Carlos A. Romero-D az - UCMgpd.sip.ucm.es/rafa/docencia/epl/qlp-overview.pdf · 2009. 1. 12. ·...

(Similarity-based) Qualified Logic ProgrammingA General Overview

Carlos A. Romero-D́ıaz

Grupo de Programación DeclarativaDSIC - Universidad Complutense de Madrid, Spain

[email protected]

Università di Salerno (Italy)December 2008

Carlos A. Romero-D́ıaz (GPD / UCM) Qualified Logic Programming December, 2008 1 / 1

Introduction

Motivation:I Work out an expressive framework for Constraint Funcional-Logic

Programming with uncertainty.I In particular, extend the multiparadigm language T OY with support

for uncertainty and maybe more.

Planning:I Two main phases to develop:

1 Framework for (Constraint) Logic Programming.2 Extend the previous framework with non-deterministic lazy functions

and constraints (that is, Constraint Functional-Logic Programming).

This talk is focused on the actual results of the first phase.

Introduction

Motivation:I Work out an expressive framework for Constraint Funcional-Logic

Programming with uncertainty.I In particular, extend the multiparadigm language T OY with support

for uncertainty and maybe more.

Planning:I Two main phases to develop:

1 Framework for (Constraint) Logic Programming.2 Extend the previous framework with non-deterministic lazy functions

and constraints (that is, Constraint Functional-Logic Programming).

This talk is focused on the actual results of the first phase.

Introduction (II): Starting the work

1 Search for fundamental works in the field to start our investigation:taken as base a seminal paper by van Emden based in a previous oneby Shapiro.

2 Revise the early proposal by van Emden (1986): Quantitative LogicProgramming and improve it in two ways:

I generalizing van Emden’s QLP to a generic scheme parametrized by aqualification domain D.

I generalizing van Emden’s results by providing stronger ones, both indeclarative semantics and goal solving.

Obtaining as result a basic QLP(D).3 Extending QLP(D) with characteristics of other related approaches as

similarity-based reasoning, and improving its expresiveness withthreshold constraints (annotations in the body of clauses).

Obtaining as result a basic QLP(D).

3 Extending QLP(D) with characteristics of other related approaches assimilarity-based reasoning, and improving its expresiveness withthreshold constraints (annotations in the body of clauses).

Obtaining as result a basic QLP(D).3 Extending QLP(D) with characteristics of other related approaches as

similarity-based reasoning, and improving its expresiveness withthreshold constraints (annotations in the body of clauses).

Introduction (and III): Van Emden’s proposal

Van Emden considered Quantitative Logic Programs as sets of clausesof the form:

A← f − B1, . . . ,Bnfor which the certainty being propagated to a clause head was f × bwhere:

f - attenuation factor with f ∈ (0, 1]b - minimum of the certainty factors known for the body atoms

This approach led to:

- general results on model theoretic and fixpoint semantics (similar tothose for classical LP)

- procedure for computing the certainty factors in the least Herbrandmodel of a given program by an alpha-beta heruristic (only for groundatoms with finite search trees)

Introduction (and III): Van Emden’s proposal

Van Emden considered Quantitative Logic Programs as sets of clausesof the form:

A← f − B1, . . . ,Bnfor which the certainty being propagated to a clause head was f × bwhere:

f - attenuation factor with f ∈ (0, 1]b - minimum of the certainty factors known for the body atoms

This approach led to:

- general results on model theoretic and fixpoint semantics (similar tothose for classical LP)

- procedure for computing the certainty factors in the least Herbrandmodel of a given program by an alpha-beta heruristic (only for groundatoms with finite search trees)

Contents

Qualification Domains

Qualification Domains: Axioms

Structure D = 〈D,v,⊥,>, ◦〉 such that:- 〈D,v,⊥,>〉 is a lattice with extreme points ⊥ and > w.r.t. v.- ◦ : D × D → D, called attenuation operation, verifies:

(a) ◦ is associative, commutative and monotonic w.r.t. v.(b) ∀d ∈ D : d ◦ > = d .(c) ∀d ∈ D : d ◦ ⊥ = ⊥ .(d) ∀d , e ∈ D \ {⊥,>} : d ◦ e @ e .(e) ∀d , e1, e2 ∈ D : d ◦ (e1 u e2) = d ◦ e1 u d ◦ e2 .

We generalize van Emden’s QLP to QLP(D) where:- qualification values d ∈ D \ {⊥} instead of d ∈ (0, 1] .- the glb operator

dinstead of min .

- ◦ instead of × (product).

Qualification Domains: Axioms

Structure D = 〈D,v,⊥,>, ◦〉 such that:- 〈D,v,⊥,>〉 is a lattice with extreme points ⊥ and > w.r.t. v.- ◦ : D × D → D, called attenuation operation, verifies:

(a) ◦ is associative, commutative and monotonic w.r.t. v.(b) ∀d ∈ D : d ◦ > = d .(c) ∀d ∈ D : d ◦ ⊥ = ⊥ .(d) ∀d , e ∈ D \ {⊥,>} : d ◦ e @ e .(e) ∀d , e1, e2 ∈ D : d ◦ (e1 u e2) = d ◦ e1 u d ◦ e2 .

We generalize van Emden’s QLP to QLP(D) where:- qualification values d ∈ D \ {⊥} instead of d ∈ (0, 1] .- the glb operator

dinstead of min .

- ◦ instead of × (product).

Qualification Domains (and II): Instances

B = ({0, 1},≤, 0, 1,∧) (Classical Boolean Values).U = ([0, 1],≤, 0, 1,×) (van Emden’s Certainty Degrees).W = ([0,∞],≥,∞, 0,+) (Proof trees’ depths).

The cartesian product D = D1×D2 of two given qualificationdomains is always another qualification domain.

Note: U×W qualifies both certainties and proof trees’ depths.

Qualified Logic Programming

Syntax: Programs

Program: set of qualified definite Horn clauses of the form

A←α− B1, . . . ,Bk

with α ∈ D \ {⊥}.

Example 1

a) Instance U :cruel(X)

Syntax (II): Program Example

Example 2

domestic(cat)

Example 2

domestic(cat)

Example 2

domestic(cat)

Example 2

domestic(cat)

Example 2

domestic(cat)#0.8

Example 2

domestic(cat)#0.8

Example 2

domestic(cat)#0.8

Syntax (and III): Interpretations and Models

- D-qualified atom: A ] d .- Open D-qualified atom: A ]W (W is intended to take values over

D \ {⊥}).

D-entailment relation: is defined as A ] d

Declarative Semantics: Qualified Horn Logic

Qualified Horn Logic over D: is defined as a deductive systemconsisting just of one inference rule QMP(D). If there is someinstance rule (A←α− B1, . . . ,Bk) ∈ [C ] for some C ∈ P andd v α ◦

d{d1, . . . , dk}, the following inference step is allowed:

B1 ] d1 · · · Bk ] dkA ] d

Least Herbrand model of P:

MP = {A ] d | P `QHL(D) A ] d}

Declarative Semantics: Qualified Horn Logic

Qualified Horn Logic over D: is defined as a deductive systemconsisting just of one inference rule QMP(D). If there is someinstance rule (A←α− B1, . . . ,Bk) ∈ [C ] for some C ∈ P andd v α ◦

d{d1, . . . , dk}, the following inference step is allowed:

B1 ] d1 · · · Bk ] dkA ] d

MP = {A ] d | P `QHL(D) A ] d}

Declarative Semantics (and II)

Example 3

cruel(X)

Goals

Initial goals:

· · · ,Ai ]Wi , · · · 8 � 8 · · · ,Wi w βi , · · ·

General goals:

· · · ,Ai ]Wi , · · · 8 σ 8 · · · ,W ′ = d ◦l{W ′m}, · · · , d ◦Wi w βi , · · ·

Solved goals:

8 σ 8 · · · ,W ′ = d ◦l{W ′m}, · · ·

Goals

Initial goals:

· · · ,Ai ]Wi , · · · 8 � 8 · · · ,Wi w βi , · · ·

General goals:

Solved goals:

8 σ 8 · · · ,W ′ = d ◦l{W ′m}, · · ·

Goals

Initial goals:

· · · ,Ai ]Wi , · · · 8 � 8 · · · ,Wi w βi , · · ·

General goals:

Solved goals:

8 σ 8 · · · ,W ′ = d ◦l{W ′m}, · · ·

QSLD Resolution

Resolution step:

· · · ,A ]W , · · · 8 σ 8 · · · , d ◦W w β, · · · C1,σ1

(· · · ,B1 ]W1, · · · ,Bk ]Wk , · · · )σ1 8 σσ1 8 · · · ,α ◦ d ◦W1 w β, · · · , α ◦ d ◦Wk w β, · · · ,W = α ◦

d{W1, . . . ,Wk}, · · ·

where C1 ≡ (H ←α− B1, . . . ,Bk) ∈ P and σ1 the m.g.u. (A,H).

Resolution computations:

G0 σ1 G1 σ2 · · · σn Gn ≡ σ1σ2 · · ·σn 8 ∆n

with G0 initial and Gn solved, and the computed answer (σ, µ) whereσ = σ1σ2 · · ·σn and µ(W ) computed according to ∆n.

QSLD Resolution (and II)

Example 4

eats(X,Y)#W | {} | W >= 0.20 eats.2,{X7→mother(X’)}

eats(X’,Y)#W1 | {X 7→ mother(X’)} |W = 0.70 * min {W1},0.70 * W1 >= 0.20 eats.1,{X’7→eve}

animal(Y)#W2 | {X 7→ mother(eve)} |W = 0.70 * min {W1},W1 = 0.300.30 * 0.70 * W2 >= 0.20 animal,{Y7→bird}

| {X 7→ mother(eve), Y 7→ bird} |W = 0.70 * min {W1},W1 = 0.30W2 = 1.0

Computed answer: {X 7→ mother(eve),Y 7→ bird} 8 {W 7→ 0.21}

Example 4

eats(X,Y)#W | {} | W >= 0.20 eats.2,{X7→mother(X’)}eats(X’,Y)#W1 | {X 7→ mother(X’)} |

W = 0.70 * min {W1},0.70 * W1 >= 0.20 eats.1,{X’7→eve}

Example 4

W = 0.70 * min {W1},0.70 * W1 >= 0.20 eats.1,{X’7→eve}

Example 4

W = 0.70 * min {W1},0.70 * W1 >= 0.20 eats.1,{X’7→eve}

Example 4

W = 0.70 * min {W1},0.70 * W1 >= 0.20 eats.1,{X’7→eve}

Properties of QSLD Resolution

Definition of Solution: A pair of substitutions (θ, ρ) is called solutionof a goal G ≡ A 8 σ 8 ∆ iff:

(i) θ = σθ .(ii) ρ ∈ Sol(∆) .(iii) P `QHL(D) Aθ ]W ρ for every annotated atom in A.

Soundness: Assume G0 ∗ G and G = σ 8 ∆ solved. Let (σ, µ) be thecomputed answer to G . Then (σ, µ) is a solution of G0.

Strong Completeness: Assume a given solution (θ, ρ) for G0 and any fixedstrategy for choosing the selected atom at each resolutionstep. Then there is some computed answer (σ, µ) for G0which is a more general solution than (θ, ρ).

Properties of QSLD Resolution

Definition of Solution: A pair of substitutions (θ, ρ) is called solutionof a goal G ≡ A 8 σ 8 ∆ iff:

(i) θ = σθ .(ii) ρ ∈ Sol(∆) .(iii) P `QHL(D) Aθ ]W ρ for every annotated atom in A.

Soundness: Assume G0 ∗ G and G = σ 8 ∆ solved. Let (σ, µ) be thecomputed answer to G . Then (σ, µ) is a solution of G0.

Strong Completeness: Assume a given solution (θ, ρ) for G0 and any fixedstrategy for choosing the selected atom at each resolutionstep. Then there is some computed answer (σ, µ) for G0which is a more general solution than (θ, ρ).

Implementation: translating to T OY

Requirement CLP or CFLP system with support for CD constraints.

C ≡ a(t)←α− b1(s1), . . . , bk(sk)

C t ≡ a(t,D,W ,Beta) ← α ◦ D w Beta,b1(s1, α ◦ D,W1,Beta),...bk(sk , α ◦ D,Wk ,Beta),W = α ◦

d{W1, . . . ,Wk}

G ≡ a1(t1) ]W1, . . . , am(tm) ]Wm 8 W1 w β1, . . . ,Wm w βm

G t ≡ a1(t1,>,W1, β1), . . . , am(tm,>,Wm, βm)

Implementation: translating to T OY

Requirement CLP or CFLP system with support for CD constraints.

C ≡ a(t)←α− b1(s1), . . . , bk(sk)

C t ≡ a(t,D,W ,Beta) ← α ◦ D w Beta,b1(s1, α ◦ D,W1,Beta),...bk(sk , α ◦ D,Wk ,Beta),W = α ◦

d{W1, . . . ,Wk}

G ≡ a1(t1) ]W1, . . . , am(tm) ]Wm 8 W1 w β1, . . . ,Wm w βm

G t ≡ a1(t1,>,W1, β1), . . . , am(tm,>,Wm, βm)

Similarity-basedQualified Logic Programming

Similarity-based Reasoning

Motivation:I Incorporating similarity-based reasoning in QLP.I Defining a global scheme for programming with qualifications and

similarities.I Showing the possibility of reducing similarities to qualifications.

Results:1 Declarative semantics for a Similarity-based Qualified Logic

Programming (SQLP) scheme.2 A SQLP program transformation to an equivalent QLP program ⇒

Goal Solving Procedure for QLP can be reused.

Similarity-based Reasoning

Motivation:I Incorporating similarity-based reasoning in QLP.I Defining a global scheme for programming with qualifications and

similarities.I Showing the possibility of reducing similarities to qualifications.

Results:1 Declarative semantics for a Similarity-based Qualified Logic

Programming (SQLP) scheme.2 A SQLP program transformation to an equivalent QLP program ⇒

Goal Solving Procedure for QLP can be reused.

Similarity Relations

- Traditional approach in SLP: R : S × S → [0, 1] defined as ageneralization of classical equivalence relations.

- R(x , y) : similarity degree between x and y .- For instance: R(farm, domestic) = 0.3.

D-valued similarity relation over S :

R : S × S → D

s.t. the following hold:

(a) Reflexivity: R(x , y) = >.(b) Symmetry: R(x , y) = R(y , x).(c) Transitivity: R(x , z) w R(x , y) uR(y , z).

Similarity Relations

- Traditional approach in SLP: R : S × S → [0, 1] defined as ageneralization of classical equivalence relations.

- R(x , y) : similarity degree between x and y .- For instance: R(farm, domestic) = 0.3.

D-valued similarity relation over S :

R : S × S → D

s.t. the following hold:

(a) Reflexivity: R(x , y) = >.(b) Symmetry: R(x , y) = R(y , x).(c) Transitivity: R(x , z) w R(x , y) uR(y , z).

Admissible relation R over a set of symbols S

If S = Var ] CS ] PS , and:

I R restricted to Var : R(X ,X ) = > and R(X ,Y ) = ⊥ if X 6= Y .I R(x , y) 6= ⊥ ⇒

(a) x and y are the same variable, or(b) x , y ∈ CSn (for arity n ≥ 0), or(c) x , y ∈ PSn (for arity n ≥ 0).

Definition: Extension of R to act over terms.

R(X ,X ) = > and R(X , t) = R(t,X ) = ⊥ if X 6= t.R(c(tn), c ′(t ′m)) = ⊥ if n 6= m.R(c(tn), c ′(t ′n)) = R(c, c ′) uR(t1, t ′1) u . . . uR(tn, t ′n).

(Analogously for atoms and clauses).

If S = Var ] CS ] PS , and:I R restricted to Var : R(X ,X ) = > and R(X ,Y ) = ⊥ if X 6= Y .

I R(x , y) 6= ⊥ ⇒(a) x and y are the same variable, or(b) x , y ∈ CSn (for arity n ≥ 0), or(c) x , y ∈ PSn (for arity n ≥ 0).

If S = Var ] CS ] PS , and:I R restricted to Var : R(X ,X ) = > and R(X ,Y ) = ⊥ if X 6= Y .I R(x , y) 6= ⊥ ⇒

Definition: Extension of R to act over terms.R(X ,X ) = > and R(X , t) = R(t,X ) = ⊥ if X 6= t.

R(c(tn), c ′(t ′m)) = ⊥ if n 6= m.R(c(tn), c ′(t ′n)) = R(c, c ′) uR(t1, t ′1) u . . . uR(tn, t ′n).

Definition: Extension of R to act over terms.R(X ,X ) = > and R(X , t) = R(t,X ) = ⊥ if X 6= t.R(c(tn), c ′(t ′m)) = ⊥ if n 6= m.

R(c(tn), c ′(t ′n)) = R(c, c ′) uR(t1, t ′1) u . . . uR(tn, t ′n).

Definition: Extension of R to act over terms.R(X ,X ) = > and R(X , t) = R(t,X ) = ⊥ if X 6= t.R(c(tn), c ′(t ′m)) = ⊥ if n 6= m.R(c(tn), c ′(t ′n)) = R(c, c ′) uR(t1, t ′1) u . . . uR(tn, t ′n).

Similarity-based Qualified Logic Programming

Example 5

farm(cow)

Example 5

farm(cow)

Example 5

farm(cow)

Example 5

farm(cow)

Example 5

farm(cow)

Example 5

farm(cow)

Example 5

farm(cow)

SQLP Declarative Semantics

Similarity-based Qualified Horn Logic over (R,D): If there issome ((A←α− B1, . . . ,Bk), δ) ∈ [C ]R for some C ∈ P. Then, forany d ∈ D \ {⊥} s.t. d v α ◦

d{δ, d1, . . . , dk}:

B1 ] d1 · · · Bk ] dkA ] d

{A ] d | P `SQHL(R,D) A ] d}

SQLP Declarative Semantics

Similarity-based Qualified Horn Logic over (R,D): If there issome ((A←α− B1, . . . ,Bk), δ) ∈ [C ]R for some C ∈ P. Then, forany d ∈ D \ {⊥} s.t. d v α ◦

d{δ, d1, . . . , dk}:

B1 ] d1 · · · Bk ] dkA ] d

{A ] d | P `SQHL(R,D) A ] d}

Goal Solving in SQLP

Two possibilities:

(1) Change unification to similarity-based unification (done in some SLPworks).

(2) Reduce a SQLP program to an equivalent QLP one and solve there(some transformations from SLP to LP proposed by other authors).

We take possibility (2), so

SQLP(R,D) program

P =⇒QLP(D) program

SR(P)

Proving as main result that

P `SQHL(R,D) A ] d ⇐⇒ SR(P) `QHL(D) A ] d .

Two possibilities:

SQLP(R,D) program

SR(P)

Two possibilities:

SQLP(R,D) program

SR(P)

Reducing Similarities to Qualifications:A Program Transformation

SR(P) = PS ∪ P∼ ∪ Ppay

PS For every (H ←α− B) ∈ P and for each H ′ such that R(H`,H ′) 6= ⊥:

(H ′ ←α− payR(H`,H′),S`,B) ∈ PS

where (H`,S`) = lin(H).Linearization of clause heads is a novelty in our approach.

P∼ = {X ∼ X←>−}∪ {(c(Xn) ∼ c′(Yn)←>− payR(c,c′), Xn ∼ Yn) |c, c′ ∈ CSn, R(c, c′) 6= ⊥}.

Ppay = {payw ←w− | for each atom payw occurring in PS ∪ P∼}.

(H ′ ←α− payR(H`,H′),S`,B) ∈ PS

Why to linearize?

To let equal variables in the head atom take similar (and not only thesame) values:

Example 6

P ≡ p(X, X)←1.0−, R(c, d) = 0.7

We want to prove: p(c,c)#1.0 and p(d,d)#1.0,and also p(c,d)#0.7 and p(d,c)#0.7.

Definition: lin(H) = (H`, S`)

H` : linear version of H (now without repeated variables).

S` : set of similarity conditions X ∼ Xi (1 ≤ i ≤ n) between thevariables that replaced the n additional occurrences of the samevariable in H.

Example: lin(p(X, X)) = (p(X, X1), {X ∼ X1}).

Why to linearize?

Example 6

P ≡ p(X, X)←1.0−, R(c, d) = 0.7

Why to linearize?

Example 6

P ≡ p(X, X)←1.0−, R(c, d) = 0.7

We want to prove: p(c,c)#1.0 and p(d,d)#1.0,

and also p(c,d)#0.7 and p(d,c)#0.7.

Why to linearize?

Example 6

P ≡ p(X, X)←1.0−, R(c, d) = 0.7

Why to linearize?

Example 6

P ≡ p(X, X)←1.0−, R(c, d) = 0.7

Why to linearize?

Example 6

P ≡ p(X, X)←1.0−, R(c, d) = 0.7

Why to linearize?

Example 6

P ≡ p(X, X)←1.0−, R(c, d) = 0.7

Why to linearize?

Example 6

P ≡ p(X, X)←1.0−, R(c, d) = 0.7

(H ′ ←α− payR(H`,H′),S`,B) ∈ PS

Program transformation in action

Example 7: SQLP(R,U) programfarm(cow)

Example 7

* farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7

farm(cow)

Example 7: QLP(U) programfarm(cow)

Improving the semantics

Motivation:I Incorporating to QLP a simple treatement of negation.I Adding further expressivity to QLP with threshold constraints in clause

bodies.

Results:1 An adapted declarative semantics for a Qualified Logic Programming

with Bivalued Predicates (BQLP) scheme.2 Improved expressivity (from QLP) with a simple kind of negation and

threshold constraints in clause bodies.3 Sound and complete goal solving precedure (via QSLD(D) resolution).4 Easy implementation based on the existent QLP implementation.

Motivation:I Incorporating to QLP a simple treatement of negation.I Adding further expressivity to QLP with threshold constraints in clause

bodies.

Results:1 An adapted declarative semantics for a Qualified Logic Programming

with Bivalued Predicates (BQLP) scheme.2 Improved expressivity (from QLP) with a simple kind of negation and

threshold constraints in clause bodies.3 Sound and complete goal solving precedure (via QSLD(D) resolution).4 Easy implementation based on the existent QLP implementation.

BQLP Program Example

Example 8

animal(bird)#tt

Example 8

animal(bird)#tt

Example 8

animal(bird)#tt

Example 8

animal(bird)#tt

Example 8

animal(bird)#(tt,1.0)

Example 8

BQLP Declarative Semantics

Qualified Horn Logic over D: If there is some instance rule (A ] v←α− B1 ] (v1,w1), . . . , Bk ] (vk ,wk)) ∈ [C ] with C ∈ P, and somed1, . . . , dk ∈ D \ {⊥} such that di w? wi for all 1 ≤ i ≤ k , then thefollowing inference step is allowed for any d v α ◦

d{d1, . . . , dk}:

B1 ] (v1, d1) · · · Bk ] (vk , dk)A ] (v , d)

- d w? w ⇐⇒def w =? or w 6=? and d w w .

{A ] (v , d) | P `QHL(D) A ] (v , d)}

BQLP Declarative Semantics

Qualified Horn Logic over D: If there is some instance rule (A ] v←α− B1 ] (v1,w1), . . . , Bk ] (vk ,wk)) ∈ [C ] with C ∈ P, and somed1, . . . , dk ∈ D \ {⊥} such that di w? wi for all 1 ≤ i ≤ k , then thefollowing inference step is allowed for any d v α ◦

d{d1, . . . , dk}:

B1 ] (v1, d1) · · · Bk ] (vk , dk)A ] (v , d)

- d w? w ⇐⇒def w =? or w 6=? and d w w .Least Herbrand model of P:

{A ] (v , d) | P `QHL(D) A ] (v , d)}

BQLP Declarative Semantics (and II)

Example 9

cruel(X)#ff

Ongoing & Future Work

Ongoing Work1 Improvement of the actual prototype supporting threshold constraints

and making an extensive use of constraint solvers for qualificationconstraints managing.

2 Extension of QLP to a first-order Qualified Constraint Functional-LogicProgramming Language.

Future WorkI Goal Solving by Similarity-based QSLD and its implementation both via

a program transformation and similarity-based primitive unification.I Develop of methods and tools for defining complex similarity relations.I Search for new instances and interesting examples.

To think about:I Similarity between terms as if they were constants?I Even more general Qualification Domains without losing usefulness?