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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 IRSESunder 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 Programming2.1 Positive Logic Programs2.2 Minimal Model Semantics
3. Stable Logic Programming
4. Extensions4.1 Disjunction4.2 Integrity Constraints
5. Answer Set for the Semantic Web5.1 DL-Programs5.2 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 solvingapproach
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 auniform way
Minh Dao-Tran DERI, July 2011 2/33
An Answer Set Programming Tutorial 1. Introduction
Knapsack Problem
Minh Dao-Tran DERI, July 2011 3/33
An Answer Set Programming Tutorial 1. Introduction
Sudoku
6 1 4 58 3 5 6
2 18 4 7 6
6 37 9 1 45 2
7 2 6 94 5 8 7
Task:
Fill in the grid so that every row, every column, and every 3x3 boxcontains the digits 1 through 9.
Minh Dao-Tran DERI, July 2011 4/33
An Answer Set Programming Tutorial 1. Introduction
Wanted!
A general-purpose, declarative approach for modeling and solvingthese and many other problems.
Declarative:• “What is the problem?”
instead of
• “How to solve the problem?”
Proposal:
Answer Set Programming (ASP) paradigm!
Minh Dao-Tran DERI, July 2011 5/33
An Answer Set Programming Tutorial 1. Introduction
Wanted!
A general-purpose, declarative approach for modeling and solvingthese and many other problems.
Declarative:• “What is the problem?”
instead of
• “How to solve the problem?”
Proposal:
Answer Set Programming (ASP) paradigm!
Minh Dao-Tran DERI, July 2011 5/33
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)
Exampleprof (supervisor(X)) ← phd_student(X).
poor(X) ← phd_student(X).phd_student(Y) ← phd_student(X), couple(X,Y).
Minh Dao-Tran DERI, July 2011 6/33
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)
Exampleprof (supervisor(X)) ← phd_student(X).
poor(X) ← phd_student(X).phd_student(Y) ← phd_student(X), couple(X,Y).
Minh Dao-Tran DERI, July 2011 6/33
An Answer Set Programming Tutorial 2. Horn Logic Programming 2.1 Positive Logic Programs
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).phd_student(Y) ← phd_student(X), couple(X, Y).
Constant symbols: peter, jane; Function symbols: supervisor.
Herbrand Universe
HU(P) ={
peter, supervisor(peter), supervisor(supervisor(peter)), . . . ,jane, supervisor(jane), supervisor(supervisor(jane)), . . .
}
Herbrand Base
HB(P) =
phd_student(peter), phd_student(jane), phd_student(supervisor(peter)), . . . ,prof (peter), prof (jane), prof (supervisor(peter)), prof (supervisor(jane)), . . . ,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))}
Minh Dao-Tran DERI, July 2011 7/33
An Answer Set Programming Tutorial 2. Horn Logic Programming 2.1 Positive Logic Programs
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).phd_student(Y) ← phd_student(X), couple(X, Y).
Constant symbols: peter, jane; Function symbols: supervisor.
Herbrand Universe
HU(P) ={
peter, supervisor(peter), supervisor(supervisor(peter)), . . . ,jane, supervisor(jane), supervisor(supervisor(jane)), . . .
}
Herbrand Base
HB(P) =
phd_student(peter), phd_student(jane), phd_student(supervisor(peter)), . . . ,prof (peter), prof (jane), prof (supervisor(peter)), prof (supervisor(jane)), . . . ,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))}
Minh Dao-Tran DERI, July 2011 7/33
An Answer Set Programming Tutorial 2. Horn Logic Programming 2.1 Positive Logic Programs
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).phd_student(Y) ← phd_student(X), couple(X, Y).
Constant symbols: peter, jane; Function symbols: supervisor.
Herbrand Universe
HU(P) ={
peter, supervisor(peter), supervisor(supervisor(peter)), . . . ,jane, supervisor(jane), supervisor(supervisor(jane)), . . .
}
Herbrand Base
HB(P) =
phd_student(peter), phd_student(jane), phd_student(supervisor(peter)), . . . ,prof (peter), prof (jane), prof (supervisor(peter)), prof (supervisor(jane)), . . . ,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))}
Minh Dao-Tran DERI, July 2011 7/33
An Answer Set Programming Tutorial 2. Horn Logic Programming 2.1 Positive Logic Programs
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).phd_student(Y) ← phd_student(X), couple(X, Y).
Constant symbols: peter, jane; Function symbols: supervisor.
Herbrand Universe
HU(P) ={
peter, supervisor(peter), supervisor(supervisor(peter)), . . . ,jane, supervisor(jane), supervisor(supervisor(jane)), . . .
}
Herbrand Base
HB(P) =
phd_student(peter), phd_student(jane), phd_student(supervisor(peter)), . . . ,prof (peter), prof (jane), prof (supervisor(peter)), prof (supervisor(jane)), . . . ,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))}
Minh Dao-Tran DERI, July 2011 7/33
An Answer Set Programming Tutorial 2. Horn Logic Programming 2.1 Positive Logic Programs
Grounding
Grounding a rule
r = poor(X)← phd_student(X).
A ground instance of r is obtained by replacing its variables by terms from HUP.
grnd(r) =
poor(peter) ← phd_student(peter).poor(jane) ← phd_student(jane).
poor(supervisor(peter)) ← phd_student(supervisor(peter))....
Grounding a program
grnd(P) =⋃
r∈P grnd(r)
Minh Dao-Tran DERI, July 2011 8/33
An Answer Set Programming Tutorial 2. Horn Logic Programming 2.1 Positive Logic Programs
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)
Minh Dao-Tran DERI, July 2011 9/33
An Answer Set Programming Tutorial 2. Horn Logic Programming 2.1 Positive Logic Programs
Example (Program P)prof (supervisor(X)) ← phd_student(X). phd_student(peter).
poor(X) ← phd_student(X). couple(peter, jane).phd_student(Y) ← phd_student(X), couple(X,Y).
Which of the following interpretations are models of P?
I1 = ∅
no
I2 = HB(P)
yes
I3 =
{phd_student(peter), couple(peter, jane), phd_student(jane),poor(peter), poor(jane), prof (supervisor(peter))
}
no
Minh Dao-Tran DERI, July 2011 10/33
An Answer Set Programming Tutorial 2. Horn Logic Programming 2.1 Positive Logic Programs
Example (Program P)prof (supervisor(X)) ← phd_student(X). phd_student(peter).
poor(X) ← phd_student(X). couple(peter, jane).phd_student(Y) ← phd_student(X), couple(X,Y).
Which of the following interpretations are models of P?
I1 = ∅ no
I2 = HB(P)
yes
I3 =
{phd_student(peter), couple(peter, jane), phd_student(jane),poor(peter), poor(jane), prof (supervisor(peter))
}
no
Minh Dao-Tran DERI, July 2011 10/33
An Answer Set Programming Tutorial 2. Horn Logic Programming 2.1 Positive Logic Programs
Example (Program P)prof (supervisor(X)) ← phd_student(X). phd_student(peter).
poor(X) ← phd_student(X). couple(peter, jane).phd_student(Y) ← phd_student(X), couple(X,Y).
Which of the following interpretations are models of P?
I1 = ∅ no
I2 = HB(P) yes
I3 =
{phd_student(peter), couple(peter, jane), phd_student(jane),poor(peter), poor(jane), prof (supervisor(peter))
}
no
Minh Dao-Tran DERI, July 2011 10/33
An Answer Set Programming Tutorial 2. Horn Logic Programming 2.1 Positive Logic Programs
Example (Program P)prof (supervisor(X)) ← phd_student(X). phd_student(peter).
poor(X) ← phd_student(X). couple(peter, jane).phd_student(Y) ← phd_student(X), couple(X,Y).
Which of the following interpretations are models of P?
I1 = ∅ no
I2 = HB(P) yes
I3 =
{phd_student(peter), couple(peter, jane), phd_student(jane),poor(peter), poor(jane), prof (supervisor(peter))
}no
Minh Dao-Tran DERI, July 2011 10/33
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.
Theorem
Every logic program P has a single minimal model (called the leastmodel), denoted LM(P).
Minh Dao-Tran DERI, July 2011 11/33
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.
Theorem
Every logic program P has a single minimal model (called the leastmodel), denoted LM(P).
Minh Dao-Tran DERI, July 2011 11/33
An Answer Set Programming Tutorial 2. Horn Logic Programming 2.2 Minimal Model Semantics
Computation
The minimal model can be computed via fixpoint iteration.
Exampleprof (supervisor(X)) ← phd_student(X). phd_student(peter).
poor(X) ← phd_student(X). couple(peter, jane).phd_student(Y) ← phd_student(X), couple(X,Y).
T0P = ∅
T1P = {phd_student(peter), couple(peter, jane)}
T2P =
{phd_student(peter), couple(peter, jane), phd_student(jane),poor(peter), prof (supervisor(peter))
}
T3P =
phd_student(peter), couple(peter, jane), phd_student(jane),poor(peter), prof (supervisor(peter)), poor(jane),prof (supervisor(jane))
T4
P = T3P
Minh Dao-Tran DERI, July 2011 12/33
An Answer Set Programming Tutorial 2. Horn Logic Programming 2.2 Minimal Model Semantics
Computation
The minimal model can be computed via fixpoint iteration.
Exampleprof (supervisor(X)) ← phd_student(X). phd_student(peter).
poor(X) ← phd_student(X). couple(peter, jane).phd_student(Y) ← phd_student(X), couple(X,Y).
T0P = ∅
T1P = {phd_student(peter), couple(peter, jane)}
T2P =
{phd_student(peter), couple(peter, jane), phd_student(jane),poor(peter), prof (supervisor(peter))
}
T3P =
phd_student(peter), couple(peter, jane), phd_student(jane),poor(peter), prof (supervisor(peter)), poor(jane),prof (supervisor(jane))
T4
P = T3P
Minh Dao-Tran DERI, July 2011 12/33
An Answer Set Programming Tutorial 2. Horn Logic Programming 2.2 Minimal Model Semantics
Computation
The minimal model can be computed via fixpoint iteration.
Exampleprof (supervisor(X)) ← phd_student(X). phd_student(peter).
poor(X) ← phd_student(X). couple(peter, jane).phd_student(Y) ← phd_student(X), couple(X,Y).
T0P = ∅
T1P = {phd_student(peter), couple(peter, jane)}
T2P =
{phd_student(peter), couple(peter, jane), phd_student(jane),poor(peter), prof (supervisor(peter))
}
T3P =
phd_student(peter), couple(peter, jane), phd_student(jane),poor(peter), prof (supervisor(peter)), poor(jane),prof (supervisor(jane))
T4
P = T3P
Minh Dao-Tran DERI, July 2011 12/33
An Answer Set Programming Tutorial 2. Horn Logic Programming 2.2 Minimal Model Semantics
Computation
The minimal model can be computed via fixpoint iteration.
Exampleprof (supervisor(X)) ← phd_student(X). phd_student(peter).
poor(X) ← phd_student(X). couple(peter, jane).phd_student(Y) ← phd_student(X), couple(X,Y).
T0P = ∅
T1P = {phd_student(peter), couple(peter, jane)}
T2P =
{phd_student(peter), couple(peter, jane), phd_student(jane),poor(peter), prof (supervisor(peter))
}
T3P =
phd_student(peter), couple(peter, jane), phd_student(jane),poor(peter), prof (supervisor(peter)), poor(jane),prof (supervisor(jane))
T4P = T3
P
Minh Dao-Tran DERI, July 2011 12/33
An Answer Set Programming Tutorial 2. Horn Logic Programming 2.2 Minimal Model Semantics
Computation
The minimal model can be computed via fixpoint iteration.
Exampleprof (supervisor(X)) ← phd_student(X). phd_student(peter).
poor(X) ← phd_student(X). couple(peter, jane).phd_student(Y) ← phd_student(X), couple(X,Y).
T0P = ∅
T1P = {phd_student(peter), couple(peter, jane)}
T2P =
{phd_student(peter), couple(peter, jane), phd_student(jane),poor(peter), prof (supervisor(peter))
}
T3P =
phd_student(peter), couple(peter, jane), phd_student(jane),poor(peter), prof (supervisor(peter)), poor(jane),prof (supervisor(jane))
T4
P = T3P
Minh Dao-Tran DERI, July 2011 12/33
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.”
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!
Minh Dao-Tran DERI, July 2011 13/33
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.”
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!Minh Dao-Tran DERI, July 2011 13/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
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)
M′ = {man(dilbert)}, get {man(dilbert), single(dilbert), husband(dilbert)}
M′′ = {man(dilbert), single(dilbert), husband(dilbert)}, get {man(dilbert)}.
Minh Dao-Tran DERI, July 2011 14/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
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)
M′ = {man(dilbert)}, get {man(dilbert), single(dilbert), husband(dilbert)}
M′′ = {man(dilbert), single(dilbert), husband(dilbert)}, get {man(dilbert)}.
Minh Dao-Tran DERI, July 2011 14/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
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)
M′ = {man(dilbert)}, get {man(dilbert), single(dilbert), husband(dilbert)}
M′′ = {man(dilbert), single(dilbert), husband(dilbert)}, get {man(dilbert)}.
Minh Dao-Tran DERI, July 2011 14/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
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)
M′ = {man(dilbert)}, get {man(dilbert), single(dilbert), husband(dilbert)}
M′′ = {man(dilbert), single(dilbert), husband(dilbert)}, get {man(dilbert)}.
Minh Dao-Tran DERI, July 2011 14/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
Stable Models
Definition (Gelfond-Lifschitz Reduct PM 1988)The GL-reduct (simply reduct) of a ground program P w.r.t. aninterpretation 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.
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).
Minh Dao-Tran DERI, July 2011 15/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
Stable Models
Definition (Gelfond-Lifschitz Reduct PM 1988)The GL-reduct (simply reduct) of a ground program P w.r.t. aninterpretation 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.
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).
Minh Dao-Tran DERI, July 2011 15/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
Stable Models
Definition (Gelfond-Lifschitz Reduct PM 1988)The GL-reduct (simply reduct) of a ground program P w.r.t. aninterpretation 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.
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).
Minh Dao-Tran DERI, July 2011 15/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
Example (P2 cont’d)
man(dilbert). (f1)
single(dilbert)← man(dilbert), not husband(dilbert). (r1)
husband(dilbert)← man(dilbert), not single(dilbert). (r2)
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.
Minh Dao-Tran DERI, July 2011 16/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
Example (P2 cont’d)
man(dilbert). (f1)
single(dilbert)← man(dilbert), not husband(dilbert). (r1)
husband(dilbert)← man(dilbert), not single(dilbert). (r2)
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.
Minh Dao-Tran DERI, July 2011 16/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
Example (P2 cont’d)
man(dilbert). (f1)
single(dilbert)← man(dilbert), not husband(dilbert). (r1)
husband(dilbert)← man(dilbert), not single(dilbert). (r2)
M1 = {man(dilbert), single(dilbert)}:
reduct PM12 :
man(dilbert).
single(dilbert)← man(dilbert).
The least model of PM12 is {man(dilbert), single(dilbert)} = M1.
M2 = {man(dilbert), husband(dilbert)}: by symmetry of husband andsingle, also M2 is stable.
Minh Dao-Tran DERI, July 2011 17/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
Example (P2 cont’d)
man(dilbert). (f1)
single(dilbert)← man(dilbert), not husband(dilbert). (r1)
husband(dilbert)← man(dilbert), not single(dilbert). (r2)
M1 = {man(dilbert), single(dilbert)}:
reduct PM12 :
man(dilbert).
single(dilbert)← man(dilbert).
The least model of PM12 is {man(dilbert), single(dilbert)} = M1.
M2 = {man(dilbert), husband(dilbert)}: by symmetry of husband andsingle, also M2 is stable.
Minh Dao-Tran DERI, July 2011 17/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
Example (P2 cont’d)
man(dilbert). (f1)
single(dilbert)← man(dilbert), not husband(dilbert). (r1)
husband(dilbert)← man(dilbert), not single(dilbert). (r2)
M1 = {man(dilbert), single(dilbert)}:
reduct PM12 :
man(dilbert).
single(dilbert)← man(dilbert).
The least model of PM12 is {man(dilbert), single(dilbert)} = M1.
M2 = {man(dilbert), husband(dilbert)}: by symmetry of husband andsingle, also M2 is stable.
Minh Dao-Tran DERI, July 2011 17/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
Example (P2 cont’d)
man(dilbert). (f1)
single(dilbert)← man(dilbert), not husband(dilbert). (r1)
husband(dilbert)← man(dilbert), not single(dilbert). (r2)
M3 = {man(dilbert), single(dilbert), husband(dilbert)}:
PM32 is
man(dilbert).
LM(PM32 ) = {man(dilbert)} 6= M3.
M4 = {man(dilbert)}:
PM42 is
man(dilbert).
single(dilbert)← man(dilbert).
husband(dilbert)← man(dilbert).
LM(PM42 ) = {man(dilbert), single(dilbert), husband(dilbert)} 6= M4.
Minh Dao-Tran DERI, July 2011 18/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
Example (P2 cont’d)
man(dilbert). (f1)
single(dilbert)← man(dilbert), not husband(dilbert). (r1)
husband(dilbert)← man(dilbert), not single(dilbert). (r2)
M3 = {man(dilbert), single(dilbert), husband(dilbert)}:
PM32 is
man(dilbert).
LM(PM32 ) = {man(dilbert)} 6= M3.
M4 = {man(dilbert)}:
PM42 is
man(dilbert).
single(dilbert)← man(dilbert).
husband(dilbert)← man(dilbert).
LM(PM42 ) = {man(dilbert), single(dilbert), husband(dilbert)} 6= M4.
Minh Dao-Tran DERI, July 2011 18/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
Example (P2 cont’d)
man(dilbert). (f1)
single(dilbert)← man(dilbert), not husband(dilbert). (r1)
husband(dilbert)← man(dilbert), not single(dilbert). (r2)
M3 = {man(dilbert), single(dilbert), husband(dilbert)}:
PM32 is
man(dilbert).
LM(PM32 ) = {man(dilbert)} 6= M3.
M4 = {man(dilbert)}:
PM42 is
man(dilbert).
single(dilbert)← man(dilbert).
husband(dilbert)← man(dilbert).
LM(PM42 ) = {man(dilbert), single(dilbert), husband(dilbert)} 6= M4.
Minh Dao-Tran DERI, July 2011 18/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
Example (P2 cont’d)
man(dilbert). (f1)
single(dilbert)← man(dilbert), not husband(dilbert). (r1)
husband(dilbert)← man(dilbert), not single(dilbert). (r2)
M3 = {man(dilbert), single(dilbert), husband(dilbert)}:
PM32 is
man(dilbert).
LM(PM32 ) = {man(dilbert)} 6= M3.
M4 = {man(dilbert)}:
PM42 is
man(dilbert).
single(dilbert)← man(dilbert).
husband(dilbert)← man(dilbert).
LM(PM42 ) = {man(dilbert), single(dilbert), husband(dilbert)} 6= M4.
Minh Dao-Tran DERI, July 2011 18/33
An Answer Set Programming Tutorial 3. Stable Logic Programming
Programs with Variables
As for positive programs, view a program rule as a shorthand for allits 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 ofgrnd(P).
Minh Dao-Tran DERI, July 2011 19/33
An Answer Set Programming Tutorial 4. Extensions
Extensions
Many extensions exist, partly motivated by applicationsSome are syntactic sugar, other strictly add expressivenessIncomplete 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]
Minh Dao-Tran DERI, July 2011 20/33
An Answer Set Programming Tutorial 4. Extensions
Extensions
Many extensions exist, partly motivated by applicationsSome are syntactic sugar, other strictly add expressivenessIncomplete 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]
Minh Dao-Tran DERI, July 2011 20/33
An Answer Set Programming Tutorial 4. Extensions 4.1 Disjunction
Disjunction
The use of disjunction is natural to express indefinite knowledge.
Example
female(X) ∨ male(X)← person(X).
single(dilbert) ∨ husband(dilbert)← man(dilbert).
Disjunction is natural for expressing a “guess” and to createnon-determinism
Example
ok(C) ∨ notok(C)← component(C).
Minh Dao-Tran DERI, July 2011 21/33
An Answer Set Programming Tutorial 4. Extensions 4.1 Disjunction
Disjunction
The use of disjunction is natural to express indefinite knowledge.
Example
female(X) ∨ male(X)← person(X).
single(dilbert) ∨ husband(dilbert)← man(dilbert).
Disjunction is natural for expressing a “guess” and to createnon-determinism
Example
ok(C) ∨ notok(C)← component(C).
Minh Dao-Tran DERI, July 2011 21/33
An Answer Set Programming Tutorial 4. Extensions 4.1 Disjunction
Disjunction
The use of disjunction is natural to express indefinite knowledge.
Example
female(X) ∨ male(X)← person(X).
single(dilbert) ∨ husband(dilbert)← man(dilbert).
Disjunction is natural for expressing a “guess” and to createnon-determinism
Example
ok(C) ∨ notok(C)← component(C).
Minh Dao-Tran DERI, July 2011 21/33
An Answer Set Programming Tutorial 4. Extensions 4.1 Disjunction
Disjunction
The use of disjunction is natural to express indefinite knowledge.
Example
female(X) ∨ male(X)← person(X).
single(dilbert) ∨ husband(dilbert)← man(dilbert).
Disjunction is natural for expressing a “guess” and to createnon-determinism
Example
ok(C) ∨ notok(C)← component(C).
Minh Dao-Tran DERI, July 2011 21/33
An Answer Set Programming Tutorial 4. Extensions 4.2 Integrity Constraints
Integrity Constraints
Addingfail← 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 satisfyintegrity constraints.
Short:
Integrity Constraint
← b1, . . . , bm, not c1, . . . , not cn.
Minh Dao-Tran DERI, July 2011 22/33
An Answer Set Programming Tutorial 4. Extensions 4.2 Integrity Constraints
Integrity Constraints
Addingfail← 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 satisfyintegrity constraints.
Short:
Integrity Constraint
← b1, . . . , bm, not c1, . . . , not cn.
Minh Dao-Tran DERI, July 2011 22/33
An Answer Set Programming Tutorial 4. Extensions 4.2 Integrity Constraints
Example (P2 cont’d)
man(dilbert). (f1)
single(dilbert) ∨ husband(dilbert)← man(dilbert). (r′1)
← husband(X), not wedding_ring(X). (c1)
The constraint c1 eliminates models in which there is no evidence fora husband having a wedding ring.
Single stable model: M1 = {man(dilbert), single(dilbert)}
Minh Dao-Tran DERI, July 2011 23/33
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 XSBaspps 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-Complex2+ Smodels_cc
Minh Dao-Tran DERI, July 2011 24/33
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 gringowill be built in
More details athttp://www.kr.tuwien.ac.at/staff/tkren/deb.html
Minh Dao-Tran DERI, July 2011 25/33
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 programsunder 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
Minh Dao-Tran DERI, July 2011 26/33
An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web 5.1 DL-Programs
The famous tweety example
L =
Flier v ¬NonFlier,Penguin v Bird,Penguin v NonFlier,—————–Bird v FlierBird(tweety),Penguin(joe)
P =
{flies(X) ← DL[Flier ] flies;Bird](X),
not DL[Flier ] flies;¬Flier](X).
}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).
Minh Dao-Tran DERI, July 2011 27/33
An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web 5.1 DL-Programs
The famous tweety example
L =
Flier v ¬NonFlier,Penguin v Bird,Penguin v NonFlier,—————–Bird v FlierBird(tweety),Penguin(joe)
P =
{flies(X) ← DL[Flier ] flies;Bird](X),
not DL[Flier ] flies;¬Flier](X).
}
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).
Minh Dao-Tran DERI, July 2011 27/33
An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web 5.1 DL-Programs
The famous tweety example
L =
Flier v ¬NonFlier,Penguin v Bird,Penguin v NonFlier,—————–Bird v FlierBird(tweety),Penguin(joe)
P =
{flies(X) ← DL[Flier ] flies;Bird](X),
not DL[Flier ] flies;¬Flier](X).
}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).
Minh Dao-Tran DERI, July 2011 27/33
An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web 5.1 DL-Programs
The famous tweety example
L =
Flier v ¬NonFlier,Penguin v Bird,Penguin v NonFlier,—————–Bird v FlierBird(tweety),Penguin(joe)
P =
{flies(X) ← DL[Flier ] flies;Bird](X),
not DL[Flier ] flies;¬Flier](X).
}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).
Minh Dao-Tran DERI, July 2011 27/33
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.
P can interface multiple external sources of knowledge of any sortvia so called external atoms
P has higher order atoms
Minh Dao-Tran DERI, July 2011 28/33
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.
P can interface multiple external sources of knowledge of any sortvia so called external atoms
P has higher order atoms
Minh Dao-Tran DERI, July 2011 28/33
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.
P can interface multiple external sources of knowledge of any sortvia so called external atoms
P has higher order atoms
Minh Dao-Tran DERI, July 2011 28/33
An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web 5.2 HEX-Programs
An example
subRelation(brotherOf , relativeOf ).
brotherOf (john, al).
relativeOf (john, joe).
brotherOf (al,mick).
invites(john,X) ∨ skip(X)← X 6= john,&reach[relativeOf , john](X).
R(X,Y)← subRelation(P,R),P(X,Y).
someInvited ← invites(john,X).
← not someInvited.
← °s[invites](Min,Max),Max > 2.
Example
Input: Some data about John’s neighborhoodOutput: Possible picks for persons John might want to invite, according to someconstraints
Minh Dao-Tran DERI, July 2011 29/33
An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web 5.2 HEX-Programs
An example
subRelation(brotherOf , relativeOf ).
brotherOf (john, al).
relativeOf (john, joe).
brotherOf (al,mick).
invites(john,X) ∨ skip(X)← X 6= john,&reach[relativeOf , john](X).
R(X,Y)← subRelation(P,R),P(X,Y).
someInvited ← invites(john,X).
← not someInvited.
← °s[invites](Min,Max),Max > 2.
Example
Input: Some data about John’s neighborhoodOutput: Possible picks for persons John might want to invite, according to someconstraints
Minh Dao-Tran DERI, July 2011 29/33
An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web 5.2 HEX-Programs
An example
subRelation(brotherOf , relativeOf ).
brotherOf (john, al).
relativeOf (john, joe).
brotherOf (al,mick).
invites(john,X) ∨ skip(X)← X 6= john,&reach[relativeOf , john](X).
R(X,Y)← subRelation(P,R),P(X,Y).
someInvited ← invites(john,X).
← not someInvited.
← °s[invites](Min,Max),Max > 2.
Example
Input: Some data about John’s neighborhoodOutput: Possible picks for persons John might want to invite, according to someconstraints
Minh Dao-Tran DERI, July 2011 29/33
An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web 5.2 HEX-Programs
An example
subRelation(brotherOf , relativeOf ).
brotherOf (john, al).
relativeOf (john, joe).
brotherOf (al,mick).
invites(john,X) ∨ skip(X)← X 6= john,&reach[relativeOf , john](X).
R(X,Y)← subRelation(P,R),P(X,Y).
someInvited ← invites(john,X).
← not someInvited.
← °s[invites](Min,Max),Max > 2.
Example
Input: Some data about John’s neighborhoodOutput: Possible picks for persons John might want to invite, according to someconstraints
Minh Dao-Tran DERI, July 2011 29/33
An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web 5.2 HEX-Programs
Higher order atoms
subRelation(brotherOf , relativeOf ).
R(X,Y)← subRelation(P,R),P(X,Y).
brotherOf (john, al).
relativeOf (john, joe).
brotherOf (al,mick).
The device of higher order atoms
Predicate names can be variables
Constants can appear both as terms values or as predicate values
Allows (comfortable) meta-reasoning
subRelation(brotherOf , relativeOf ). ⇒ relativeOf (X,Y)← brotherOf (X,Y).
R(X,Y)← subRelation(P,R),P(X,Y).
Minh Dao-Tran DERI, July 2011 30/33
An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web 5.2 HEX-Programs
Higher order atoms
subRelation(brotherOf , relativeOf ).
R(X,Y)← subRelation(P,R),P(X,Y).
brotherOf (john, al).
relativeOf (john, joe).
brotherOf (al,mick).
The device of higher order atoms
Predicate names can be variables
Constants can appear both as terms values or as predicate values
Allows (comfortable) meta-reasoning
subRelation(brotherOf , relativeOf ). ⇒ relativeOf (X,Y)← brotherOf (X,Y).
R(X,Y)← subRelation(P,R),P(X,Y).
Minh Dao-Tran DERI, July 2011 30/33
An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web 5.2 HEX-Programs
Higher order atoms
subRelation(brotherOf , relativeOf ).
R(X,Y)← subRelation(P,R),P(X,Y).
brotherOf (john, al).
relativeOf (john, joe).
brotherOf (al,mick).
The device of higher order atoms
Predicate names can be variables
Constants can appear both as terms values or as predicate values
Allows (comfortable) meta-reasoning
subRelation(brotherOf , relativeOf ). ⇒ relativeOf (X,Y)← brotherOf (X,Y).
R(X,Y)← subRelation(P,R),P(X,Y).
Minh Dao-Tran DERI, July 2011 30/33
An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web 5.2 HEX-Programs
Higher order atoms
subRelation(brotherOf , relativeOf ).
R(X,Y)← subRelation(P,R),P(X,Y).
brotherOf (john, al).
relativeOf (john, joe).
brotherOf (al,mick).
The device of higher order atoms
Predicate names can be variables
Constants can appear both as terms values or as predicate values
Allows (comfortable) meta-reasoning
subRelation(brotherOf , relativeOf ).
⇒ relativeOf (X,Y)← brotherOf (X,Y).
R(X,Y)← subRelation(P,R),P(X,Y).
Minh Dao-Tran DERI, July 2011 30/33
An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web 5.2 HEX-Programs
Higher order atoms
subRelation(brotherOf , relativeOf ).
R(X,Y)← subRelation(P,R),P(X,Y).
brotherOf (john, al).
relativeOf (john, joe).
brotherOf (al,mick).
The device of higher order atoms
Predicate names can be variables
Constants can appear both as terms values or as predicate values
Allows (comfortable) meta-reasoning
subRelation(brotherOf , relativeOf ). ⇒ relativeOf (X,Y)← brotherOf (X,Y).
R(X,Y)← subRelation(P,R),P(X,Y).
Minh Dao-Tran DERI, July 2011 30/33
An Answer Set Programming Tutorial 5. Answer Set for the Semantic Web 5.2 HEX-Programs
External atoms
&reach[relativeOf , john](X) (2)
°s[invites](Min,Max) (3)
The device of external atoms
Each external predicate is tied to a corresponding evaluationfunction
E.g. &reach corresponds to f&reach.
For a given interpretation I, I |= &reach[relativeOf , john](x) iff&reach(I, relativeOf , john) = 1
Minh Dao-Tran DERI, July 2011 31/33
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 andJanhunen, 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-databaseevaluation [Terracina et al., 2008]
See the SEA workshop series http://sea07.cs.bath.ac.uk/
Minh Dao-Tran DERI, July 2011 32/33
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 andJanhunen, 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-databaseevaluation [Terracina et al., 2008]
See the SEA workshop series http://sea07.cs.bath.ac.uk/
Minh Dao-Tran DERI, July 2011 32/33
An Answer Set Programming Tutorial 6. Future Directions of ASP
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 competitionhttp://www.cs.kuleuven.be/∼dtai/events/ASP-competition/https://www.mat.unical.it/aspcomp2011
Minh Dao-Tran DERI, July 2011 33/33
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).
Minh Dao-Tran DERI, July 2011 34/33
An Answer Set Programming Tutorial 7. Playing with ASP
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).
Minh Dao-Tran DERI, July 2011 35/33
An Answer Set Programming Tutorial 7. Playing with ASP
Hamilton Path/Cycle
inPath(X,Y) ∨ outPath(X,Y) ← edge(X,Y).
← inPath(X,Y), inPath(X,Y1), Y 6= Y1.← inPath(X,Y), inPath(X1,Y), X 6= 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).
Minh Dao-Tran DERI, July 2011 36/33
An Answer Set Programming Tutorial 7. Playing with ASP
Hamilton Path/Cycle
inPath(X,Y) ∨ outPath(X,Y) ← edge(X,Y).
← inPath(X,Y), inPath(X,Y1), Y 6= Y1.← inPath(X,Y), inPath(X1,Y), X 6= 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).
Minh Dao-Tran DERI, July 2011 36/33
An Answer Set Programming Tutorial 7. Playing with ASP
Choosing wine
Assume that we have a Wine ontology and specification of who preferwhich type of wine.
person(“axel”). preferredWine(“axel”, “SweetWine”).person(“gibbi”). preferredWine(“gibbi”, “DessertWine”).person(“roman”). preferredWine(“roman”, “ItalianWine”).
isA(X, “SweetWine”) ← DL[SweetWine](X).isA(X, “DessertWine”) ← DL[DessertWine](X).isA(X, “ItalianWine”) ← DL[ItalianWine](X).
compliantBottle(X,Z)← preferredWine(X,Y), isA(Z,Y).
bottleChosen(X) ∨ nonbottleChosen(X)← compliantBottle(_,X).
hasBottleChosen(X)← bottleChosen(Z), compliantBottle(X,Z).
← person(X), not hasBottleChosen(X).
← bottleChosen(X).[1 : 1]
Minh Dao-Tran DERI, July 2011 37/33
An Answer Set Programming Tutorial 7. Playing with ASP
Choosing wine
Assume that we have a Wine ontology and specification of who preferwhich type of wine.
person(“axel”). preferredWine(“axel”, “SweetWine”).person(“gibbi”). preferredWine(“gibbi”, “DessertWine”).person(“roman”). preferredWine(“roman”, “ItalianWine”).
isA(X, “SweetWine”) ← DL[SweetWine](X).isA(X, “DessertWine”) ← DL[DessertWine](X).isA(X, “ItalianWine”) ← DL[ItalianWine](X).
compliantBottle(X,Z)← preferredWine(X,Y), isA(Z,Y).
bottleChosen(X) ∨ nonbottleChosen(X)← compliantBottle(_,X).
hasBottleChosen(X)← bottleChosen(Z), compliantBottle(X,Z).
← person(X), not hasBottleChosen(X).
← bottleChosen(X).[1 : 1]
Minh Dao-Tran DERI, July 2011 37/33
An Answer Set Programming Tutorial 7. Playing with ASP
Choosing wine
Assume that we have a Wine ontology and specification of who preferwhich type of wine.
person(“axel”). preferredWine(“axel”, “SweetWine”).person(“gibbi”). preferredWine(“gibbi”, “DessertWine”).person(“roman”). preferredWine(“roman”, “ItalianWine”).
isA(X, “SweetWine”) ← DL[SweetWine](X).isA(X, “DessertWine”) ← DL[DessertWine](X).isA(X, “ItalianWine”) ← DL[ItalianWine](X).
compliantBottle(X,Z)← preferredWine(X,Y), isA(Z,Y).
bottleChosen(X) ∨ nonbottleChosen(X)← compliantBottle(_,X).
hasBottleChosen(X)← bottleChosen(Z), compliantBottle(X,Z).
← person(X), not hasBottleChosen(X).
← bottleChosen(X).[1 : 1]
Minh Dao-Tran DERI, July 2011 37/33
An Answer Set Programming Tutorial 7. Playing with ASP
Choosing wine
Assume that we have a Wine ontology and specification of who preferwhich type of wine.
person(“axel”). preferredWine(“axel”, “SweetWine”).person(“gibbi”). preferredWine(“gibbi”, “DessertWine”).person(“roman”). preferredWine(“roman”, “ItalianWine”).
isA(X, “SweetWine”) ← DL[SweetWine](X).isA(X, “DessertWine”) ← DL[DessertWine](X).isA(X, “ItalianWine”) ← DL[ItalianWine](X).
compliantBottle(X,Z)← preferredWine(X,Y), isA(Z,Y).
bottleChosen(X) ∨ nonbottleChosen(X)← compliantBottle(_,X).
hasBottleChosen(X)← bottleChosen(Z), compliantBottle(X,Z).
← person(X), not hasBottleChosen(X).
← bottleChosen(X).[1 : 1]
Minh Dao-Tran DERI, July 2011 37/33
An Answer Set Programming Tutorial 7. Playing with ASP
Choosing wine
Assume that we have a Wine ontology and specification of who preferwhich type of wine.
person(“axel”). preferredWine(“axel”, “SweetWine”).person(“gibbi”). preferredWine(“gibbi”, “DessertWine”).person(“roman”). preferredWine(“roman”, “ItalianWine”).
isA(X, “SweetWine”) ← DL[SweetWine](X).isA(X, “DessertWine”) ← DL[DessertWine](X).isA(X, “ItalianWine”) ← DL[ItalianWine](X).
compliantBottle(X,Z)← preferredWine(X,Y), isA(Z,Y).
bottleChosen(X) ∨ nonbottleChosen(X)← compliantBottle(_,X).
hasBottleChosen(X)← bottleChosen(Z), compliantBottle(X,Z).
← person(X), not hasBottleChosen(X).
← bottleChosen(X).[1 : 1]
Minh Dao-Tran DERI, July 2011 37/33
An Answer Set Programming Tutorial 7. Playing with ASP
Choosing wine
Assume that we have a Wine ontology and specification of who preferwhich type of wine.
person(“axel”). preferredWine(“axel”, “SweetWine”).person(“gibbi”). preferredWine(“gibbi”, “DessertWine”).person(“roman”). preferredWine(“roman”, “ItalianWine”).
isA(X, “SweetWine”) ← DL[SweetWine](X).isA(X, “DessertWine”) ← DL[DessertWine](X).isA(X, “ItalianWine”) ← DL[ItalianWine](X).
compliantBottle(X,Z)← preferredWine(X,Y), isA(Z,Y).
bottleChosen(X) ∨ nonbottleChosen(X)← compliantBottle(_,X).
hasBottleChosen(X)← bottleChosen(Z), compliantBottle(X,Z).
← person(X), not hasBottleChosen(X).
← bottleChosen(X).[1 : 1]
Minh Dao-Tran DERI, July 2011 37/33
An Answer Set Programming Tutorial 7. Playing with ASP
Choosing wine
Assume that we have a Wine ontology and specification of who preferwhich type of wine.
person(“axel”). preferredWine(“axel”, “SweetWine”).person(“gibbi”). preferredWine(“gibbi”, “DessertWine”).person(“roman”). preferredWine(“roman”, “ItalianWine”).
isA(X, “SweetWine”) ← DL[SweetWine](X).isA(X, “DessertWine”) ← DL[DessertWine](X).isA(X, “ItalianWine”) ← DL[ItalianWine](X).
compliantBottle(X,Z)← preferredWine(X,Y), isA(Z,Y).
bottleChosen(X) ∨ nonbottleChosen(X)← compliantBottle(_,X).
hasBottleChosen(X)← bottleChosen(Z), compliantBottle(X,Z).
← person(X), not hasBottleChosen(X).
← bottleChosen(X).[1 : 1]Minh Dao-Tran DERI, July 2011 37/33
An Answer Set Programming Tutorial 7. Playing with ASP
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).
Minh Dao-Tran DERI, July 2011 38/33
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 ofKnowledge 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 the1st International Workshop on Software Engineering for Answer SetProgramming, 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 SetProgramming.
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 operationalsemantics.
AI Communications, 19(3):193–206, 2006.
Francesco Calimeri, Susanna Cozza, Giovambattista Ianni, and NicolaLeone.
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, pages407–424. 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 InternationalConference 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 JointConference 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 7thInternational Conference on Logic Programming and NonmonotonicReasoning (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 AnswerSet 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 24thInternational Conference on Logic Programming (ICLP 2008), number 5366in LNCS, pages 190–205. Springer, 2008.
Martin Gebser, Orkunt Sabuncu, and Torsten Schaub.
An incremental answer set programming based system for finite modelcomputation.
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 KnowledgeRepresentation, chapter 7, pages 285–316. Elsevier, 2008.
References VII
Tomi Janhunen, Emilia Oikarinen, Hans Tompits, and Stefan Woltran.
Modularity Aspects of Disjunctive Stable Models.
In Proceedings of the 9th International Conference on Logic Programmingand 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 Smodelsprograms.
Theory and Practice of Logic Programming, 8(5–6):717–761, November2008.
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 fromdeclarative 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 (ESWC2006), volume 4011 of LNCS, pages 332–347. Springer, 2006.
Mantas Šimkus and Thomas Eiter.
FDNC: Decidable non-monotonic disjunctive logic programs with functionsymbols.
In N. Dershowitz and A. Voronkov, editors, Proceedings 14th InternationalConference 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.
In Proceedings of the 6th International Conference on Logic Programmingand Nonmonotonic Reasoning, Vienna, Austria, September 2001.Springer-Verlag.
Luis Tari, Chitta Baral, and Saadat Anwar.
A Language for Modular Answer Set Programming: Application to ACCTournament Scheduling.
In Proceedings of the 3Proceedings of the 3rd International ASP’05Workshop, Bath, UK, 27th–29th July 2005, volume 142 of CEUR WorkshopProceedings, pages 277–293. CEUR WS, July 2005.
Giorgio Terracina, Nicola Leone, Vincenzino Lio, and Claudio Panetta.
Experimenting with recursive queries in database and logic programmingsystems.
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 inanswer-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
1Many thanks to Giovambattista Ianni for insightful analysis of the results.Minh Dao-Tran DERI, July 2011 50/33
An Answer Set Programming Tutorial 8. References 8.1 ASP Performance
The 3rd ASP Competition - Scoring
Minh Dao-Tran DERI, July 2011 51/33
An Answer Set Programming Tutorial 8. References 8.1 ASP Performance
clasp Benchmarking
http://www.cs.uni-potsdam.de/clasp/?page=experiments
Check some of the well-known problems: HamiltonianCycle,HamiltonianPath, Su-DoKu, TowersOfHanoiCompetition
Minh Dao-Tran DERI, July 2011 52/33
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 ofdata):
• For instance, auctions, bio-informatics, computer-aided verification,configuration, constraint satisfaction, diagnosis, informationintegration, 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 supportsystem 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
Minh Dao-Tran DERI, July 2011 53/33
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
Minh Dao-Tran DERI, July 2011 54/33
An Answer Set Programming Tutorial 8. References 8.3 What is the difference compared to Prolog?
Prolog – Truly Declarative Programming?
Exampleparent(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)
Minh Dao-Tran DERI, July 2011 55/33
An Answer Set Programming Tutorial 8. References 8.3 What is the difference compared to Prolog?
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
Minh Dao-Tran DERI, July 2011 56/33
An Answer Set Programming Tutorial 8. References 8.3 What is the difference compared to Prolog?
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
Minh Dao-Tran DERI, July 2011 56/33
An Answer Set Programming Tutorial 8. References 8.3 What is the difference compared to Prolog?
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), nothusband(X).husband(X) : − fail.
Query:?− single(X).
Answer:X = dilbert.
Minh Dao-Tran DERI, July 2011 57/33
An Answer Set Programming Tutorial 8. References 8.3 What is the difference compared to Prolog?
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), nothusband(X).husband(X) : − fail.
Query:?− single(X).
Answer:X = dilbert.
Minh Dao-Tran DERI, July 2011 57/33
An Answer Set Programming Tutorial 8. References 8.3 What is the difference compared to Prolog?
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), nothusband(X).husband(X) : − fail.
Query:?− single(X).
Answer:X = dilbert.
Minh Dao-Tran DERI, July 2011 57/33
An Answer Set Programming Tutorial 8. References 8.3 What is the difference compared to Prolog?
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), nothusband(X).husband(X) : − fail.
Query:?− single(X).
Answer:X = dilbert.
Minh Dao-Tran DERI, July 2011 57/33
An Answer Set Programming Tutorial 8. References 8.3 What is the difference compared to Prolog?
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 ????
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?
Minh Dao-Tran DERI, July 2011 58/33
An Answer Set Programming Tutorial 8. References 8.3 What is the difference compared to Prolog?
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 ????
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?
Minh Dao-Tran DERI, July 2011 58/33
An Answer Set Programming Tutorial 8. References 8.3 What is the difference compared to Prolog?
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 ????
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?
Minh Dao-Tran DERI, July 2011 58/33
An Answer Set Programming Tutorial 8. References 8.4 Semantics of Logic Programs With Negation
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
• 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
Minh Dao-Tran DERI, July 2011 59/33
An Answer Set Programming Tutorial 8. References 8.4 Semantics of Logic Programs With Negation
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
• 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
Minh Dao-Tran DERI, July 2011 59/33
An Answer Set Programming Tutorial 8. References 8.4 Semantics of Logic Programs With Negation
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
• 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
Minh Dao-Tran DERI, July 2011 59/33
An Answer Set Programming Tutorial 8. References 8.4 Semantics of Logic Programs With Negation
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
• 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
Minh Dao-Tran DERI, July 2011 59/33
An Answer Set Programming Tutorial 8. References 8.4 Semantics of Logic Programs With Negation
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
• 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
Minh Dao-Tran DERI, July 2011 59/33
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}.
actually subset minimal:
a ∨ b. a ∨ c.
Minimal models: {a} and {b, 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}.
Minh Dao-Tran DERI, July 2011 60/33
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}.
actually subset minimal:
a ∨ b. a ∨ c.
Minimal models: {a} and {b, 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}.
Minh Dao-Tran DERI, July 2011 60/33
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}.
actually subset minimal:
a ∨ b. a ∨ c.
Minimal models: {a} and {b, 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}.
Minh Dao-Tran DERI, July 2011 60/33
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}.
actually subset minimal:
a ∨ b. a ∨ c.
Minimal models: {a} and {b, 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}.Minh Dao-Tran DERI, July 2011 60/33
An Answer Set Programming Tutorial 8. References 8.5 Minimality
Minimal Model Semantics
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
GivenP2 = {a← b. b← c. c},
truth of a in the model I = {a, b, c} is “founded.”
GivenP2 = {a← b. b← a. c},
truth of a in the model I = {a, b, c} is not founded.
Minh Dao-Tran DERI, July 2011 61/33
An Answer Set Programming Tutorial 8. References 8.5 Minimality
Minimal Model Semantics
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
GivenP2 = {a← b. b← c. c},
truth of a in the model I = {a, b, c} is “founded.”
GivenP2 = {a← b. b← a. c},
truth of a in the model I = {a, b, c} is not founded.
Minh Dao-Tran DERI, July 2011 61/33
An Answer Set Programming Tutorial 8. References 8.5 Minimality
Minimal Model Semantics
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
GivenP2 = {a← b. b← c. c},
truth of a in the model I = {a, b, c} is “founded.”
GivenP2 = {a← b. b← a. c},
truth of a in the model I = {a, b, c} is not founded.
Minh Dao-Tran DERI, July 2011 61/33
An Answer Set Programming Tutorial 8. References 8.5 Minimality
Minimal Model Semantics (cont’d)
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.
Theorem
Every logic program P has a single minimal model (called the leastmodel), denoted LM(P).
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}.
Minh Dao-Tran DERI, July 2011 62/33
An Answer Set Programming Tutorial 8. References 8.5 Minimality
Minimal Model Semantics (cont’d)
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.
Theorem
Every logic program P has a single minimal model (called the leastmodel), denoted LM(P).
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}.
Minh Dao-Tran DERI, July 2011 62/33
An Answer Set Programming Tutorial 8. References 8.5 Minimality
Minimal Model Semantics (cont’d)
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.
Theorem
Every logic program P has a single minimal model (called the leastmodel), denoted LM(P).
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}.
Minh Dao-Tran DERI, July 2011 62/33
An Answer Set Programming Tutorial 8. References 8.6 TP operator
Computation
The minimal model can be computed via fixpoint iteration.
Definition (TP Operator)
Let TP : 2HB(P) → 2HB(P) be defined as
TP(I) ={
a∣∣∣∣ there exists some a← b1, . . . , bm
in grnd(P) such that {b1, . . . , bm} ⊆ I
}.
We let denote T0P = ∅, T i+1
P = TP(T iP), i ≥ 0.
Fundamental result:
Theorem
TP has a least fixpoint, lfp(TP), and the sequence 〈T iP〉, i ≥ 0,
converges to lfp(TP).
Minh Dao-Tran DERI, July 2011 63/33
An Answer Set Programming Tutorial 8. References 8.6 TP operator
Computation
The minimal model can be computed via fixpoint iteration.
Definition (TP Operator)
Let TP : 2HB(P) → 2HB(P) be defined as
TP(I) ={
a∣∣∣∣ there exists some a← b1, . . . , bm
in grnd(P) such that {b1, . . . , bm} ⊆ I
}.
We let denote T0P = ∅, T i+1
P = TP(T iP), i ≥ 0.
Fundamental result:
Theorem
TP has a least fixpoint, lfp(TP), and the sequence 〈T iP〉, i ≥ 0,
converges to lfp(TP).
Minh Dao-Tran DERI, July 2011 63/33
An Answer Set Programming Tutorial 8. References 8.7 Answer Set Programming Systems for Debian/Ubuntu
Answer Set Programming Systems for Debian/Ubuntu
Minh Dao-Tran DERI, July 2011 64/33
An Answer Set Programming Tutorial 8. References 8.7 Answer Set Programming Systems for Debian/Ubuntu
Answer Set Programming Systems for Debian/Ubuntu
Minh Dao-Tran DERI, July 2011 65/33
An Answer Set Programming Tutorial 8. References 8.7 Answer Set Programming Systems for Debian/Ubuntu
Answer Set Programming Systems for Debian/Ubuntu
Minh Dao-Tran DERI, July 2011 66/33
An Answer Set Programming Tutorial 8. References 8.7 Answer Set Programming Systems for Debian/Ubuntu
Answer Set Programming Systems for Debian/Ubuntu
Minh Dao-Tran DERI, July 2011 67/33
An Answer Set Programming Tutorial 8. References 8.7 Answer Set Programming Systems for Debian/Ubuntu
Answer Set Programming Systems for Debian/Ubuntu
Minh Dao-Tran DERI, July 2011 68/33
An Answer Set Programming Tutorial 8. References 8.7 Answer Set Programming Systems for Debian/Ubuntu
Answer Set Programming Systems for Debian/Ubuntu
Minh Dao-Tran DERI, July 2011 69/33