unit 2 -- asp extensions › asptut › slides › unit2.pdf · 2008-01-30 · intro weak...
TRANSCRIPT
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
Answer Set Programming for the Semantic Web
Tutorial
Thomas Eiter, Roman Schindlauer (TU Wien)Giovambattista Ianni (TU Wien, Univ. della Calabria)
Axel Polleres (Univ. Rey Juan Carlos, Madrid)
Supported by IST REWERSE, FWF Project P17212-N04, CICyT project TIC-2003-9001-C02.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
Unit 2 � ASP Extensions
G. Ianni
Dipartimento di Matematica - Università della Calabria
European Semantic Web Conference 2006
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
Unit Outline
1 Introduction
2 Weak constraints
3 Aggregate Atoms
4 Frame Logic Syntax
5 Template Predicates
6 References
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
Logic Programming Extensions
• Besides disjunction and strong negation, many extensions ofnormal logic programs have been proposed
• Some of these extensions are motivated by applications
• Some of these extensions are syntactic sugar, other strictly addexpressiveness
• Comprehensive survey of extensions:
See http://www.tcs.hut.fi/Research/Logic/wasp/wp3/
• Here, we consider some DLV speci�c extensions.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
Logic Programming Extensions
• Besides disjunction and strong negation, many extensions ofnormal logic programs have been proposed
• Some of these extensions are motivated by applications
• Some of these extensions are syntactic sugar, other strictly addexpressiveness
• Comprehensive survey of extensions:
See http://www.tcs.hut.fi/Research/Logic/wasp/wp3/
• Here, we consider some DLV speci�c extensions.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
Logic Programming Extensions
• Besides disjunction and strong negation, many extensions ofnormal logic programs have been proposed
• Some of these extensions are motivated by applications
• Some of these extensions are syntactic sugar, other strictly addexpressiveness
• Comprehensive survey of extensions:
See http://www.tcs.hut.fi/Research/Logic/wasp/wp3/
• Here, we consider some DLV speci�c extensions.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
Logic Programming Extensions
• Besides disjunction and strong negation, many extensions ofnormal logic programs have been proposed
• Some of these extensions are motivated by applications
• Some of these extensions are syntactic sugar, other strictly addexpressiveness
• Comprehensive survey of extensions:
See http://www.tcs.hut.fi/Research/Logic/wasp/wp3/
• Here, we consider some DLV speci�c extensions.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
Logic Programming Extensions
• Besides disjunction and strong negation, many extensions ofnormal logic programs have been proposed
• Some of these extensions are motivated by applications
• Some of these extensions are syntactic sugar, other strictly addexpressiveness
• Comprehensive survey of extensions:
See http://www.tcs.hut.fi/Research/Logic/wasp/wp3/
• Here, we consider some DLV speci�c extensions.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints
• Allow the formalization of optimization problems in an easyand natural way.
• Constraints vs. weak constraints:• Constraints �kill� unwanted models;• Weak constraints express desiderata which should be satis�ed,
if possible.
• The answer sets of a program P with a set W of weakconstraints are those answer sets of P which minimize thenumber of violated constraints.
• Such answer sets are called optimal or best models of (P,W ).
• Other solvers feature similar constructs.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints
• Allow the formalization of optimization problems in an easyand natural way.
• Constraints vs. weak constraints:• Constraints �kill� unwanted models;• Weak constraints express desiderata which should be satis�ed,
if possible.
• The answer sets of a program P with a set W of weakconstraints are those answer sets of P which minimize thenumber of violated constraints.
• Such answer sets are called optimal or best models of (P,W ).
• Other solvers feature similar constructs.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints
• Allow the formalization of optimization problems in an easyand natural way.
• Constraints vs. weak constraints:• Constraints �kill� unwanted models;• Weak constraints express desiderata which should be satis�ed,
if possible.
• The answer sets of a program P with a set W of weakconstraints are those answer sets of P which minimize thenumber of violated constraints.
• Such answer sets are called optimal or best models of (P,W ).
• Other solvers feature similar constructs.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints
• Allow the formalization of optimization problems in an easyand natural way.
• Constraints vs. weak constraints:• Constraints �kill� unwanted models;• Weak constraints express desiderata which should be satis�ed,
if possible.
• The answer sets of a program P with a set W of weakconstraints are those answer sets of P which minimize thenumber of violated constraints.
• Such answer sets are called optimal or best models of (P,W ).
• Other solvers feature similar constructs.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints
• Allow the formalization of optimization problems in an easyand natural way.
• Constraints vs. weak constraints:• Constraints �kill� unwanted models;• Weak constraints express desiderata which should be satis�ed,
if possible.
• The answer sets of a program P with a set W of weakconstraints are those answer sets of P which minimize thenumber of violated constraints.
• Such answer sets are called optimal or best models of (P,W ).
• Other solvers feature similar constructs.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints
• Allow the formalization of optimization problems in an easyand natural way.
• Constraints vs. weak constraints:• Constraints �kill� unwanted models;• Weak constraints express desiderata which should be satis�ed,
if possible.
• The answer sets of a program P with a set W of weakconstraints are those answer sets of P which minimize thenumber of violated constraints.
• Such answer sets are called optimal or best models of (P,W ).
• Other solvers feature similar constructs.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints
• Allow the formalization of optimization problems in an easyand natural way.
• Constraints vs. weak constraints:• Constraints �kill� unwanted models;• Weak constraints express desiderata which should be satis�ed,
if possible.
• The answer sets of a program P with a set W of weakconstraints are those answer sets of P which minimize thenumber of violated constraints.
• Such answer sets are called optimal or best models of (P,W ).
• Other solvers feature similar constructs.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Syntax and Semantics
• Syntax:
: ~ b1, · · · , bk, not bk+1, · · · , not bm. [Weight : Level]
• In the presence of weights, best models minimize the sum ofthe weights of violated constraints.
• Semantics: minimizes the violation of constraints with highestpriority level �rst; then with the lower priority levels indescending order.
• Level part is syntactic sugar, can be compiled into weights.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Syntax and Semantics
• Syntax:
: ~ b1, · · · , bk, not bk+1, · · · , not bm. [Weight : Level]
• In the presence of weights, best models minimize the sum ofthe weights of violated constraints.
• Semantics: minimizes the violation of constraints with highestpriority level �rst; then with the lower priority levels indescending order.
• Level part is syntactic sugar, can be compiled into weights.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Syntax and Semantics
• Syntax:
: ~ b1, · · · , bk, not bk+1, · · · , not bm. [Weight : Level]
• In the presence of weights, best models minimize the sum ofthe weights of violated constraints.
• Semantics: minimizes the violation of constraints with highestpriority level �rst; then with the lower priority levels indescending order.
• Level part is syntactic sugar, can be compiled into weights.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Syntax and Semantics
• Syntax:
: ~ b1, · · · , bk, not bk+1, · · · , not bm. [Weight : Level]
• In the presence of weights, best models minimize the sum ofthe weights of violated constraints.
• Semantics: minimizes the violation of constraints with highestpriority level �rst; then with the lower priority levels indescending order.
• Level part is syntactic sugar, can be compiled into weights.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints: Examples
a v b.
c :- b.
:~ a.
:~ b.
:~ c.
Best model: a Cost ([Weight:Level]): <[1:1]> Answer set {b, c}is discarded because it violates two weak constraints!
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints: Examples
a v b.
c :- b.
:~ a.
:~ b.
:~ c.
Best model: a Cost ([Weight:Level]): <[1:1]> Answer set {b, c}is discarded because it violates two weak constraints!
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints: Examples
a v b.
c :- b.
:~ a.
:~ b.
:~ c.
Best model: a Cost ([Weight:Level]): <[1:1]> Answer set {b, c}is discarded because it violates two weak constraints!
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints: Examples /2
a v b.
:~ a. [1:] :~ a. [1:] :~ b. [2:]
Best model: b Cost ([Weight:Level]): <[2:1]>Best model: a Cost ([Weight:Level]): <[2:1]>
a v b1 v b2.
:~ a. [:1] :~ b1. [:2] :~ b2. [:2]
Best model: a Cost ([Weight:Level]): <[1:1],[0:2]>
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints: Examples /2
a v b.
:~ a. [1:] :~ a. [1:] :~ b. [2:]
Best model: b Cost ([Weight:Level]): <[2:1]>Best model: a Cost ([Weight:Level]): <[2:1]>
a v b1 v b2.
:~ a. [:1] :~ b1. [:2] :~ b2. [:2]
Best model: a Cost ([Weight:Level]): <[1:1],[0:2]>
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints: Examples /2
a v b.
:~ a. [1:] :~ a. [1:] :~ b. [2:]
Best model: b Cost ([Weight:Level]): <[2:1]>Best model: a Cost ([Weight:Level]): <[2:1]>
a v b1 v b2.
:~ a. [:1] :~ b1. [:2] :~ b2. [:2]
Best model: a Cost ([Weight:Level]): <[1:1],[0:2]>
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints: Examples /2
a v b.
:~ a. [1:] :~ a. [1:] :~ b. [2:]
Best model: b Cost ([Weight:Level]): <[2:1]>Best model: a Cost ([Weight:Level]): <[2:1]>
a v b1 v b2.
:~ a. [:1] :~ b1. [:2] :~ b2. [:2]
Best model: a Cost ([Weight:Level]): <[1:1],[0:2]>
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
A bigger example - Employee Assignment
• Goal: Divide employees in two project groups p1 and p21.
1 Skills of group members should be di�erent.2 Persons in the same group should not be married to each other.3 Members of a group should possibly know each other.
• Requirement 1) is more important than 2) and 3), which areequally important
• Layers express the relative importance of the requirements.
assign(X,p1) v assign(X,p2) :- employee(X).
:~ assign(X,P), assign(Y,P), same_skill(X,Y). [:2]
:~ assign(X,P), assign(Y,P), married(X,Y). [:1]
:~ assign(X,P), assign(Y,P), X!=Y, not know(X,Y).[:1]
1Example assignment.dlvG. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
A bigger example - Employee Assignment
• Goal: Divide employees in two project groups p1 and p21.
1 Skills of group members should be di�erent.2 Persons in the same group should not be married to each other.3 Members of a group should possibly know each other.
• Requirement 1) is more important than 2) and 3), which areequally important
• Layers express the relative importance of the requirements.
assign(X,p1) v assign(X,p2) :- employee(X).
:~ assign(X,P), assign(Y,P), same_skill(X,Y). [:2]
:~ assign(X,P), assign(Y,P), married(X,Y). [:1]
:~ assign(X,P), assign(Y,P), X!=Y, not know(X,Y).[:1]
1Example assignment.dlvG. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
A bigger example - Employee Assignment
• Goal: Divide employees in two project groups p1 and p21.
1 Skills of group members should be di�erent.2 Persons in the same group should not be married to each other.3 Members of a group should possibly know each other.
• Requirement 1) is more important than 2) and 3), which areequally important
• Layers express the relative importance of the requirements.
assign(X,p1) v assign(X,p2) :- employee(X).
:~ assign(X,P), assign(Y,P), same_skill(X,Y). [:2]
:~ assign(X,P), assign(Y,P), married(X,Y). [:1]
:~ assign(X,P), assign(Y,P), X!=Y, not know(X,Y).[:1]
1Example assignment.dlvG. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
A bigger example - Employee Assignment
• Goal: Divide employees in two project groups p1 and p21.
1 Skills of group members should be di�erent.2 Persons in the same group should not be married to each other.3 Members of a group should possibly know each other.
• Requirement 1) is more important than 2) and 3), which areequally important
• Layers express the relative importance of the requirements.
assign(X,p1) v assign(X,p2) :- employee(X).
:~ assign(X,P), assign(Y,P), same_skill(X,Y). [:2]
:~ assign(X,P), assign(Y,P), married(X,Y). [:1]
:~ assign(X,P), assign(Y,P), X!=Y, not know(X,Y).[:1]
1Example assignment.dlvG. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
A bigger example - Employee Assignment
• Goal: Divide employees in two project groups p1 and p21.
1 Skills of group members should be di�erent.2 Persons in the same group should not be married to each other.3 Members of a group should possibly know each other.
• Requirement 1) is more important than 2) and 3), which areequally important
• Layers express the relative importance of the requirements.
assign(X,p1) v assign(X,p2) :- employee(X).
:~ assign(X,P), assign(Y,P), same_skill(X,Y). [:2]
:~ assign(X,P), assign(Y,P), married(X,Y). [:1]
:~ assign(X,P), assign(Y,P), X!=Y, not know(X,Y).[:1]
1Example assignment.dlvG. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
A bigger example - Employee Assignment
• Goal: Divide employees in two project groups p1 and p21.
1 Skills of group members should be di�erent.2 Persons in the same group should not be married to each other.3 Members of a group should possibly know each other.
• Requirement 1) is more important than 2) and 3), which areequally important
• Layers express the relative importance of the requirements.
assign(X,p1) v assign(X,p2) :- employee(X).
:~ assign(X,P), assign(Y,P), same_skill(X,Y). [:2]
:~ assign(X,P), assign(Y,P), married(X,Y). [:1]
:~ assign(X,P), assign(Y,P), X!=Y, not know(X,Y).[:1]
1Example assignment.dlvG. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
A bigger example - Employee Assignment
• Goal: Divide employees in two project groups p1 and p21.
1 Skills of group members should be di�erent.2 Persons in the same group should not be married to each other.3 Members of a group should possibly know each other.
• Requirement 1) is more important than 2) and 3), which areequally important
• Layers express the relative importance of the requirements.
assign(X,p1) v assign(X,p2) :- employee(X).
:~ assign(X,P), assign(Y,P), same_skill(X,Y). [:2]
:~ assign(X,P), assign(Y,P), married(X,Y). [:1]
:~ assign(X,P), assign(Y,P), X!=Y, not know(X,Y).[:1]
1Example assignment.dlvG. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
A bigger example - Employee Assignment
• Goal: Divide employees in two project groups p1 and p21.
1 Skills of group members should be di�erent.2 Persons in the same group should not be married to each other.3 Members of a group should possibly know each other.
• Requirement 1) is more important than 2) and 3), which areequally important
• Layers express the relative importance of the requirements.
assign(X,p1) v assign(X,p2) :- employee(X).
:~ assign(X,P), assign(Y,P), same_skill(X,Y). [:2]
:~ assign(X,P), assign(Y,P), married(X,Y). [:1]
:~ assign(X,P), assign(Y,P), X!=Y, not know(X,Y).[:1]
1Example assignment.dlvG. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
A bigger example - Employee Assignment
• Goal: Divide employees in two project groups p1 and p21.
1 Skills of group members should be di�erent.2 Persons in the same group should not be married to each other.3 Members of a group should possibly know each other.
• Requirement 1) is more important than 2) and 3), which areequally important
• Layers express the relative importance of the requirements.
assign(X,p1) v assign(X,p2) :- employee(X).
:~ assign(X,P), assign(Y,P), same_skill(X,Y). [:2]
:~ assign(X,P), assign(Y,P), married(X,Y). [:1]
:~ assign(X,P), assign(Y,P), X!=Y, not know(X,Y).[:1]
1Example assignment.dlvG. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
A bigger example - Employee Assignment
• Goal: Divide employees in two project groups p1 and p21.
1 Skills of group members should be di�erent.2 Persons in the same group should not be married to each other.3 Members of a group should possibly know each other.
• Requirement 1) is more important than 2) and 3), which areequally important
• Layers express the relative importance of the requirements.
assign(X,p1) v assign(X,p2) :- employee(X).
:~ assign(X,P), assign(Y,P), same_skill(X,Y). [:2]
:~ assign(X,P), assign(Y,P), married(X,Y). [:1]
:~ assign(X,P), assign(Y,P), X!=Y, not know(X,Y).[:1]
1Example assignment.dlvG. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
A bigger example - Employee Assignment
• Goal: Divide employees in two project groups p1 and p21.
1 Skills of group members should be di�erent.2 Persons in the same group should not be married to each other.3 Members of a group should possibly know each other.
• Requirement 1) is more important than 2) and 3), which areequally important
• Layers express the relative importance of the requirements.
assign(X,p1) v assign(X,p2) :- employee(X).
:~ assign(X,P), assign(Y,P), same_skill(X,Y). [:2]
:~ assign(X,P), assign(Y,P), married(X,Y). [:1]
:~ assign(X,P), assign(Y,P), X!=Y, not know(X,Y).[:1]
1Example assignment.dlvG. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
A bigger example - Employee Assignment
• Goal: Divide employees in two project groups p1 and p21.
1 Skills of group members should be di�erent.2 Persons in the same group should not be married to each other.3 Members of a group should possibly know each other.
• Requirement 1) is more important than 2) and 3), which areequally important
• Layers express the relative importance of the requirements.
assign(X,p1) v assign(X,p2) :- employee(X).
:~ assign(X,P), assign(Y,P), same_skill(X,Y). [:2]
:~ assign(X,P), assign(Y,P), married(X,Y). [:1]
:~ assign(X,P), assign(Y,P), X!=Y, not know(X,Y).[:1]
1Example assignment.dlvG. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Guess-Check-Optimize Methodology
• Extend the �Guess & Check� Methodology
• Use weak constraints to �lter out best (optimal) solutions
�Guess-Check-Optimize� : Divide P into three main parts:
Guessing Part
G ⊆ P: Answer_Sets(G ∪ FI ) represent �solution candidates� for instance I .
Checking Part (optional)
C ⊆ P: Answer_Sets(G ∪ C ∪ FI ) represent the admissible solutions for I .
Optimization Part (optional)
The optimization part O ⊆ P consists of weak constraints, and implicitlyde�nes an objective function f : Answer_Sets(G ∪ C ∪ FI ) → NThose answer sets minimizing f are selected.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Guess-Check-Optimize Methodology
• Extend the �Guess & Check� Methodology
• Use weak constraints to �lter out best (optimal) solutions
�Guess-Check-Optimize� : Divide P into three main parts:
Guessing Part
G ⊆ P: Answer_Sets(G ∪ FI ) represent �solution candidates� for instance I .
Checking Part (optional)
C ⊆ P: Answer_Sets(G ∪ C ∪ FI ) represent the admissible solutions for I .
Optimization Part (optional)
The optimization part O ⊆ P consists of weak constraints, and implicitlyde�nes an objective function f : Answer_Sets(G ∪ C ∪ FI ) → NThose answer sets minimizing f are selected.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Guess-Check-Optimize Methodology
• Extend the �Guess & Check� Methodology
• Use weak constraints to �lter out best (optimal) solutions
�Guess-Check-Optimize� : Divide P into three main parts:
Guessing Part
G ⊆ P: Answer_Sets(G ∪ FI ) represent �solution candidates� for instance I .
Checking Part (optional)
C ⊆ P: Answer_Sets(G ∪ C ∪ FI ) represent the admissible solutions for I .
Optimization Part (optional)
The optimization part O ⊆ P consists of weak constraints, and implicitlyde�nes an objective function f : Answer_Sets(G ∪ C ∪ FI ) → NThose answer sets minimizing f are selected.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Guess-Check-Optimize Methodology
• Extend the �Guess & Check� Methodology
• Use weak constraints to �lter out best (optimal) solutions
�Guess-Check-Optimize� : Divide P into three main parts:
Guessing Part
G ⊆ P: Answer_Sets(G ∪ FI ) represent �solution candidates� for instance I .
Checking Part (optional)
C ⊆ P: Answer_Sets(G ∪ C ∪ FI ) represent the admissible solutions for I .
Optimization Part (optional)
The optimization part O ⊆ P consists of weak constraints, and implicitlyde�nes an objective function f : Answer_Sets(G ∪ C ∪ FI ) → NThose answer sets minimizing f are selected.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Guess-Check-Optimize Methodology
• Extend the �Guess & Check� Methodology
• Use weak constraints to �lter out best (optimal) solutions
�Guess-Check-Optimize� : Divide P into three main parts:
Guessing Part
G ⊆ P: Answer_Sets(G ∪ FI ) represent �solution candidates� for instance I .
Checking Part (optional)
C ⊆ P: Answer_Sets(G ∪ C ∪ FI ) represent the admissible solutions for I .
Optimization Part (optional)
The optimization part O ⊆ P consists of weak constraints, and implicitlyde�nes an objective function f : Answer_Sets(G ∪ C ∪ FI ) → NThose answer sets minimizing f are selected.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Guess-Check-Optimize Methodology
• Extend the �Guess & Check� Methodology
• Use weak constraints to �lter out best (optimal) solutions
�Guess-Check-Optimize� : Divide P into three main parts:
Guessing Part
G ⊆ P: Answer_Sets(G ∪ FI ) represent �solution candidates� for instance I .
Checking Part (optional)
C ⊆ P: Answer_Sets(G ∪ C ∪ FI ) represent the admissible solutions for I .
Optimization Part (optional)
The optimization part O ⊆ P consists of weak constraints, and implicitlyde�nes an objective function f : Answer_Sets(G ∪ C ∪ FI ) → NThose answer sets minimizing f are selected.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Guess-Check-Optimize Methodology
• Extend the �Guess & Check� Methodology
• Use weak constraints to �lter out best (optimal) solutions
�Guess-Check-Optimize� : Divide P into three main parts:
Guessing Part
G ⊆ P: Answer_Sets(G ∪ FI ) represent �solution candidates� for instance I .
Checking Part (optional)
C ⊆ P: Answer_Sets(G ∪ C ∪ FI ) represent the admissible solutions for I .
Optimization Part (optional)
The optimization part O ⊆ P consists of weak constraints, and implicitlyde�nes an objective function f : Answer_Sets(G ∪ C ∪ FI ) → NThose answer sets minimizing f are selected.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Social Dinner III
Task
Now that we have de�ned bottleChosen as the solution predicate,is there a way to select only the smallest sets of wines? Try toexpand wineCover4.dlv
?
Solution available as wineCover5.dlv
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Social Dinner III
Task
Now that we have de�ned bottleChosen as the solution predicate,is there a way to select only the smallest sets of wines? Try toexpand wineCover4.dlv
?
Solution available as wineCover5.dlv
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Social Dinner III
Task
Now that we have de�ned bottleChosen as the solution predicate,is there a way to select only the smallest sets of wines? Try toexpand wineCover4.dlv
:~ bottleChosen(X). [1:1]
Solution available as wineCover5.dlv
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Social Dinner III
Task
Now that we have de�ned bottleChosen as the solution predicate,is there a way to select only the smallest sets of wines? Try toexpand wineCover4.dlv
:~ bottleChosen(X). [1:1]
Solution available as wineCover5.dlv
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints with Weights
• A single weak constraints in some layer n is more importantthan all weak constraints in lower layers (n − 1, n − 2,...)together!
• Weak constraints are weighted to make �ner distinctionsamong elements of the same priority::˜ G1.[3.5:1] :˜ G2.[4.6:1]
• The weights of violated weak constraints are summed up foreach layer.
• Example: High School Time Tabling ProblemStructural Requirements > Pedagogical Requirements >Personal Wishes
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints with Weights
• A single weak constraints in some layer n is more importantthan all weak constraints in lower layers (n − 1, n − 2,...)together!
• Weak constraints are weighted to make �ner distinctionsamong elements of the same priority::˜ G1.[3.5:1] :˜ G2.[4.6:1]
• The weights of violated weak constraints are summed up foreach layer.
• Example: High School Time Tabling ProblemStructural Requirements > Pedagogical Requirements >Personal Wishes
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints with Weights
• A single weak constraints in some layer n is more importantthan all weak constraints in lower layers (n − 1, n − 2,...)together!
• Weak constraints are weighted to make �ner distinctionsamong elements of the same priority::˜ G1.[3.5:1] :˜ G2.[4.6:1]
• The weights of violated weak constraints are summed up foreach layer.
• Example: High School Time Tabling ProblemStructural Requirements > Pedagogical Requirements >Personal Wishes
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Weak Constraints with Weights
• A single weak constraints in some layer n is more importantthan all weak constraints in lower layers (n − 1, n − 2,...)together!
• Weak constraints are weighted to make �ner distinctionsamong elements of the same priority::˜ G1.[3.5:1] :˜ G2.[4.6:1]
• The weights of violated weak constraints are summed up foreach layer.
• Example: High School Time Tabling ProblemStructural Requirements > Pedagogical Requirements >Personal Wishes
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Traveling Salesperson
Given: Weighted directed graph G = (V ,E ,C) and a node a ∈ V of this graph.Task: Find a minimum-cost cycle (closed path) in G starting at a and going througheach node in V exactly once2.
• G stored by facts over predicates node(X) and arc(X,Y).• Starting node a is speci�ed by the predicate start (unary).
Guess:
inPath(X,Y,C) v outPath(X,Y,C) :- start(X), arc(X,Y,C).
inPath(X,Y,C) v outPath(X,Y,C) :- reached(X), arc(X,Y,C).
reached(X):- inPath(Y,X,C).
Check:
:- inPath(X,Y,_), inPath(X,Y1,_), Y <> Y1.
:- inPath(X,Y,_), inPath(X1,Y,_), X <> X1.
:- node(X), not reached(X).
Optimize:
:~ inPath(X,Y,C). [C:1]
2Example tsp.dlvG. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Traveling Salesperson
Given: Weighted directed graph G = (V ,E ,C) and a node a ∈ V of this graph.Task: Find a minimum-cost cycle (closed path) in G starting at a and going througheach node in V exactly once2.
• G stored by facts over predicates node(X) and arc(X,Y).• Starting node a is speci�ed by the predicate start (unary).
Guess:
inPath(X,Y,C) v outPath(X,Y,C) :- start(X), arc(X,Y,C).
inPath(X,Y,C) v outPath(X,Y,C) :- reached(X), arc(X,Y,C).
reached(X):- inPath(Y,X,C).
Check:
:- inPath(X,Y,_), inPath(X,Y1,_), Y <> Y1.
:- inPath(X,Y,_), inPath(X1,Y,_), X <> X1.
:- node(X), not reached(X).
Optimize:
:~ inPath(X,Y,C). [C:1]
2Example tsp.dlvG. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Traveling Salesperson
Given: Weighted directed graph G = (V ,E ,C) and a node a ∈ V of this graph.Task: Find a minimum-cost cycle (closed path) in G starting at a and going througheach node in V exactly once2.
• G stored by facts over predicates node(X) and arc(X,Y).• Starting node a is speci�ed by the predicate start (unary).
Guess:
inPath(X,Y,C) v outPath(X,Y,C) :- start(X), arc(X,Y,C).
inPath(X,Y,C) v outPath(X,Y,C) :- reached(X), arc(X,Y,C).
reached(X):- inPath(Y,X,C).
Check:
:- inPath(X,Y,_), inPath(X,Y1,_), Y <> Y1.
:- inPath(X,Y,_), inPath(X1,Y,_), X <> X1.
:- node(X), not reached(X).
Optimize:
:~ inPath(X,Y,C). [C:1]
2Example tsp.dlvG. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Traveling Salesperson
Given: Weighted directed graph G = (V ,E ,C) and a node a ∈ V of this graph.Task: Find a minimum-cost cycle (closed path) in G starting at a and going througheach node in V exactly once2.
• G stored by facts over predicates node(X) and arc(X,Y).• Starting node a is speci�ed by the predicate start (unary).
Guess:
inPath(X,Y,C) v outPath(X,Y,C) :- start(X), arc(X,Y,C).
inPath(X,Y,C) v outPath(X,Y,C) :- reached(X), arc(X,Y,C).
reached(X):- inPath(Y,X,C).
Check:
:- inPath(X,Y,_), inPath(X,Y1,_), Y <> Y1.
:- inPath(X,Y,_), inPath(X1,Y,_), X <> X1.
:- node(X), not reached(X).
Optimize:
:~ inPath(X,Y,C). [C:1]
2Example tsp.dlvG. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Social Dinner IV
Task
Let each wine bootle have a price encoded byprice(bottle,value). Modify wineCover5b.dlv and try tochoose the best cost selection of bottles.
?
Solution available at wineCover5c.dlv
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSemanticsExamplesThe Guess-Check-Optimize patternSocial Dinner
Social Dinner IV
Task
Let each wine bootle have a price encoded byprice(bottle,value). Modify wineCover5b.dlv and try tochoose the best cost selection of bottles.
:~ bottleChosen(X),prize(X,N). [N:1]
Solution available at wineCover5c.dlv
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregates
• Compute aggregate functions over a set of values, similar as inSQL (count, min, max, sum)
• A few examples:
:- actiontime(T), #count{ move(B,L,T) } >= 4.
small :- #max{ X : f(A,X,C), b(C,G) } < 3.
ok_price :- 30 <= #sum{ Price :
bought(Good),
price(Good,Price) } < 50.
• other solvers (e.g. Smodels) o�er similar constructs(cardinality atoms, weight constraints).
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregates
• Compute aggregate functions over a set of values, similar as inSQL (count, min, max, sum)
• A few examples:
:- actiontime(T), #count{ move(B,L,T) } >= 4.
small :- #max{ X : f(A,X,C), b(C,G) } < 3.
ok_price :- 30 <= #sum{ Price :
bought(Good),
price(Good,Price) } < 50.
• other solvers (e.g. Smodels) o�er similar constructs(cardinality atoms, weight constraints).
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregates
• Compute aggregate functions over a set of values, similar as inSQL (count, min, max, sum)
• A few examples:
:- actiontime(T), #count{ move(B,L,T) } >= 4.
small :- #max{ X : f(A,X,C), b(C,G) } < 3.
ok_price :- 30 <= #sum{ Price :
bought(Good),
price(Good,Price) } < 50.
• other solvers (e.g. Smodels) o�er similar constructs(cardinality atoms, weight constraints).
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregates
• Compute aggregate functions over a set of values, similar as inSQL (count, min, max, sum)
• A few examples:
:- actiontime(T), #count{ move(B,L,T) } >= 4.
small :- #max{ X : f(A,X,C), b(C,G) } < 3.
ok_price :- 30 <= #sum{ Price :
bought(Good),
price(Good,Price) } < 50.
• other solvers (e.g. Smodels) o�er similar constructs(cardinality atoms, weight constraints).
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregates
• Compute aggregate functions over a set of values, similar as inSQL (count, min, max, sum)
• A few examples:
:- actiontime(T), #count{ move(B,L,T) } >= 4.
small :- #max{ X : f(A,X,C), b(C,G) } < 3.
ok_price :- 30 <= #sum{ Price :
bought(Good),
price(Good,Price) } < 50.
• other solvers (e.g. Smodels) o�er similar constructs(cardinality atoms, weight constraints).
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregates
• Compute aggregate functions over a set of values, similar as inSQL (count, min, max, sum)
• A few examples:
:- actiontime(T), #count{ move(B,L,T) } >= 4.
small :- #max{ X : f(A,X,C), b(C,G) } < 3.
ok_price :- 30 <= #sum{ Price :
bought(Good),
price(Good,Price) } < 50.
• other solvers (e.g. Smodels) o�er similar constructs(cardinality atoms, weight constraints).
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Syntax
• Symbolic Set: Expression
{Vars : Conj}of a list Vars of variables and a list Conj of literals (safetyrequired) (e.g. { X : f(A,X,C), b(C,G) }).
• Aggregate Function: Expression
f {Vars : Conj}where• f ∈ {#count,#min,#max ,#sum,#times}, and• {Vars : Conj} is a symbolic set
(e.g. #max{ X : f(A,X,C), b(C,G) } })
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Syntax
• Symbolic Set: Expression
{Vars : Conj}of a list Vars of variables and a list Conj of literals (safetyrequired) (e.g. { X : f(A,X,C), b(C,G) }).
• Aggregate Function: Expression
f {Vars : Conj}where• f ∈ {#count,#min,#max ,#sum,#times}, and• {Vars : Conj} is a symbolic set
(e.g. #max{ X : f(A,X,C), b(C,G) } })
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Syntax
• Symbolic Set: Expression
{Vars : Conj}of a list Vars of variables and a list Conj of literals (safetyrequired) (e.g. { X : f(A,X,C), b(C,G) }).
• Aggregate Function: Expression
f {Vars : Conj}where• f ∈ {#count,#min,#max ,#sum,#times}, and• {Vars : Conj} is a symbolic set
(e.g. #max{ X : f(A,X,C), b(C,G) } })
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Syntax
• Symbolic Set: Expression
{Vars : Conj}of a list Vars of variables and a list Conj of literals (safetyrequired) (e.g. { X : f(A,X,C), b(C,G) }).
• Aggregate Function: Expression
f {Vars : Conj}where• f ∈ {#count,#min,#max ,#sum,#times}, and• {Vars : Conj} is a symbolic set
(e.g. #max{ X : f(A,X,C), b(C,G) } })
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Syntax
• Symbolic Set: Expression
{Vars : Conj}of a list Vars of variables and a list Conj of literals (safetyrequired) (e.g. { X : f(A,X,C), b(C,G) }).
• Aggregate Function: Expression
f {Vars : Conj}where• f ∈ {#count,#min,#max ,#sum,#times}, and• {Vars : Conj} is a symbolic set
(e.g. #max{ X : f(A,X,C), b(C,G) } })
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Syntax
• Symbolic Set: Expression
{Vars : Conj}of a list Vars of variables and a list Conj of literals (safetyrequired) (e.g. { X : f(A,X,C), b(C,G) }).
• Aggregate Function: Expression
f {Vars : Conj}where• f ∈ {#count,#min,#max ,#sum,#times}, and• {Vars : Conj} is a symbolic set
(e.g. #max{ X : f(A,X,C), b(C,G) } })
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Syntax /2
• Aggregate Atom: Expression
Agg_Atom ::= val } f {Vars : Conj}| f {Vars : Conj} } val| vall }l f {Vars : Conj} }r valu
where• val , vall , valu are constants or variables,• } ∈ {<,>,≤,≥,= },• }l ,}r ∈ {<,≤}, and• f {Vars : Conj} is an aggregate function
(e.g. #max{ X : f(A,X,C), b(C,G) } < 3)
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Syntax /2
• Aggregate Atom: Expression
Agg_Atom ::= val } f {Vars : Conj}| f {Vars : Conj} } val| vall }l f {Vars : Conj} }r valu
where• val , vall , valu are constants or variables,• } ∈ {<,>,≤,≥,= },• }l ,}r ∈ {<,≤}, and• f {Vars : Conj} is an aggregate function
(e.g. #max{ X : f(A,X,C), b(C,G) } < 3)
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Syntax /2
• Aggregate Atom: Expression
Agg_Atom ::= val } f {Vars : Conj}| f {Vars : Conj} } val| vall }l f {Vars : Conj} }r valu
where• val , vall , valu are constants or variables,• } ∈ {<,>,≤,≥,= },• }l ,}r ∈ {<,≤}, and• f {Vars : Conj} is an aggregate function
(e.g. #max{ X : f(A,X,C), b(C,G) } < 3)
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Syntax /2
• Aggregate Atom: Expression
Agg_Atom ::= val } f {Vars : Conj}| f {Vars : Conj} } val| vall }l f {Vars : Conj} }r valu
where• val , vall , valu are constants or variables,• } ∈ {<,>,≤,≥,= },• }l ,}r ∈ {<,≤}, and• f {Vars : Conj} is an aggregate function
(e.g. #max{ X : f(A,X,C), b(C,G) } < 3)
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Syntax /2
• Aggregate Atom: Expression
Agg_Atom ::= val } f {Vars : Conj}| f {Vars : Conj} } val| vall }l f {Vars : Conj} }r valu
where• val , vall , valu are constants or variables,• } ∈ {<,>,≤,≥,= },• }l ,}r ∈ {<,≤}, and• f {Vars : Conj} is an aggregate function
(e.g. #max{ X : f(A,X,C), b(C,G) } < 3)
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Syntax /2
• Aggregate Atom: Expression
Agg_Atom ::= val } f {Vars : Conj}| f {Vars : Conj} } val| vall }l f {Vars : Conj} }r valu
where• val , vall , valu are constants or variables,• } ∈ {<,>,≤,≥,= },• }l ,}r ∈ {<,≤}, and• f {Vars : Conj} is an aggregate function
(e.g. #max{ X : f(A,X,C), b(C,G) } < 3)
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Semantics
• Informally:Suppose I is an interpretation.
• Evaluate symbolic set {Vars : Conj} with respect to I : Collectall instances of Vars for which Conj is true in I (Result:SemSet).
• Apply f on SemSet (Result: v = f (SemSet)).• Evaluate comparison val θ v resp. vall θl v ∧ v θr valu with
(instantiated) value val resp. values vall , valu.
• Appealing formal de�nition of semantics is a bit tricky
• Widely acknowledged proposal: Faber et al. [32].
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Semantics
• Informally:Suppose I is an interpretation.
• Evaluate symbolic set {Vars : Conj} with respect to I : Collectall instances of Vars for which Conj is true in I (Result:SemSet).
• Apply f on SemSet (Result: v = f (SemSet)).• Evaluate comparison val θ v resp. vall θl v ∧ v θr valu with
(instantiated) value val resp. values vall , valu.
• Appealing formal de�nition of semantics is a bit tricky
• Widely acknowledged proposal: Faber et al. [32].
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Semantics
• Informally:Suppose I is an interpretation.
• Evaluate symbolic set {Vars : Conj} with respect to I : Collectall instances of Vars for which Conj is true in I (Result:SemSet).
• Apply f on SemSet (Result: v = f (SemSet)).• Evaluate comparison val θ v resp. vall θl v ∧ v θr valu with
(instantiated) value val resp. values vall , valu.
• Appealing formal de�nition of semantics is a bit tricky
• Widely acknowledged proposal: Faber et al. [32].
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Semantics
• Informally:Suppose I is an interpretation.
• Evaluate symbolic set {Vars : Conj} with respect to I : Collectall instances of Vars for which Conj is true in I (Result:SemSet).
• Apply f on SemSet (Result: v = f (SemSet)).• Evaluate comparison val θ v resp. vall θl v ∧ v θr valu with
(instantiated) value val resp. values vall , valu.
• Appealing formal de�nition of semantics is a bit tricky
• Widely acknowledged proposal: Faber et al. [32].
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Semantics
• Informally:Suppose I is an interpretation.
• Evaluate symbolic set {Vars : Conj} with respect to I : Collectall instances of Vars for which Conj is true in I (Result:SemSet).
• Apply f on SemSet (Result: v = f (SemSet)).• Evaluate comparison val θ v resp. vall θl v ∧ v θr valu with
(instantiated) value val resp. values vall , valu.
• Appealing formal de�nition of semantics is a bit tricky
• Widely acknowledged proposal: Faber et al. [32].
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Aggregate Atoms � Semantics
• Informally:Suppose I is an interpretation.
• Evaluate symbolic set {Vars : Conj} with respect to I : Collectall instances of Vars for which Conj is true in I (Result:SemSet).
• Apply f on SemSet (Result: v = f (SemSet)).• Evaluate comparison val θ v resp. vall θl v ∧ v θr valu with
(instantiated) value val resp. values vall , valu.
• Appealing formal de�nition of semantics is a bit tricky
• Widely acknowledged proposal: Faber et al. [32].
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Social Dinner V
Task
Modify wineCover5c.dlv so that the weak constraint
:~ bottleChosen(X),prize(X,N). [N:1]
can be changed in
:~ totalcost(N). [N:1]
totalcost(N) :- #int(N),
? .
Solution at wineCover6.dlv
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Social Dinner V
Task
Modify wineCover5c.dlv so that the weak constraint
:~ bottleChosen(X),prize(X,N). [N:1]
can be changed in
:~ totalcost(N). [N:1]
totalcost(N) :- #int(N),
? .
Solution at wineCover6.dlv
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Social Dinner V
Task
Modify wineCover5c.dlv so that the weak constraint
:~ bottleChosen(X),prize(X,N). [N:1]
can be changed in
:~ totalcost(N). [N:1]
totalcost(N) :- #int(N),
? .
Solution at wineCover6.dlv
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Social Dinner V
Task
Modify wineCover5c.dlv so that the weak constraint
:~ bottleChosen(X),prize(X,N). [N:1]
can be changed in
:~ totalcost(N). [N:1]
totalcost(N) :- #int(N),
? .
Solution at wineCover6.dlv
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Social Dinner V
Task
Modify wineCover5c.dlv so that the weak constraint
:~ bottleChosen(X),prize(X,N). [N:1]
can be changed in
:~ totalcost(N). [N:1]
totalcost(N) :- #int(N),
#sum{ Y : bottleChosen(X),prize(X,Y) } = N.
Solution at wineCover6.dlv
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner
Social Dinner V
Task
Modify wineCover5c.dlv so that the weak constraint
:~ bottleChosen(X),prize(X,N). [N:1]
can be changed in
:~ totalcost(N). [N:1]
totalcost(N) :- #int(N),
#sum{ Y : bottleChosen(X),prize(X,Y) } = N.
Solution at wineCover6.dlv
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Frame logic: the idea
The molecular syntax typical of F-logic is quite useful formanipulating triple stores and complex join patterns:
Datalog Syntax
wineBottle("Brachetto"). isA("Brachetto","RedWine"),
isA("Brachetto","SweetWine"). prize("Brachetto",10).
F-Logic Syntax
"Brachetto" : wineBottle[isA-�{"RedWine","SweetWine"},
prize->10].
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Frame syntax: the idea
The molecular syntax typical of F-logic is quite useful formanipulating triple stores and complex join patterns:
Datalog Syntax
mainEntity(M) :- "foaf:PersonalProfileDocument"(X),
"foaf:primaryTopic"(X,M).
F-Logic Syntax
M : mainEntity :-
X:"foaf:PersonalProfileDocument"["foaf:primaryTopic"->M].
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Informal Syntax and Semantics
F-Logic molecule
subject : type[predicate1->object, ...,
predicate2-> >{ object1, ..., objectn },
...]
It is a syntactic shortcut to
Datalog conjunction of facts
type(subject),
predicate1(subject,object), ... , predicate2(subject,object1),
... , predicate2(subject, objectn)
• Objects can be nested frames (only atomic frames in rules' heads)• Subjects and Objects unify with terms of the language. Under higher order
extensions (see Unit 5), also Predicates and Types do.• F-Logic semantic features (inheritance, etc.) are not currently implemented, this
is only syntactic sugar.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Informal Syntax and Semantics
F-Logic molecule
subject : type[predicate1->object, ...,
predicate2-> >{ object1, ..., objectn },
...]
It is a syntactic shortcut to
Datalog conjunction of facts
type(subject),
predicate1(subject,object), ... , predicate2(subject,object1),
... , predicate2(subject, objectn)
• Objects can be nested frames (only atomic frames in rules' heads)• Subjects and Objects unify with terms of the language. Under higher order
extensions (see Unit 5), also Predicates and Types do.• F-Logic semantic features (inheritance, etc.) are not currently implemented, this
is only syntactic sugar.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Informal Syntax and Semantics
F-Logic molecule
subject : type[predicate1->object, ...,
predicate2-> >{ object1, ..., objectn },
...]
It is a syntactic shortcut to
Datalog conjunction of facts
type(subject),
predicate1(subject,object), ... , predicate2(subject,object1),
... , predicate2(subject, objectn)
• Objects can be nested frames (only atomic frames in rules' heads)• Subjects and Objects unify with terms of the language. Under higher order
extensions (see Unit 5), also Predicates and Types do.• F-Logic semantic features (inheritance, etc.) are not currently implemented, this
is only syntactic sugar.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Informal Syntax and Semantics
F-Logic molecule
subject : type[predicate1->object, ...,
predicate2-> >{ object1, ..., objectn },
...]
It is a syntactic shortcut to
Datalog conjunction of facts
type(subject),
predicate1(subject,object), ... , predicate2(subject,object1),
... , predicate2(subject, objectn)
• Objects can be nested frames (only atomic frames in rules' heads)• Subjects and Objects unify with terms of the language. Under higher order
extensions (see Unit 5), also Predicates and Types do.• F-Logic semantic features (inheritance, etc.) are not currently implemented, this
is only syntactic sugar.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Informal Syntax and Semantics
F-Logic molecule
subject : type[predicate1->object, ...,
predicate2-> >{ object1, ..., objectn },
...]
It is a syntactic shortcut to
Datalog conjunction of facts
type(subject),
predicate1(subject,object), ... , predicate2(subject,object1),
... , predicate2(subject, objectn)
• Objects can be nested frames (only atomic frames in rules' heads)• Subjects and Objects unify with terms of the language. Under higher order
extensions (see Unit 5), also Predicates and Types do.• F-Logic semantic features (inheritance, etc.) are not currently implemented, this
is only syntactic sugar.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Informal Syntax and Semantics
F-Logic molecule
subject : type[predicate1->object, ...,
predicate2-> >{ object1, ..., objectn },
...]
It is a syntactic shortcut to
Datalog conjunction of facts
type(subject),
predicate1(subject,object), ... , predicate2(subject,object1),
... , predicate2(subject, objectn)
• Objects can be nested frames (only atomic frames in rules' heads)• Subjects and Objects unify with terms of the language. Under higher order
extensions (see Unit 5), also Predicates and Types do.• F-Logic semantic features (inheritance, etc.) are not currently implemented, this
is only syntactic sugar.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Informal Syntax and Semantics
F-Logic molecule
subject : type[predicate1->object, ...,
predicate2-> >{ object1, ..., objectn },
...]
It is a syntactic shortcut to
Datalog conjunction of facts
type(subject),
predicate1(subject,object), ... , predicate2(subject,object1),
... , predicate2(subject, objectn)
• Objects can be nested frames (only atomic frames in rules' heads)• Subjects and Objects unify with terms of the language. Under higher order
extensions (see Unit 5), also Predicates and Types do.• F-Logic semantic features (inheritance, etc.) are not currently implemented, this
is only syntactic sugar.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Informal Syntax and Semantics
F-Logic molecule
subject : type[predicate1->object, ...,
predicate2-> >{ object1, ..., objectn },
...]
It is a syntactic shortcut to
Datalog conjunction of facts
type(subject),
predicate1(subject,object), ... , predicate2(subject,object1),
... , predicate2(subject, objectn)
• Objects can be nested frames (only atomic frames in rules' heads)• Subjects and Objects unify with terms of the language. Under higher order
extensions (see Unit 5), also Predicates and Types do.• F-Logic semantic features (inheritance, etc.) are not currently implemented, this
is only syntactic sugar.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Informal Syntax and Semantics
F-Logic molecule
subject : type[predicate1->object, ...,
predicate2-> >{ object1, ..., objectn },
...]
It is a syntactic shortcut to
Datalog conjunction of facts
type(subject),
predicate1(subject,object), ... , predicate2(subject,object1),
... , predicate2(subject, objectn)
• Objects can be nested frames (only atomic frames in rules' heads)• Subjects and Objects unify with terms of the language. Under higher order
extensions (see Unit 5), also Predicates and Types do.• F-Logic semantic features (inheritance, etc.) are not currently implemented, this
is only syntactic sugar.
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Frame Spaces
A Frame Space directive tells how frames are mapped to regularatoms
@triple.
A[brother->B] :- A[father->Y],
B[father->Y].
Maps to:
brother(A,B,triple) :-
father(A,Y,triple),
father(B,Y,triple).
@.
A[brother->B] :- A[father->Y],
B[father->Y].
Maps to:
brother(A,B) :- father(A,Y),
father(B,Y).
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Frame Spaces
A Frame Space directive tells how frames are mapped to regularatoms
@triple.
A[brother->B] :- A[father->Y],
B[father->Y].
Maps to:
brother(A,B,triple) :-
father(A,Y,triple),
father(B,Y,triple).
@.
A[brother->B] :- A[father->Y],
B[father->Y].
Maps to:
brother(A,B) :- father(A,Y),
father(B,Y).
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Frame Spaces
A Frame Space directive tells how frames are mapped to regularatoms
@triple.
A[brother->B] :- A[father->Y],
B[father->Y].
Maps to:
brother(A,B,triple) :-
father(A,Y,triple),
father(B,Y,triple).
@.
A[brother->B] :- A[father->Y],
B[father->Y].
Maps to:
brother(A,B) :- father(A,Y),
father(B,Y).
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Social Dinner VII
Task
Take wineCover7a.dlt. It is partially in frame syntax. Put thefollowing rule in frame logic syntax:
compliantBottle(X,Z) :- preferredWine(X,Y), isA(Z,Y).
Solution at wineCover7b.dlt
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Social Dinner VII
Task
Take wineCover7a.dlt. It is partially in frame syntax. Put thefollowing rule in frame logic syntax:
X[compliantBottle->Z] :- X[preferredWine->Y], Z[isA->Y].
Solution at wineCover7b.dlt
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsSocial Dinner Example
Social Dinner VII
Task
Take wineCover7a.dlt. It is partially in frame syntax. Put thefollowing rule in frame logic syntax:
X[compliantBottle->Z] :- X[preferredWine->Y], Z[isA->Y].
Solution at wineCover7b.dlt
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
The idea of templates
Imagine you want to encode all the possible permutations of agiven predicate p (assume maxint = |X : p(X )|)
First, I guess worlds of permutations
permutation(X,N) v -permutation(X,N) :- p(X),#int(N).
Then, I cut worlds I don't like
:- permutation(X,A),permutation(Z,A), Z <> X.
:- permutation(X,A),permutation(X,B), A <> B.
Also, each element must be in the partition
covered(X) :- permutation(X,A).
:- p(X), not covered(X).
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
The idea of templates
Imagine you want to encode all the possible permutations of agiven predicate p (assume maxint = |X : p(X )|)
First, I guess worlds of permutations
permutation(X,N) v -permutation(X,N) :- p(X),#int(N).
Then, I cut worlds I don't like
:- permutation(X,A),permutation(Z,A), Z <> X.
:- permutation(X,A),permutation(X,B), A <> B.
Also, each element must be in the partition
covered(X) :- permutation(X,A).
:- p(X), not covered(X).
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
The idea of templates
Imagine you want to encode all the possible permutations of agiven predicate p (assume maxint = |X : p(X )|)
First, I guess worlds of permutations
permutation(X,N) v -permutation(X,N) :- p(X),#int(N).
Then, I cut worlds I don't like
:- permutation(X,A),permutation(Z,A), Z <> X.
:- permutation(X,A),permutation(X,B), A <> B.
Also, each element must be in the partition
covered(X) :- permutation(X,A).
:- p(X), not covered(X).
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
The idea of templates
Imagine you want to encode all the possible permutations of agiven predicate p (assume maxint = |X : p(X )|)
First, I guess worlds of permutations
permutation(X,N) v -permutation(X,N) :- p(X),#int(N).
Then, I cut worlds I don't like
:- permutation(X,A),permutation(Z,A), Z <> X.
:- permutation(X,A),permutation(X,B), A <> B.
Also, each element must be in the partition
covered(X) :- permutation(X,A).
:- p(X), not covered(X).
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
The idea of templates - 2
• Thus, this �small� program encodes a search space ofpermutations
• But it can be reused and put in a library (let maxint bigenough here)
#template permutation{p(1)}(2)
{
permutation(X,N) v -permutation(X,N)
:- p(X),#int(N),
#count{ Y : p(Y) } = N1,
N <= N1, N > 0.
:- permutation(X,A),permutation(Z,A), Z <> X.
:- permutation(X,A),permutation(X,B), A <> B.
covered(X) :- permutation(X,A).
:- p(X), not covered(X).
}
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
The idea of templates - 2
• Thus, this �small� program encodes a search space ofpermutations
• But it can be reused and put in a library (let maxint bigenough here)
#template permutation{p(1)}(2)
{
permutation(X,N) v -permutation(X,N)
:- p(X),#int(N),
#count{ Y : p(Y) } = N1,
N <= N1, N > 0.
:- permutation(X,A),permutation(Z,A), Z <> X.
:- permutation(X,A),permutation(X,B), A <> B.
covered(X) :- permutation(X,A).
:- p(X), not covered(X).
}
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Syntax and Semantics
Template de�nition:
#template closure{e(2)}(2)
{
closure(X,Y) :- e(X,Y).
closure(X,Y) :- e(X,Z),
closure(Z,Y).
}
#template max{p(1)}(1)
{
exceeded(X) :- p(X),p(Y), Y > X.
max(X) :- p(X),
not exceeded(X).
}
• e(2), p(1) = formal parameter list
• ..}(2), ..}(1) = output predicate arities
• closure, max = output predicate names
• exceeded = local predicate name
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Syntax and Semantics
Template de�nition:
#template closure{e(2)}(2)
{
closure(X,Y) :- e(X,Y).
closure(X,Y) :- e(X,Z),
closure(Z,Y).
}
#template max{p(1)}(1)
{
exceeded(X) :- p(X),p(Y), Y > X.
max(X) :- p(X),
not exceeded(X).
}
• e(2), p(1) = formal parameter list
• ..}(2), ..}(1) = output predicate arities
• closure, max = output predicate names
• exceeded = local predicate name
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Syntax and Semantics
Template de�nition:
#template closure{e(2)}(2)
{
closure(X,Y) :- e(X,Y).
closure(X,Y) :- e(X,Z),
closure(Z,Y).
}
#template max{p(1)}(1)
{
exceeded(X) :- p(X),p(Y), Y > X.
max(X) :- p(X),
not exceeded(X).
}
• e(2), p(1) = formal parameter list
• ..}(2), ..}(1) = output predicate arities
• closure, max = output predicate names
• exceeded = local predicate name
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Syntax and Semantics
Template de�nition:
#template closure{e(2)}(2)
{
closure(X,Y) :- e(X,Y).
closure(X,Y) :- e(X,Z),
closure(Z,Y).
}
#template max{p(1)}(1)
{
exceeded(X) :- p(X),p(Y), Y > X.
max(X) :- p(X),
not exceeded(X).
}
• e(2), p(1) = formal parameter list
• ..}(2), ..}(1) = output predicate arities
• closure, max = output predicate names
• exceeded = local predicate name
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Syntax and Semantics
Template de�nition:
#template closure{e(2)}(2)
{
closure(X,Y) :- e(X,Y).
closure(X,Y) :- e(X,Z),
closure(Z,Y).
}
#template max{p(1)}(1)
{
exceeded(X) :- p(X),p(Y), Y > X.
max(X) :- p(X),
not exceeded(X).
}
• e(2), p(1) = formal parameter list
• ..}(2), ..}(1) = output predicate arities
• closure, max = output predicate names
• exceeded = local predicate name
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Syntax and Semantics
Template de�nition:
#template closure{e(2)}(2)
{
closure(X,Y) :- e(X,Y).
closure(X,Y) :- e(X,Z),
closure(Z,Y).
}
#template max{p(1)}(1)
{
exceeded(X) :- p(X),p(Y), Y > X.
max(X) :- p(X),
not exceeded(X).
}
• e(2), p(1) = formal parameter list
• ..}(2), ..}(1) = output predicate arities
• closure, max = output predicate names
• exceeded = local predicate name
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Syntax and Semantics - 2
Template atoms:
clo(X,Y) :- closure{ edge(*,*) }(X,Y).
inPath(X,N) :- permutation{ clo(*,$) }(X,N).
maxAgePerSex(S,A) :- max{ person($,S,*) }(A).
• edge(*,*), clo(*,$), person($,S,*) = actualparameters
• closure{ edge(*,*) }(X,Y) = a template atom
• * = input terms
• $ = projection terms
• S = group-by (quanti�cation) term
• (X,Y), (X,N), S..A = output terms
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Syntax and Semantics - 2
Template atoms:
clo(X,Y) :- closure{ edge(*,*) }(X,Y).
inPath(X,N) :- permutation{ clo(*,$) }(X,N).
maxAgePerSex(S,A) :- max{ person($,S,*) }(A).
• edge(*,*), clo(*,$), person($,S,*) = actualparameters
• closure{ edge(*,*) }(X,Y) = a template atom
• * = input terms
• $ = projection terms
• S = group-by (quanti�cation) term
• (X,Y), (X,N), S..A = output terms
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Syntax and Semantics - 2
Template atoms:
clo(X,Y) :- closure{ edge(*,*) }(X,Y).
inPath(X,N) :- permutation{ clo(*,$) }(X,N).
maxAgePerSex(S,A) :- max{ person($,S,*) }(A).
• edge(*,*), clo(*,$), person($,S,*) = actualparameters
• closure{ edge(*,*) }(X,Y) = a template atom
• * = input terms
• $ = projection terms
• S = group-by (quanti�cation) term
• (X,Y), (X,N), S..A = output terms
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Syntax and Semantics - 2
Template atoms:
clo(X,Y) :- closure{ edge(*,*) }(X,Y).
inPath(X,N) :- permutation{ clo(*,$) }(X,N).
maxAgePerSex(S,A) :- max{ person($,S,*) }(A).
• edge(*,*), clo(*,$), person($,S,*) = actualparameters
• closure{ edge(*,*) }(X,Y) = a template atom
• * = input terms
• $ = projection terms
• S = group-by (quanti�cation) term
• (X,Y), (X,N), S..A = output terms
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Syntax and Semantics - 2
Template atoms:
clo(X,Y) :- closure{ edge(*,*) }(X,Y).
inPath(X,N) :- permutation{ clo(*,$) }(X,N).
maxAgePerSex(S,A) :- max{ person($,S,*) }(A).
• edge(*,*), clo(*,$), person($,S,*) = actualparameters
• closure{ edge(*,*) }(X,Y) = a template atom
• * = input terms
• $ = projection terms
• S = group-by (quanti�cation) term
• (X,Y), (X,N), S..A = output terms
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Syntax and Semantics - 2
Template atoms:
clo(X,Y) :- closure{ edge(*,*) }(X,Y).
inPath(X,N) :- permutation{ clo(*,$) }(X,N).
maxAgePerSex(S,A) :- max{ person($,S,*) }(A).
• edge(*,*), clo(*,$), person($,S,*) = actualparameters
• closure{ edge(*,*) }(X,Y) = a template atom
• * = input terms
• $ = projection terms
• S = group-by (quanti�cation) term
• (X,Y), (X,N), S..A = output terms
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Syntax and Semantics - 2
Template atoms:
clo(X,Y) :- closure{ edge(*,*) }(X,Y).
inPath(X,N) :- permutation{ clo(*,$) }(X,N).
maxAgePerSex(S,A) :- max{ person($,S,*) }(A).
• edge(*,*), clo(*,$), person($,S,*) = actualparameters
• closure{ edge(*,*) }(X,Y) = a template atom
• * = input terms
• $ = projection terms
• S = group-by (quanti�cation) term
• (X,Y), (X,N), S..A = output terms
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Syntax and Semantics - 2
Template atoms:
clo(X,Y) :- closure{ edge(*,*) }(X,Y).
inPath(X,N) :- permutation{ clo(*,$) }(X,N).
maxAgePerSex(S,A) :- max{ person($,S,*) }(A).
• edge(*,*), clo(*,$), person($,S,*) = actualparameters
• closure{ edge(*,*) }(X,Y) = a template atom
• * = input terms
• $ = projection terms
• S = group-by (quanti�cation) term
• (X,Y), (X,N), S..A = output terms
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
The Hamiltonian Path problem
HP: �nd a path between nodes of a graph s.t. I cross each nodeexactly once. (permutation.dlt)
If I want to encode the HP problem with templates, I can do thisway:
path(X,N) :- permutation{node(*)}(X,N).
:- path(X,M), path(Y,N), not edge(X,Y), M = N+1.
Also, I can use permutation taking input predicates other thanunary:
path(X,N) :- permutation{edge(*,$)}(X,N).
• * = parameter
• $ = projection
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
The Hamiltonian Path problem
HP: �nd a path between nodes of a graph s.t. I cross each nodeexactly once. (permutation.dlt)
If I want to encode the HP problem with templates, I can do thisway:
path(X,N) :- permutation{node(*)}(X,N).
:- path(X,M), path(Y,N), not edge(X,Y), M = N+1.
Also, I can use permutation taking input predicates other thanunary:
path(X,N) :- permutation{edge(*,$)}(X,N).
• * = parameter
• $ = projection
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
The Hamiltonian Path problem
HP: �nd a path between nodes of a graph s.t. I cross each nodeexactly once. (permutation.dlt)
If I want to encode the HP problem with templates, I can do thisway:
path(X,N) :- permutation{node(*)}(X,N).
:- path(X,M), path(Y,N), not edge(X,Y), M = N+1.
Also, I can use permutation taking input predicates other thanunary:
path(X,N) :- permutation{edge(*,$)}(X,N).
• * = parameter
• $ = projection
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
The Hamiltonian Path problem
HP: �nd a path between nodes of a graph s.t. I cross each nodeexactly once. (permutation.dlt)
If I want to encode the HP problem with templates, I can do thisway:
path(X,N) :- permutation{node(*)}(X,N).
:- path(X,M), path(Y,N), not edge(X,Y), M = N+1.
Also, I can use permutation taking input predicates other thanunary:
path(X,N) :- permutation{edge(*,$)}(X,N).
• * = parameter
• $ = projection
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
The Hamiltonian Path problem
HP: �nd a path between nodes of a graph s.t. I cross each nodeexactly once. (permutation.dlt)
If I want to encode the HP problem with templates, I can do thisway:
path(X,N) :- permutation{node(*)}(X,N).
:- path(X,M), path(Y,N), not edge(X,Y), M = N+1.
Also, I can use permutation taking input predicates other thanunary:
path(X,N) :- permutation{edge(*,$)}(X,N).
• * = parameter
• $ = projection
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
The Hamiltonian Path problem
HP: �nd a path between nodes of a graph s.t. I cross each nodeexactly once. (permutation.dlt)
If I want to encode the HP problem with templates, I can do thisway:
path(X,N) :- permutation{node(*)}(X,N).
:- path(X,M), path(Y,N), not edge(X,Y), M = N+1.
Also, I can use permutation taking input predicates other thanunary:
path(X,N) :- permutation{edge(*,$)}(X,N).
• * = parameter
• $ = projection
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Social Dinner VIII
Task
Try to expand wineCover7.dlt: de�ne a template subset forspecifying the search space of minimum cardinality subsets of wines.
#template subset{ p(1) }(1)
{
?
?
}
bottleChosen(X) :- ?
Solution at wineCover8.dlt
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Social Dinner VIII
Task
Try to expand wineCover7.dlt: de�ne a template subset forspecifying the search space of minimum cardinality subsets of wines.
#template subset{ p(1) }(1)
{
?
?
}
bottleChosen(X) :- ?
Solution at wineCover8.dlt
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Social Dinner VIII
Task
Try to expand wineCover7.dlt: de�ne a template subset forspecifying the search space of minimum cardinality subsets of wines.
#template subset{ p(1) }(1)
{
subset(X) v nonsubset(X) :- p(X).
:~ subset(X). [1:1]
}
bottleChosen(X) :- subset{compliantBottle($,*)}(X).
Solution at wineCover8.dlt
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
IdeaSyntax and SemanticsExamplesSocial Dinner Example
Social Dinner VIII
Task
Try to expand wineCover7.dlt: de�ne a template subset forspecifying the search space of minimum cardinality subsets of wines.
#template subset{ p(1) }(1)
{
subset(X) v nonsubset(X) :- p(X).
:~ subset(X). [1:1]
}
bottleChosen(X) :- subset{compliantBottle($,*)}(X).
Solution at wineCover8.dlt
G. Ianni Unit 2 � ASP Extensions
IntroWeak constraints
AggregatesFrame Syntax
TemplatesReferences
References
1 Weak Constraints: [11]2 Aggregates: [32]3 Templates: [13]4 Frame Logic: [48]5 Other extensions:http://www.tcs.hut.fi/Research/Logic/wasp/wp3/
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G. Ianni Unit 2 � ASP Extensions