Steffen Staab [email protected] Advanced Data Modeling 1 of 32 WeST Häufungspunkte Bifurkation: x n+1 = r x n (1-x n ) Startwert x 0 = 0,25

Steffen [email protected]

Advanced Data Modeling1 of 32

WeST

Häufungspunkte Bifurkation: xn+1 = r xn (1-xn)

Startwert x0 = 0,25

Web Science & Technologies

University of Koblenz ▪ Landau, Germany

Implementing DLV



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DLV System Architecture



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RULE HANDLING AND INTELLIGENT GROUNDING



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Intelligent Grounding

Ground Programm has exactly the same answer sets as the original programme

For some syntactically restricted classes of programs (e.g. stratified programs), the “Intelligent Grounding” module already computes the corresponding answer sets.

We assume “safe programmes”, i.e. no variables in the head that are not bound by positive literals in the body.

We assume function-free programmes here.

Thomas Eiter, Nicola Leone, Cristinel Mateis, Gerald Pfeifer, Franceseo Scarcello. A Deductive System for Non-Monotonic Reasoning. In: Logic Programming And Nonmonotonic Reasoning. Lecture Notes in Computer Science, 1997, Volume 1265/1997, 363-374, Springer.



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Rules and Graphs Handler

Motivation: Overall problem of EDLP is co-NP hard, BUT many of its parts are much easier (e.g. stratified programms are polynomially computable)



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Ground Dependency Graph (GG)

Nodes: atoms Edges: E+, E-, E Ground Programme:

p(2,3) :-

t(2) :-

q(1) q(3) :- p(2,3), NOT t(2).

t(3) :- q(3), p(2,3).



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Ground Collapsed Positive Dependency Graph (GG+)

Strongly connected components form modules and become one node (not in the example here)



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Non-Ground Collapsed Positive Dependency Graph (G+ P)



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Summarizing Steps for Analyzing Rules

Given programme P, it computes G+ P for the intelligent

grounding module (IGM) Given from IGM it computes GG

From and GG it sets apart

the modules, syntactical properties (e.g. stratification)

for the Model Generator

It computes GG+ for Model Checker to reduce search

space For each component of GG+

it analyzes syntactical properties (e.g. headcycle-freeness) for use by Model Checker



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Grounding

Init

NF = EDBP % Grounded facts initially= DB extension

= % Grow grounded program from G+

P % Analyze the collapsed ng rule graph

Key steps of procedure:

1.Grow rule grounding from node C in G+P without incoming

edges (i.e. non-recursive)

2.Grow rule grounding using positive literals (and remove them from the grounded rules!)



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Example Programme

Programme P

p(1,2) p(2,3) :-

t(X) :- a(X).

q(X) q(Z) :- p(X,Y), p(Y,Z), NOT t(Y).

t(X) :- q(X), p(Y,X)

a(2) :-

Grounded Programme p(1,2) p(2,3) :-

t(2) :-

q(1) q(3) :- p(1,2), p(2,3), NOT t(2).

t(3) :- q(3), p(2,3)

a(2) :-

Note: ground(P) contains 40 rule instances, but Instantiate generates only 4.



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Theorem

Let P be a safe disjunctive datalog program, and be the ground program generated by Instantiat(P). Then, P and have the same stable models.



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Summary

Grounding is a bit like forward chaining (applying immediate consequence operator).

However, because of disjunction in the heads it is overgenerating and it is also not retracting implications that fail because of negation.



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MODEL GENERATOR



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Model Generator

1. Derive what is definitely derivable,

2. Make ‘educated’ guess for one of those literals which have not been decided yet.

3. IF further guesses can be madeTHEN GOTO 2ELSE Call MODEL-CHECKER

If candidate model is inconsistent or not stable then

backtrack, taking knowledge about previous guesses

into account to avoid redundancies

Nicola Leone, Pasquale Rullo, Francesco Scarcello. Disjunctive Stable Models: Unfounded Sets, Fixpoint Semantics, and Computation. In: Information and Computation, 135(2): 69-112, 1997.



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Note

The number of candidate models can be exponential Smart generation of candidate models is crucial

Prune the search space! Special algorithms for:

• Polynomial-time algorithm for normal stratified programs

• Special algorithm for headcycle-free disjunctive programs

• General algorithm



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Unfounded Sets

IntuitivelyAn unfounded set for a disjunctive program P w.r.t. an interpretation I is a set of positive literals that cannot be derived from P assuming the facts in I.

[generalization of a technique from well-founded semantics!]



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Unfounded Sets

Formally:Let I be an interpretation for a program P. A set X of ground atoms is an unfounded set for P w.r.t. I if, for each a∈X,

for each rule r∈ground(P) such that a∈H(r), at least one of the following conditions holds:

• B(r) ∩ .I ≠ , that is, the body of r is false w.r.t. I

• B+(r)∩ X ≠ , that is, some positive body literal belongs to X

• (H(r)-X) ∩ I ≠ , that is, an atom in the head of r, distinct from a and other elements in X, is true w.r.t. I

[generalization of a technique from well-founded semantics!]



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Unfounded Sets: Example

Program P:

a b :-

I = {b}

Then {a} is an unfounded set of P wrt I

(criterion 3 applies)

Intuitively: there is no reason to believe that a is true given I



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Program P:

a b :-

I = {a,b}

Then both {a} and {b} are unfounded sets of P wrt I

(i.e. we should not believe both!)

However, {a,b} is not an unfounded set!



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Program P:

a b :-

a :- b.

b :- a.

I = {a,b}

Then is the only unfounded set of P wrt I



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Definition & Proposition

DefinitionLet I be an interpretation for a program P.I is unfounded-free if I ∩ X = for each unfounded set X for P w.r.t. I.

Proposition: Let I be an unfounded-free interpretation for a program P. ThenP has the greatest unfounded set GUSP(I)

GUSP(I) is computable in polynomial time



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Monotonic Operator WP

Plan:Compute WP

()

Either this is a unique stable model Otherwise move from WP

() towards the stable models



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Immediate Consequence Operator TP

Definition:TP(I) = {a∈BP | ∃ r∈ground(P)

s.t. a∈H(r), H(r) – {a} ⊆.I, and B(r) ⊆ I}

Note: TP is deterministic!

Definition:WP (I) = TP(I) .GUSP(I)



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Theorem

TheoremLet M be a total interpretation for program P. M is a stable model for P iff M is a fixpoint of WP.

Proposition: Given a propositional program P, WP

() is computable in polynomial time.



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Examples

Program P:

a b :-

WP() = , i.e. not a model!

Program P:

b :- NOT c.

a b :-

WP() = {NOT c, b, NOT a},

i.e. total, i.e. stable model



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Computing the Stable Models: Main

Possibly true literals



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Computing the Stable Models: Compute-Stable



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FINISThis slide deck is not part of the material for examinations

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Steffen Staab [email protected] Advanced Data Modeling 1 of 32 WeST Häufungspunkte Bifurkation: x n+1 = r x n (1-x n ) Startwert x 0 = 0,25