Šimkus and Eiter, 2007],[Eiter and Šimkus,
2009]
Modularity [Dao-Tran et al., 2009],[Janhunen et al., 2007],[Oikarinen and
Janhunen, 2008],[Tari et al., 2005],[Balduccini, 2007],[Baral et al.,
2006],[Calimeri and Ianni, 2006],[Polleres et al., 2006],[Analyti et al., 2008]
Study of equivalence [Lifschitz et al., 2001],[Gelfond, 2008],[Eiter et al.,
2007],[Eiter et al., 2004],[Woltran, 2008]

Engineering

Debuggers [El-Khatib et al., 2005],[Brain and Vos, 2005] and in-database
evaluation [Terracina et al., 2008]
See the SEA workshop series http://sea07.cs.bath.ac.uk/



Current state-of-the-art II

Scalability

Intelligent and lazy grounders [Calimeri et al., 2008],[Gebser et al.,
2008],[Palù et al., 2008], [Lefévre and Nicolas, 2008]
Incremental reasoning [Gebser et al., 2011]
See the Answer Set Programming competition
http://www.cs.kuleuven.be/∼dtai/events/ASP-competition/
https://www.mat.unical.it/aspcomp2011


An Answer Set Programming Tutorial 7. Playing with ASP

xkcd Knapsack

order(Food) ∨ skip(Food) ← menu(Food, _).
ok ← 825 ≤ #sum{Price : order(Food), menu(Food, Price)} ≤ 825.
← not ok.

menu(mixed_fruit, 215).
menu(french_fries, 275).
menu(side_salad, 335).
menu(host_wings, 355).
menu(mozzarella_sticks, 420).
menu(samples_place, 580).



Graph 3-Coloring

r(X) ∨ g(X) ∨ b(X) ← node(X).
← edge(X, Y), r(X), r(Y).
← edge(X, Y), g(X), g(Y).
← edge(X, Y), b(X), b(Y).

node(a). node(b). node(c).
edge(a, b). edge(b, c). edge(a, c).



Hamilton Path/Cycle

inPath(X, Y) ∨ outPath(X, Y) ← edge(X, Y).

← inPath(X, Y), inPath(X, Y1 ), Y = Y1 .
← inPath(X, Y), inPath(X1 , Y), X = X1 .
← node(X), not reached(X).

reached(X) ← start(X).
reached(X) ← reached(Y), inPath(Y, X).



Hamilton Path/Cycle

inPath(X, Y) ∨ outPath(X, Y) ← edge(X, Y).

← inPath(X, Y), inPath(X, Y1 ), Y = Y1 .
← inPath(X, Y), inPath(X1 , Y), X = X1 .
← node(X), not reached(X).
← not start_reached.

reached(X) ← start(X).
reached(X) ← reached(Y), inPath(Y, X).
start_reached ← start(Y), inPath(X, Y).



Choosing wine
Assume that we have a Wine ontology and speciﬁcation of who prefer
which type of wine.
person(“axel”). preferredWine(“axel”, “SweetWine”).
person(“gibbi”). preferredWine(“gibbi”, “DessertWine”).
person(“roman”). preferredWine(“roman”, “ItalianWine”).



Choosing wine
which type of wine.
isA(X, “SweetWine”) ← DL[SweetWine](X).
isA(X, “DessertWine”) ← DL[DessertWine](X).
isA(X, “ItalianWine”) ← DL[ItalianWine](X).



Choosing wine
which type of wine.
compliantBottle(X, Z) ← preferredWine(X, Y), isA(Z, Y).



Choosing wine
which type of wine.
bottleChosen(X) ∨ nonbottleChosen(X) ← compliantBottle(_, X).



Choosing wine
which type of wine.
hasBottleChosen(X) ← bottleChosen(Z), compliantBottle(X, Z).



Choosing wine
which type of wine.
← person(X), not hasBottleChosen(X).



Choosing wine
which type of wine.
← person(X), not hasBottleChosen(X).
← bottleChosen(X).[1 : 1]


People whom Axel knows

triple(X, Y, Z) ← &rdf [“foaf .axel.rdf ”](X, Y, Z).

knownbyAxelbyName(X) ← triple(ID, “foaf : name”, “Axel Polleres”),
triple(ID, “foaf : knows”, ID2),
triple(ID2, “foaf : name”, X).


References I

Anastasia Analyti, Grigoris Antoniou, and Carlos Viegas Damásio.
A principled framework for modular web rule bases and its semantics.
In Proceedings of the 11th International Conference on Principles of
Knowledge Representation and Reasoning (KR2008). AAAI Press,
September 2008.

Marcello Balduccini.
Modules and Signature Declarations for A-Prolog: Progress Report.
In Marina de Vos and Torsten Schaub, editors, Informal Proceedings of the
1st International Workshop on Software Engineering for Answer Set
Programming, Tempe, AZ (USA), May 2007, 2007.
Available at http://sea07.cs.bath.ac.uk/downloads/
sea07-proceedings.pdf.

References II

Chitta Baral, Juraj Dzifcak, and Hiro Takahashi.
Macros, Macro calls and Use of Ensembles in Modular Answer Set
Programming.
In Proceedings of the 22th International Conference on Logic Programming
(ICLP 2006), number 4079 in LNCS, pages 376–390. Springer, 2006.

Piero A. Bonatti.
Reasoning with inﬁnite stable models.
Artiﬁcial Intelligence, 156(1):75–111, 2004.

Martin Brain and Marina De Vos.
Debugging logic programs under the answer set semantics.
In Answer Set Programming, 2005.

References III

Francesco Calimeri and Giovambattista Ianni.
Template programs for Disjunctive Logic Programming: An operational
semantics.
AI Communications, 19(3):193–206, 2006.

Francesco Calimeri, Susanna Cozza, Giovambattista Ianni, and Nicola
Leone.
Computable Functions in ASP: Theory and Implementation.
In Logic Programming, 24th International Conference, ICLP 2008, Udine,
Italy, December 9-13 2008, Proceedings, volume 5366 of LNCS, pages
407–424. Springer, 2008.

Minh Dao-Tran, Thomas Eiter, Michael Fink, and Thomas Krennwallner.
Modular nonmonotonic logic programming revisited.
In P. Hill and D.S. Warren, editors, Proceedings 25th International
Conference on Logic Programming (ICLP 2009), volume 5649 of LNCS,
pages 145–159. Springer, July 2009.

References IV

Thomas Eiter and Mantas Šimkus.
Bidirectional answer set programs with function symbols.
In C. Boutilier, editor, Proceedings of the 21st International Joint
Conference on Artiﬁcial Intelligence (IJCAI-09). AAAI Press/IJCAI, 2009.

T. Eiter, M. Fink, H. Tompits, and S. Woltran.
Simplifying logic programs under uniform and strong equivalence.
In Ilkka Niemelä and Vladimir Lifschitz, editors, Proceedings of the 7th
International Conference on Logic Programming and Nonmonotonic
Reasoning (LPNMR 2004), number 2923 in LNCS, pages 87–99. Springer,
2004.
Thomas Eiter, Michael Fink, and Stefan Woltran.
Semantical Characterizations and Complexity of Equivalences in Answer
Set Programming.
ACM Trans. Comput. Log., 8(3), 2007.
Article 17 (53 + 11 pages).

References V

Omar El-Khatib, Enrico Pontelli, and Tran Cao Son.
Justiﬁcation and debugging of answer set programs in asp.
In AADEBUG, pages 49–58, 2005.

Martin Gebser, Roland Kaminski, Benjamin Kaufmann, Max Ostrowski,
Torsten Schaub, and Sven Thiele.
Engineering an Incremental ASP Solver.
In M.G. de La Banda and E. Pontelli, editors, Proceedings 24th
International Conference on Logic Programming (ICLP 2008), number 5366
in LNCS, pages 190–205. Springer, 2008.

Martin Gebser, Orkunt Sabuncu, and Torsten Schaub.
An incremental answer set programming based system for ﬁnite model
computation.
AI Commun., 24(2):195–212, 2011.

References VI

Michael Gelfond and Vladimir Lifschitz.
The Stable Model Semantics for Logic Programming.
In Logic Programming: Proceedings Fifth Intl Conference and Symposium,
pages 1070–1080, Cambridge, Mass., 1988. MIT Press.

Michael Gelfond and Vladimir Lifschitz.
Classical Negation in Logic Programs and Disjunctive Databases.
New Generation Computing, 9:365–385, 1991.

M. Gelfond.
Answer sets.
In B. Porter F. van Harmelen, V. Lifschitz, editor, Handbook of Knowledge
Representation, chapter 7, pages 285–316. Elsevier, 2008.

References VII

Tomi Janhunen, Emilia Oikarinen, Hans Tompits, and Stefan Woltran.
Modularity Aspects of Disjunctive Stable Models.
and Nonmonotonic Reasoning, volume 4483 of LNCS, pages 175–187.
Springer, May 2007.

Claire Lefévre and Pascal Nicolas.
Integrating grounding in the search process for answer set computing.
In ASPOCP: Answer Set Programming and Other Constraint Paradigms,
pages 89–103, 2008.

Vladimir Lifschitz, David Pearce, and Agustín Valverde.
Strongly equivalent logic programs.
ACM Trans. Comput. Log., 2(4):526–541, 2001.

References VIII

Ilkka Niemelä (ed.).
Language Extensions and Software Engineering for ASP.
Technical Report WP3, Working Group on Answer Set Programming
(WASP), IST-FET-2001-37004, September 2005.
Available at http://www.tcs.hut.fi/Research/Logic/wasp/
wp3/wasp-wp3-web/.
Emilia Oikarinen and Tomi Janhunen.
Achieving compositionality of the stable model semantics for Smodels
programs.
Theory and Practice of Logic Programming, 8(5–6):717–761, November
2008.
A. Dal Palù, A. Dovier, E. Pontelli, and G. Rossi.
Gasp: Answer set programming with lazy grounding.
In LaSh 2008: LOGIC AND SEARCH - Computation of structures from
declarative descriptions, 2008.

References IX

Axel Polleres, Cristina Feier, and Andreas Harth.
Rules with Contextually Scoped Negation.
In Proceedings of the 3rd European Conference on Semantic Web (ESWC
2006), volume 4011 of LNCS, pages 332–347. Springer, 2006.

Mantas Šimkus and Thomas Eiter.
FDNC: Decidable non-monotonic disjunctive logic programs with function
symbols.
In N. Dershowitz and A. Voronkov, editors, Proceedings 14th International
Conference on Logic for Programming, Artiﬁcial Intelligence and Reasoning
(LPAR 2007), number 4790 in LNCS, pages 514–530. Springer, 2007.
Extended Paper to appear in ACM Trans. Computational Logic.

References X

Tommi Syrjänen.
Omega-restricted logic programs.
and Nonmonotonic Reasoning, Vienna, Austria, September 2001.
Springer-Verlag.

Luis Tari, Chitta Baral, and Saadat Anwar.
A Language for Modular Answer Set Programming: Application to ACC
Tournament Scheduling.
In Proceedings of the 3Proceedings of the 3rd International ASP’05
Workshop, Bath, UK, 27th–29th July 2005, volume 142 of CEUR Workshop
Proceedings, pages 277–293. CEUR WS, July 2005.

Giorgio Terracina, Nicola Leone, Vincenzino Lio, and Claudio Panetta.
Experimenting with recursive queries in database and logic programming
systems.
TPLP, 8(2):129–165, 2008.

References XI

Allen Van Gelder, Kenneth A. Ross, and John S. Schlipf.
The Well-Founded Semantics for General Logic Programs.
Journal of the ACM, 38(3):620–650, 1991.

Stefan Woltran.
A common view on strong, uniform, and other notions of equivalence in
answer-set programming.
Theory and Practice of Logic Programming, 8(2):217–234, 2008.

An Answer Set Programming Tutorial 8. References 8.1 ASP Performance

The 3rd ASP Competition
Results from https://www.mat.unical.it/aspcomp20111

Input size (plain text of facts):
Reachability: 0.5 — 20 MB
Grammar Based Information Extraction: 1.5 — 5 MB
1
Many thanks to Giovambattista Ianni for insightful analysis of the results.


The 3rd ASP Competition - Scoring



clasp Benchmarking

http://www.cs.uni-potsdam.de/clasp/?page=experiments

Check some of the well-known problems: HamiltonianCycle,
HamiltonianPath, Su-DoKu, TowersOfHanoiCompetition


An Answer Set Programming Tutorial 8. References 8.2 What is ASP good for?

What is ASP good for?

Combinatorial search problems (some with substantial amount of
data):
• For instance, auctions, bio-informatics, computer-aided veriﬁcation,
conﬁguration, constraint satisfaction, diagnosis, information
integration, planning and scheduling, security analysis, semantic web,
wire-routing, zoology and linguistics, and many more
A favorite application : Using ASP as a basis for a decision support
system for NASA’s space shuttle (Gelfond et al., Texas Tech)
And more:
• Automatic synthesis of multiprocessor systems
• Inconsistency detection, diagnosis, repair, and prediction in large
biological networks
• Home monitoring for risk prevention in ambient assisted living
• General game playing


An Answer Set Programming Tutorial 8. References 8.3 What is the difference compared to Prolog?

Logic Programming – Prolog revisited

Prolog = “Programming in Logic”

Basic data structures: terms

Programs: rules and facts

Computing: Queries (goals)
• Proofs provide answers
• SLD-resolution
• uniﬁcation - basic mechanism to manipulate data structures

Extensive use of recursion



Prolog – Truly Declarative Programming?

Example
parent(John, Mary).
parent(Mary, James).
ancestor(X, Y) : − parent(X, Y).
ancestor(X, Z) : − parent(X, Y), ancestor(Y, Z).

vs.

parent(John, Mary).
parent(Mary, James).
ancestor(X, Z) : − ancestor(Y, Z), parent(X, Y).
ancestor(X, Y) : − parent(X, Y).

Query: ?- ancestor(John,W)



Desiderata

Relieve the programmer from several concerns.
It is desirable that

the order of program rules does not matter;

the order of subgoals in a rule does not matter;

termination is not subject to such order.



Desiderata

Relieve the programmer from several concerns.
It is desirable that

the order of program rules does not matter;

the order of subgoals in a rule does not matter;

termination is not subject to such order.

“Pure” declarative programming

Prolog does not satisfy these desiderata

Satisﬁed e.g. by the answer set semantics of logic programs



Programs with Negation

Prolog: “not X ” means “Negation as Failure (to prove to X )”
Different from negation in classical logic!





Example (Program P3 )

man(dilbert).
single(X) : − man(X), not husband(X).
husband(X) : − fail.






man(dilbert).

Query:
? − single(X).






man(dilbert).

Query:
? − single(X).
Answer:
X = dilbert.



Example (cont’d)
Modifying the last rule of P3 , we get P4 :

man(dilbert).
single(X) ← man(X), not husband(X).
husband(X) ← man(X), not single(X).

Result in Prolog ????



Example (cont’d)

man(dilbert).

Problem: not a single intuitive model!



Example (cont’d)

man(dilbert).

Problem: not a single intuitive model!
Two intuitive Herbrand models:

M1 = {man(dilbert), single(dilbert)}, and
M2 = {man(dilbert), husband(dilbert)} .

Which one to choose?


An Answer Set Programming Tutorial 8. References 8.4 Semantics of Logic Programs With Negatio

Semantics of Logic Programs With Negation

“War of Semantics” in Logic Programming (1980/90s):
Meaning of programs like the Dilbert example above





Great Schism: Single model vs. multiple model semantics





To date:
• Well-Founded Semantics [Van Gelder et al., 1991]
Partial model: man(dilbert) is true,
single(dilbert), husband(dilbert) are unknown





To date:

• Answer Set (alias Stable Model) Semantics by Gelfond and Lifschitz
[1988,1991].
Alternative models: M1 = {man(dilbert), single(dilbert)},
M2 = {man(dilbert), husband(dilbert)}.





To date:

• Answer Set (alias Stable Model) Semantics by Gelfond and Lifschitz
[1988,1991].
Alternative models: M1 = {man(dilbert), single(dilbert)},
M2 = {man(dilbert), husband(dilbert)}.
Agreement for so-called “stratiﬁed programs”
Different selection principles for non-stratiﬁed programs

An Answer Set Programming Tutorial 8. References 8.5 Minimality

Minimality
Semantics: disjunction is minimal (different from classical logic):
a ∨ b ∨ c.
Minimal models: {a}, {b}, and {c}.



Minimality
a ∨ b ∨ c.
actually subset minimal:
a ∨ b. a ∨ c.
Minimal models: {a} and {b, c}.



Minimality
a ∨ b ∨ c.
a ∨ b. a ∨ c.
a ∨ b. a←b
Models {a} and {a, b}, but only {a} is minimal.



Minimality
a ∨ b ∨ c.
a ∨ b. a ∨ c.
a ∨ b. a←b
Models {a} and {a, b}, but only {a} is minimal.
but minimality is not necessarily exclusive:
a ∨ b. b ∨ c. a ∨ c.
Minimal models: {a, b}, {a, c}, and {b, c}.



A logic program has multiple models in general.
Select one of these models as the canonical model.
Commonly accepted: truth of an atom in model I should be
“founded” by clauses.





Example
Given
P2 = {a ← b. b ← c. c},
truth of a in the model I = {a, b, c} is “founded.”





Example
Given
P2 = {a ← b. b ← c. c},
truth of a in the model I = {a, b, c} is “founded.”
Given
P2 = {a ← b. b ← a. c},
truth of a in the model I = {a, b, c} is not founded.



Minimal Model Semantics (cont’d)
Semantics: Prefer models with true-part as small as possible.

Deﬁnition




Deﬁnition

Theorem




Deﬁnition

Theorem

Example

For P1 = { a ← b. b ← c. c }, we have LM(P1 ) = {a, b, c}.
For P2 = { a ← b. b ← a. c }, we have LM(P2 ) = {c}.


An Answer Set Programming Tutorial 8. References 8.6 TP operator

Computation


Deﬁnition (TP Operator)
Let TP : 2HB(P) → 2HB(P) be deﬁned as

there exists some a ← b1 , . . . , bm
TP (I) = a .
in grnd(P) such that {b1 , . . . , bm } ⊆ I

0 i+1 i
We let denote TP = ∅, TP = TP (TP ), i ≥ 0.


An Answer Set Programming Tutorial 8. References 8.6 TP operator

Computation


Definition (TP Operator)
Let TP : 2HB(P) → 2HB(P) be defined as

there exists some a ← b1 , . . . , bm
TP (I) = a .
in grnd(P) such that {b1 , . . . , bm } ⊆ I

0 i+1 i
We let denote TP = ∅, TP = TP (TP ), i ≥ 0.

Fundamental result:
Theorem
i
TP has a least fixpoint, lfp(TP ), and the sequence TP , i ≥ 0,
converges to lfp(TP ).


An Answer Set Programming Tutorial 8. References 8.7 Answer Set Programming Systems for Debi



Answer Set Programming Tutorial: Modeling Problems Declaratively

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