tutorial: logic programming

Tutorial: logic programming

Markus Krötzsch

Institute AIFB, Universität [email protected]

Markus Krötzsch (AIFB Karlsruhe) Logic programming Oct 24 2006 1 / 37

Outline

1 Basics

2 Propositional logic programs

3 Definite logic programs

4 Normal programs

5 F-Logic

6 Summary


Outline

1 Basics



4 Normal programs

5 F-Logic

6 Summary


Rules

A (logic programming) rule is a formula of the form

B1 ∧ B2 ∧ . . . ∧ Bn︸︷︷︸

Body

→ H1 ∨ H2 ∨ . . . ∨ Hm︸︷︷︸

Head

Bi and Hj are usually literals, i.e. negated or non-negated logicalatoms


Rules


B1 ∧ B2 ∧ . . . ∧ Bn︸︷︷︸

Body

→ H1 ∨ H2 ∨ . . . ∨ Hm︸︷︷︸

Head


Disjunctions in the body are not needed:

A ∨ B → C ⇒A → CB → C


Rules


B1 ∧ B2 ∧ . . . ∧ Bn︸︷︷︸

Body

→ H1 ∨ H2 ∨ . . . ∨ Hm︸︷︷︸

Head


Disjunctions in the body are not needed:

A ∨ B → C ⇒A → CB → C

Conjunctions in the head are not needed:

A → B ∧ C ⇒A → BA → C

{ Lloyd-Topor translations


Clauses

In classical logic, implications are disjunctions:

B → H ≡ ¬B ∨ H

For a rule, we obtain:

B1 ∧ B2 ∧ . . . ∧ Bn → H1 ∨ H2 ∨ . . . ∨ Hm


Clauses


B → H ≡ ¬B ∨ H


B1 ∧ B2 ∧ . . . ∧ Bn → H1 ∨ H2 ∨ . . . ∨ Hm

≡ ¬(B1 ∧ B2 ∧ . . . ∧ Bn) ∨ H1 ∨ H2 ∨ . . . ∨ Hm


Clauses


B → H ≡ ¬B ∨ H


B1 ∧ B2 ∧ . . . ∧ Bn → H1 ∨ H2 ∨ . . . ∨ Hm

≡ ¬(B1 ∧ B2 ∧ . . . ∧ Bn) ∨ H1 ∨ H2 ∨ . . . ∨ Hm

≡ ¬B1 ∨ ¬B2 ∨ . . . ∨ ¬Bn ∨ H1 ∨ H2 ∨ . . . ∨ Hm

Disjunctions of literals are called clauses

{ rules as clauses


Outline

1 Basics



4 Normal programs

5 F-Logic

6 Summary


Propositional logic programs

A propositional logic program . . .

is a finite set of rules which are

propositional: only propositional atoms,

definite: exactly one atom in the head.

General form: B1 ∧ B2 ∧ . . . ∧ Bn → H

Common syntax: H D B1,B2, . . . ,Bn

Fact: rule with empty body ({ body vacuously true)


Example

abc D ae D b, de D b, ce D a, b, e


Semantics of propositional programs

Standard semantics of propositional logic:

An interpretation I . . .

is a mapping ·I from formulae to {t , f }, such that

(A ∧ B)I = t iff AI = t and BI = t ,

(A ∨ B)I = t iff AI = t or BI = t ,

¬AI = t iff AI = f .

{We can view interpretations as sets of (true) atoms


Semantics of propositional programs

Standard semantics of propositional logic:

An interpretation I . . .

is a mapping ·I from formulae to {t , f }, such that

(A ∧ B)I = t iff AI = t and BI = t ,

(A ∨ B)I = t iff AI = t or BI = t ,

¬AI = t iff AI = f .

{We can view interpretations as sets of (true) atoms

Take a set of formulae F (e.g. a propositional program).

I is a model of F if FI = t for all F ∈ F .

A formula G is a consequence of F if GI = t for all models I of F .


Bottom-up computation

How can we compute the above semantics?

Observation: every rule has exactly one consequence

Idea: compute all consequences bottom-up, step by step






The TP operator

Consider a program P. The function TP maps interpretations tointerpretations:

TP(I) = {H | P contains a rule H D B1, . . . ,Bn such that Bi ∈ I}






The TP operator

Consider a program P. The function TP maps interpretations tointerpretations:

TP(I) = {H | P contains a rule H D B1, . . . ,Bn such that Bi ∈ I}

Step-wise computation:

T0P = ∅ Ti+1

P = TP(TiP)


Bottom-up computation (2)

Example: P =a c D a e D b, cb e D b, d e D a, b, e

{ T0P = ∅




{ T0P = ∅, T1

P = {a, b}




{ T0P = ∅, T1

P = {a, b}, T2P = {a, b , c}




{ T0P = ∅, T1

P = {a, b}, T2P = {a, b , c}, T3

P = {a, b , c, e}




{ T0P = ∅, T1

P = {a, b}, T2P = {a, b , c}, T3

P = {a, b , c, e}{ T4

P = {a, b , c, e} = T5P = T6

P = . . .




{ T0P = ∅, T1

P = {a, b}, T2P = {a, b , c}, T3

P = {a, b , c, e}{ T4

P = {a, b , c, e} = T5P = T6

P = . . .

I is a fixed point of TP if TP(I) = I

T∞P =⋃

i≥0 TiP is always the least fixed point and the least model


Goal-directed computation

Computing all consequences is not efficient.

How can we directly check whether some consequence holds?





Reasoning by contradiction

F is a consequence of P iff ¬F ∧ P has no models (i.e. is unsatisfiable).





Reasoning by contradiction

F is a consequence of P iff ¬F ∧ P has no models (i.e. is unsatisfiable).

Strategy:

Question: does Q1 ∧ Q2 ∧ . . . ∧Qn follow from P?

Step 1: negate query ¬(Q1 ∧ . . . ∧ Qn)

Step 2: check whether P ∧ ¬(Q1 ∧ . . . ∧Qn) is satisfiable

{ A goal is a formula of the form ¬(Q1 ∧ . . . ∧ Qn){ Goals can be written as D Q1, . . . ,Qn


Resolution

How can we check unsatisfiability?{ Check whether a contradictory statements follows!


Resolution


What is a contradictory statement?


Resolution


What is a contradictory statement?Recall: an implication is a disjunctionRecall: the empty implication is always false


Resolution


What is a contradictory statement?Recall: an implication is a disjunctionRecall: the empty implication is always false{ the one and only contradictory rule is “→”

How to compute logic consequences of a program?


Resolution



How to compute logic consequences of a program? { Resolution

The resolution rule:A → B B → C

A → C


Resolution



How to compute logic consequences of a program? { Resolution

The resolution rule:A → B B → C

A → C

More generally:

A1 ∧ . . . ∧ An → Bi B1 ∧ . . . ∧ Bi ∧ . . . ∧ Bm → CA1 ∧ . . . ∧ An ∧ B1 ∧ . . . ∧ Bi−1 ∧ Bi+1 ∧ . . . ∧ Bm → C


SLD-resolution

Fact: only resolutions steps with a goal as a main premise areneeded:

A1 ∧ . . . ∧ An → Bi B1 ∧ . . . ∧ Bi ∧ . . . ∧ Bm →

A1 ∧ . . . ∧ An ∧ B1 ∧ . . . ∧ Bi−1 ∧ Bi+1 ∧ . . . ∧ Bm →

{ linear resolution: apply rules only to the latest goal

Fact: selection of Bi is “don’t care” nondeterministic{ any selectionrule can be chosen


SLD-resolution

Fact: only resolutions steps with a goal as a main premise areneeded:

A1 ∧ . . . ∧ An → Bi B1 ∧ . . . ∧ Bi ∧ . . . ∧ Bm →

A1 ∧ . . . ∧ An ∧ B1 ∧ . . . ∧ Bi−1 ∧ Bi+1 ∧ . . . ∧ Bm →

{ linear resolution: apply rules only to the latest goal

Fact: selection of Bi is “don’t care” nondeterministic{ any selectionrule can be chosen

SLD-resolutionSelection-driven, Linear resolution for Definite clauses (a “folkacronym”).


SLD-trees and backtracking

Choice of program rule to resolve upon is “don’t known” nondeterministic{ backtracking needed

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Complexity of propositional programs

Computation with propositional logic programs can be done in P (which iseasy to see from the computation of TP).


Outline

1 Basics



4 Normal programs

5 F-Logic

6 Summary


Definite logic programs

Instead of propositional atoms, we have a first order language:predicate symbols with some arity, e.g. Rconstants, e.g. a, bfunction symbols with some arity, e.g. f , gvariables, e.g. X , Y

A term is any well-formed expression built from constants, functions,and variables. Terms without variables are called ground.

An atom is an expression R(t1, . . . , tn) with R an n-ary predicate andt1, . . . , tn terms

A definite rule is a rule of the form

H D B1, . . . ,Bm

with H,B1, . . . ,Bm atoms, where m might be 0.


What do variables mean?

Variables are universally quantified on rule-level.

Example: uncle(X ,Z) D brother(X ,Y), parent(Y ,Z)

Rules with variables can be read as “macros” for rules obtained byuniformly replacing all variables with ground terms

More generally: variables can be uniformly replaced by any term{substitution


Example

number(0)number(s(X)) D number(X)

add(0,Y ,Y)add(s(X),Y ,Z) D add(X , s(Y),Z)

mul(0,Y , 0)mul(s(X),Y ,Z) D mul(X ,Y ,Z ′), add(Y ,Z ′,Z)

Example conclusions:number(s(s(s(0)))), add(s(s((0)), s(0), s(s(s(0)))), . . .


Example

number(0)number(s(X)) D number(X)

add(0,Y ,Y)add(s(X),Y ,Z) D add(X , s(Y),Z)

mul(0,Y , 0)mul(s(X),Y ,Z) D mul(X ,Y ,Z ′), add(Y ,Z ′,Z)

Example conclusions:number(s(s(s(0)))), add(s(s((0)), s(0), s(s(s(0)))), . . .

{ Definite programs can encode infinite domains (they can also enforcethem, though not in the above example).


Semantics

Definite logic program as abbreviation for infinite propositionalprogram


Semantics


Example:number(s(X)) D number(X)

{

number(s(0)) D number(0)number(s(s(0))) D number(s(0))

number(s(s(s(0)))) D number(s(s(0))). . .


Semantics


Example:number(s(X)) D number(X)

{

number(s(0)) D number(0)number(s(s(0))) D number(s(0))

number(s(s(s(0)))) D number(s(s(0))). . .

ground atoms instead of propositional atoms

interpretations as sets of (true) ground atoms{ Herbrand interpretation

TP operator defined as before (on the infinite set of ground rules)



How can we practically compute consequences?



How can we practically compute consequences?{ Compute (partially) ground rules “on demand”{ Two literals represent overlapping sets of ground atoms if they canbe unified.




Example: R(X , f(a)) and R(g(Z),W) are unifiable




Example: R(X , f(a)) and R(g(Z),W) are unifiable{ R(g(Z), f(a)) represents all ground instances both agree on{ substitution σ = {X 7→ g(Z),W 7→ f(a)}




Example: R(X , f(a)) and R(g(Z),W) are unifiable{ R(g(Z), f(a)) represents all ground instances both agree on{ substitution σ = {X 7→ g(Z),W 7→ f(a)}

Resolution on non-ground rules:

A → B B ′ → C(A → C)σ

if Bσ = B ′σ

propositional SLD-resolution can be lifted


Example: SLD resolution for definite programs

(1) add(0,Y ,Y)(2) add(s(X),Y ,Z) D add(X , s(Y),Z)

Goal: D add(s(0), s(0),W)





D add(s(0), s(0),W)





D add(s(0), s(0),W) + (2) σ1 = {X1 7→ 0,Y1 7→ s(0),Z1 7→ W }





D add(s(0), s(0),W) + (2) σ1 = {X1 7→ 0,Y1 7→ s(0),Z1 7→ W }

⇓

D add(0, s(s(0)),W)





D add(s(0), s(0),W) + (2) σ1 = {X1 7→ 0,Y1 7→ s(0),Z1 7→ W }

⇓

D add(0, s(s(0)),W) + (1) σ2 = {Y2 7→ s(s(0)),W 7→ s(s(0))}





D add(s(0), s(0),W) + (2) σ1 = {X1 7→ 0,Y1 7→ s(0),Z1 7→ W }

⇓

D add(0, s(s(0)),W) + (1) σ2 = {Y2 7→ s(s(0)),W 7→ s(s(0))}

⇓

�





D add(s(0), s(0),W) + (2) σ1 = {X1 7→ 0,Y1 7→ s(0),Z1 7→ W }

⇓

D add(0, s(s(0)),W) + (1) σ2 = {Y2 7→ s(s(0)),W 7→ s(s(0))}

⇓

�

Result: Wσ1σ2 = s(s(0))


How hard is this computation?

Extend the earlier example:pow(X , 0, s(0))

pow(X , s(Y),Z) D pow(X ,Y ,Z ′),mul(X ,Z ′,Z)

Goal: D pow(X , s(s(s(N))),XP), pow(Y , s(s(s(N))),YP),pow(Z , s(s(s(N))),ZP), add(XP,YP,ZP)






In words:“Which natural numbers satisfy the equation xn + yn = zn, for n > 2?”







Uh oh . . .







Uh oh . . .

Definite logic programming is indeed undecidable.







Uh oh . . .

Definite logic programming is indeed undecidable.But it is semi-decidable: all positive conclusions eventually appear


Decidable cases

Non-recursive definite programs

Termination guaranteed for logic programs without recursive rules.

Computational complexity: NET (just like OWL DL)


Decidable cases

Non-recursive definite programs

Termination guaranteed for logic programs without recursive rules.

Computational complexity: NET (just like OWL DL)

Datalog

A datalog progam is a definite program without function symbols.

Datalog is decidable

Computational complexity: ET (just like OWL Lite)even with just a single rule and no facts (“sirup”)

Data complexity: P{ useful for deductive databases


Outline

1 Basics



4 Normal programs

5 F-Logic

6 Summary


Logical programs with negation

Normal logic programs: first-order programs with negation in the body

typically non-monotonic negation

Example:bird(tweety)

bird(tux)penguin(tux)

fly(X) D bird(X), not penguin(X)


Negation as failure

How to compute with non-monotonic negation?


Negation as failure


Negation as failure:

negated statements are true if it their positive version cannot be derived{ start a new proof tree when encountering negated statements{ SLDNF-resolution (SLD + Negation as Failure)


Negation as failure


Negation as failure:

negated statements are true if it their positive version cannot be derived{ start a new proof tree when encountering negated statements{ SLDNF-resolution (SLD + Negation as Failure)

operational, not declarative

implemented in Prolog (usually based on incompleteSLD-implementation, making the system unsound when usingnegation)


Semantic issues

What does the following mean?p D not qq D not p


Semantic issues


Classical semantics: p ∨ q

Negation as failure: loop proof search

Intuition(?): neither p nor q “really” hold


Semantic issues


Classical semantics: p ∨ q

Negation as failure: loop proof search

Intuition(?): neither p nor q “really” hold

{ Introduction of three-valued semantics: true, false, unknown

Example: Well-founded semantics

Many different non-monotonic semantics, usually agree on morewell-behaved programs


Expressive power

How powerful are non-monotonic semantics?

Recall: in definite programs, we could ask whether there is acounterexample to Fermat’s Last Theorem


Expressive power



In normal logic programs, we can also ask whether there is no suchcounterexample{ Non-monotonic semantics usually striclty more powerful thanmonotonic variants


Expressive power



In normal logic programs, we can also ask whether there is no suchcounterexample{ Non-monotonic semantics usually striclty more powerful thanmonotonic variants

Well-founded semantics: not semidecidable (captures Π11-fragment of

second order logic)


Outline

1 Basics



4 Normal programs

5 F-Logic

6 Summary


F-Logic

“Object oriented” logic programming

different definitions of semantics: classical (first-order) and LP(non-monotonic)

no normative defintions for syntax or semantics

Implementation in ONtoobroker and Flora: assumes well-foundedsemantics (and computes unspecified parts of it){ systems cannot return all answers or all true answers, even ininfinite time


F-Logic example syntax

/* Atoms */

iokaste:woman.

laios:man.

/* Molecules */

oedipus:man[father->laios; mother->iokaste].

polyneikes:man[father->oedipus; mother->iokaste;

sister->>{antigone, ismene};

brother->>eteokles:man]

/* Rules */

FORALL X,Y X[son->>Y] <- Y:man[father->X].

FORALL X,Y X[son->>Y] <- Y:man[mother->X].

/* Queries */

FORALL X,Y <- X:man[sohn->>Y[mother->iokaste]].


F-Logic semantics

“Object oriented” notation is translated into facts:

a:C 7→ isa_(a,C)

A::B 7→ sub_(A,B)

a[B->c] 7→ att_(a,B,c)

a[B-»c] 7→ setatt_(a,B,c)

A[B=>C] 7→ atttype_(A,B,C)

A[B=»C] 7→ setatttype_(A,B,C)

Properties of auxilliary predicates such as sub_ described by axioms.

Further syntactic features treated with Lloyd-Topor translations


F-Logic semantics

“Object oriented” notation is translated into facts:

a:C 7→ isa_(a,C)

A::B 7→ sub_(A,B)

a[B->c] 7→ att_(a,B,c)

a[B-»c] 7→ setatt_(a,B,c)

A[B=>C] 7→ atttype_(A,B,C)

A[B=»C] 7→ setatttype_(A,B,C)

Properties of auxilliary predicates such as sub_ described by axioms.

Further syntactic features treated with Lloyd-Topor translations

Warning! The above is based on incomplete descriptions in Angele andLausen (2004), which merely sketch the semantics. I know of nonormative public spec of the current implementation in Ontobroker or Flora(which does not say that such documents do not exist).


Outline

1 Basics



4 Normal programs

5 F-Logic

6 Summary


Summary

Complexity overview:Formalism Complexity Var.? Func.? Neg.?

Propositional programs P — — —Datalog ET-complete × — —Definite programs r.e.-complete × × —— without recursion NET-complete × × —Well-founded semantics Π1

1-complete × × ×

Prolog is based on definite programs

F-Logic is based on well-founded semantics + F-Logic


References

J.W. Lloyd. Foundations of Logic Programming. Springer, 1987.

E. Dantsin, T. Eiter, G. Gottlob, and A. Voronkov. Complexity andexpressive power of logic programming. ACM ComputingSurveys, 33(3): 374–425, 2001.

G. A. Ringwood. SLD: a folk acronym? ACM SIGPLAN Notices,24(5): 71–75, 1989.

J. Angele and G. Lausen. Ontologies in F-logic. In S. Staab and R.Studer, editors, Handbook on Ontologies, pages 29–50. Springer,2004.


tutorial: logic programming

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