Collaboration and ManagementDana Fisman

1

Principles ofProgrammingLanguages

Lesson 21 – An Introduction to Logic Programming –Relational Logic Programming

www.cs.bgu.ac.il/~ppl172

Slides by Yaron Gonen and Dana FismanBased on Book by Mira Balaban and

3 main paradigms in PL

Imperative Functional Declarative

Turing Machine [Turing 1936]

Lambda Calculus [Church 1936]

Constructive Logic Proofs [Kowalski, 1970,Colmerauer, 1972]

How it Came to be

Kowalski's Observation (early 70s):

An axiom H B1 and B2 ... and Bn

represents a procedure

H is the procedure’s head and the Bi’s are its body

To solve (execute) H, we recursively solve B1 ... Bn

In a quest to automate the process of proving logical statements, Colmerauer and his group embodied this observation into Prolog.

Prolog and subsets of interest

RelationalLogic

Programming

DatalogLogic

Programming

Prolog

Prolog and subsets of interest

RelationalLogic

Programming

Logic Programming

Prolog

Topics

1. Relational logic programming Programs specify relations among entities

2. Logic Programming Programs with data structures: lists, binary

trees, symbolic expressions, natural numbers.

3. Meta tools Backtracking optimizations (cuts), unify, meta-

circular interpreters.

4. Prolog and more advanced programming Arithmetic, cuts, negation.

Prolog and its subsets

RelationalLogic

Programming

Logic Programming

Prolog

Simple Example

?- parent(sarah,Isaac).

true

?- parent(sarah,joseph).

false

?- parent(miriam,joseph).

false

db.pl

parent(abraham, isaac). parent(isaac, jacob).parent(sarah, isaac).parent(jacob, joseph).

male(abraham).male(isaac).male(joseph).

female(sarah).

Simple Example

parent(abraham, isaac). parent(isaac, jacob).parent(sarah, isaac).parent(jacob, joseph).

male(abraham).male(isaac).male(joseph).

female(sarah).

?- parent(sarah,X).

X = isaac

?- parent(X,isaac).

X = sarah

db.pl

X = abraham

Note the mind switch:

Working with relationsas opposed to functions

We don’t have to define multiple related

functions

Queries with vars

parent(abraham, isaac). parent(isaac, jacob).parent(sarah, isaac).parent(jacob, joseph).

male(abraham).male(isaac).male(joseph).

female(sarah).

grand(X,Z) :- parent(Y,Z), parent(X,Y).

?- grand(sarah,jacob).

true

?- grad(X,jacob).

X = sarah

X = abraham

?- grand(sarah,joseph).

false

Adding RulesSimple Example

Fact

sRule

Logic Programming –as a programming language

Every programming language has:

Syntax What expressionsare legal

Formulas (Facts, Rules)

Semantics What is the valueof a given expression

Answer to query: true / false + assignments to the variables

Operational Semantics

An algorithm to compute the semantics

Unification, Backtracking

In Logic Programming:

Logic Programming - Mind Switch

Mind Switch:

◦ Formula (Fact/Rule) ⇔ procedure declaration

◦ Query ⇔ procedure call

◦ Proving ⇔ computation

Relational Logic Programming

A computational model based on Kowalski's interpretation

The Prolog language ('70) - contains RLP + additional programming features

Relational LP –Atomic Formulas Syntax

atomic formula:

predicate_symbol(term1, ..., termn)

Predicate symbols start with lowercase

terms:

◦ symbols (representing individual constants)

◦ variables (start with uppercase or with an underscore ‘_’ for anonymous)

parent(abraham, isaac)

ancestor(adam, Person)

address(City, Street, _Number)

Predicates and constantsare distinguished by

their context

Relational LP –Atomic Formulas Examples

Relational LP –Facts Syntax

A fact is an assertion of an atomic formula

H.

where H is an atomic formula.

Variables in facts are universally quantified.

Makes it a fact

parent(abraham, isaac).

color(red).

shape(square).

ancestor(adam, Person).

related(Person, Person).

Variables in facts are universally quantified

Relational LP –Facts Examples

parent(abraham, isaac). parent(sarah, isaac).

male(abraham).

female(sarah).

?- female(abraham).

true

Capitalization!Simple Example?- female(abraham).

false

parent(abraham, isaac). parent(sarah, isaac).

male(abraham).

female(Sarah).

Rules are formulas that state conditioned relationships between entities

H :- B1 , B2 , ... , Bn .

Meaning:

If B1 and B2 and ... Bn holdthen H also holds.

Relational LP –Rules Syntax

Rule head

Rulebody

facts

H

mother(X,Y) :- parent(X,Y), female(X).father(X,Y) :- parent(X,Y), male(X).

grand(X,Z) :- parent(X,Y), parent(Y,Z).

grandmother(X) :- female(X), grand(X,_).grandfather(X) :- male(X), grand(X,_).

Relational LP –Rules Examples

o Variables in rules are universally quantified

Variables in rule heads are universally quantified

Variables appearing only in rule bodies are existentially quantified

o Variables are bound within a rule

X, Y, Z. parent(X, Y) & parent(Y, Z)

grand(X,Z).

X, Z.( Y parent(Y, Z) & parent(Y, Z))

grand(X,Z)

parent(abraham, isaac). parent(isaac, jacob).parent(sarah, isaac).parent(jacob, joseph).

male(abraham).male(isaac).male(joseph).

female(sarah).

grand(X,Z) :- parent(X,Y), parent(Y,Z).

mother(X) :- parent(X, _), female(X).

?- grand(sarah,X).

X = jacob

?- grad(sarah,_X).

true

?- mother(abraham).

false

anonymous varsSimple Example

?- mother(sarah).

true

father(isaac, jacob).

father(isaac, jacob):- true.

Relational LP –Rules Special Case

o Facts can be considered as rules with an empty body.

This fact and this rule have equivalent meaning.

trueis a zero-

arity predicate

Relational LP –Procedures Syntax & Examples

Procedures are ordered collections of rules and facts sharing the same name and the same arity

% Signature: parent(Parent, Child)/2

% Purpose: Parent is a parent of Child

parent(abraham, isaac).

parent(isaac, jacob).

parent(sarah, isaac).

parent(jacob, joseph).

% Signature: male(Person)/1

% Purpose: Person is a male

male(abraham).

male(isaac).

male(joseph).

Relational LP –Procedures Examples

% Signature: rise(CO, Time, City)/3% Purpose: CO is a celestial object that in City at Time % is seen as if it rises.rise(son, six-am, beer-sheva). rise(moon, _, beer-sheva).

% Signature: rise(S,B)/2% Purpose: B is a bigger amount than Srise(milliliter, liter).rise(gram, kilogram).rise(centimeter, decimeter).rise(centimeter, meter).

These are two different procedures.

A query has the syntax

?- Q1 , Q2 , ... , Qn .

Meaning:

Assuming the program facts and rules, do Q1 and Q2 and ... Qn hold?

Variables in queries are existentially quantified

Relational LP –Queries Syntax

Queryinvocation

Relational LP –Queries Syntax

?- parent(abraham, isaac).

?- parent(isaac, X).

?- parent(sarah, _X).

?- parent(sarah, _X), female(sarah).

Is abraham a parent of isaac?

Who are isaac sons?

Is sarah a parent of someone?

Is sarah a parent of someone, and a female?

queriesSimple Example

?- parent(abraham, isaac), parent(isaac, Y),parent(sarah, _X).

?- parent(sarah, _X), female(sarah).

true

?- parent(sarah, X), female(sarah).

X = isaac

Y = jacob

parent(abraham, isaac). parent(isaac, jacob).parent(sarah, isaac).parent(jacob, joseph).parent(jacob, dan).parent(jacob, benjamin).

male(abraham).male(isaac).male(joseph).

female(sarah).

Queries w.mult. answersSimple Example

?- parent(X, issac), parent(Y, isaac).

X = abraham, Y = abraham;X = abraham, Y = sarah; X = sarah, Y = abraham; X = sarah, Y = sarah;false.

?- parent(X, issac), parent(Y, isaac),X \= Y.

X = abraham, Y = sarah; X = sarah, Y = abraham; false.

parent(abraham, isaac). parent(isaac, jacob).parent(sarah, isaac).parent(jacob, joseph).parent(jacob, dan).parent(jacob, benjamin).

male(abraham).male(isaac).male(joseph).

female(sarah).

parent(abraham, isaac). parent(isaac, jacob).parent(sarah, isaac).parent(jacob, joseph).parent(jacob, dan).parent(jacob, benjamin).

male(abraham).male(isaac).male(joseph).

female(sarah).

Queries w.mult. answersSimple Example

?- parent(F, X), parent(F, Y),X /= Y.

F=jacob, X=joseph, Y=dan;F=jacob, X=dan, Y=joseph;F=jacob, X=joseph, Y=benjamin;F=jacob, X=benjamin, Y=joseph;F=jacob, X=dan, Y=benjamin;F=jacob, X=benjamin, Y=dan;false.

parent(abraham, isaac). parent(isaac, jacob).parent(sarah, isaac).parent(jacob, joseph).parent(jacob, dan).parent(jacob, benjamin).

male(abraham).male(isaac).male(joseph).

female(sarah).

Simple Example

?- parent(F, X), parent(F, Y),X /= Y.

F=jacob, X=joseph, Y=dan

F=jacob, X=benjamin, Y=joseph

F=jacob, X=dan, Y=joseph

F=jacob, X=joseph, Y=benjamin

F=jacob, X=dan, Y=benjamin

F=jacob, X=benjamin, Y=dan

Relational LPRecursive rules Syntax

The ancestor relationship - a recursive rule that computes the transitive closure of the parentrelationship.

% Signature: ancestor(A, D)/2% Purpose: A is an ancestor of D.ancestor(A,D) :- parent(A,D).ancestor(A,D) :- parent(A,P), ancestor(P,D).

Variables occurring in the rule body and not in the headare existentially quantified.

A, D.( P parent(A, P) & ancestor(P, D))

ancestor (A, D)

.

parent(abraham, isaac). parent(isaac, jacob).parent(sarah, isaac).parent(jacob, joseph).

% Signature: ancestor(A, D)/2%Purpose: A is an ancestor of D.ancestor(A,D) :- parent(A,D).ancestor(A,D) :- parent(A,P),

ancestor(P,D).

Recursive Rules Example?- ancestor(A, joseph).

A = jacob;A = abraham;A = isaac;A = sarah;False.

?- ancestor(sarah, D).

A = isaac;A = jacob;A = joseph;False.

parent(abraham, isaac). parent(isaac, jacob).parent(sarah, isaac).parent(jacob, joseph).

% Signature: ancestor(A, D)/2%Purpose: A is an ancestor of D.ancestor(A,D) :- parent(A,D).ancestor(A,D) :- ancestor(P,D),

parent(A,P).

Recursive Rules Example?- ancestor(A, joseph).

A = jacob;A = abraham;A = isaac;A = sarah;Out of local stack.

?- ancestor(sarah, D).

A = isaac;A = jacob;A = joseph;Out of local stack.

?- ancestor(sarah, isaac).

true;Out of local stack.

ancestor(A,D) :- parent(A,D).ancestor(A,D) :- parent(A,P),

ancestor(P,D).

Recursive Rules Example

ancestor(A,D) :- parent(A,D).ancestor(A,D) :- ancestor(P,D),

parent(A,P).

A D

P

First

A D

P

Then

Ancestor(P,D)Ancestor(P2,D)Ancestor(P3,D)Ancestor(P4,D)…

FirstThen

Relational LP – Concrete Syntax

<program> ==> <procedure>+

<procedure> ==> (<rule> | <fact>)+ with identical predicate and arity

<rule> ==> <head> : - <body> .

<fact> ==> <head> .

<head> ==> <atomic-formula>

<body> ==> (<atomic-formula> ,)* <atomic-formula>

<atomic-formula> ==> <constant> | <predicate> ( (<term>,)* <term> )

<predicate> ==> <constant>

<term> ==> <constant> | <variable>

<constant> ==> A string starting with a lower case letter.

<variable> ==> A string starting with an upper case letter.

<query> ==> ?- (<atomic-formula>,)* <atomic-formula> .

Relational LP – Abstract Syntax

<program>:

Components:

<procedure>

<procedure>:

Components:

Rule: <rule>

Fact: <atomic-formula>

Overall amount of rules and facts: >=1.

Ordered.

<rule>:

Components:

Head: <atomic-formula>

Body: <atomic-formula>

Amount: >=1.

Ordered.

<atomic-formula>:Kinds: <predication>, <constant>

<predication>:Components:

Predicate: <constant>Term: <term>

Amount: >=1. Ordered.

<query>:Components:

Goal: <atomic-formula>Amount: >=1. Ordered.

<term>:Kinds: <constant>,<variable>

Relational LP – Interim Summary

Semantics:

o Quantification of variables (existential/universal)

o Answers are partial substitutions to query variables (or true/false indications)

Concepts:

o Predicate symbol

o Term

o Facts

o Rules

o Query

Operational Semantics for LP

Input: a program P and a query Q

Output: partial substitutions to query variables (or true/false indications)

Method: by trying to build a proof tree for the given query

Proof Tree - Exampleparent(abraham, isaac). %p1parent(isaac, jacob). %p2parent(sarah, isaac). %p3parent(jacob, joseph). %p4parent(jacob, dan). %p5parent(jacob, dinah). %p6

male(abraham). %m1male(isaac). %m2male(jacob). %m3male(joseph). %m4male(dan). %m5

son(X,Y) :- parent(Y,X), %s1male(X).

son(S, jacob)

parent(jacob, X1),male(X1)

%p4

{S=X1, Y1=jacob}

male(joseph) male(dan) male(dinah)

{X1=dan} {X1=dinah}

%s1

%p5 %p6

truetrue false

{X1=joseph}

S=joseph

{}%m4 {}%m5

S=dan?- son(S,jacob).

Proof Tree - DefinitionTree T is a proof tree of query Q in program P if:

o Every node is a query (list of goals)

o The root node is the query Q

o Every edge is labeled by a substitution and a rule fact name

o If there is a transition on edge e = (Subs, Rname) from node N = (G1,…,Gn)

to node N’ then:

There exists rule Rname = (H :- B1, …, Bk) in P such that H can be unified with a goal Gi via substitution Subs

and Node N’ is obtained from node N by substituting the goal Gi with B1, …, Bkand applying the substitution Subs on the result

That is, N’ = (G1,…Gi-1, B1, …, Bk , Gi+1,…Gn) Subs

A proof tree is said to be a complete proof tree of query Q in program P if no more transitions can be added.

Proof Tree –Obtaining Answers

o A leaf with goal true, is a success node

o A leaf with goal false is a failure node

o If a leaf is marked true, in order to extract the result

we can collect all substitutions along the path to the leaf,

compose them,

and return only substitutions to the required vars

son(S, jacob)

parent(jacob, X1),male(X1)

%p4

{S=X1, Y1=jacob}

male(joseph) male(dan) male(dinah)

{X1=dan} {X1=dinah}

%s1

%p5 %p6

truetrue false

{X1=joseph}

{S=X1, Y1=jacob}{X1=joseph}{} = {S=joseph,Y1=jacob,X1=jacob}

{}%m4 {}%m5

{S=X1, Y1=jacob}{X1=dan}{} = {S=dan,Y1=jacob,X1=don}

S=joseph S=dan

Proof Tree –Terminology

A proof tree is a finite tree if it has no infinite paths.

Otherwise it is an infinite tree.

A proof tree is a success tree if it has a success leaf.

Otherwise it is a failure tree.

Thus,

o given a finite tree

we can always determine whether it is a success tree or failure tree.

o given an infinite tree,

if it has a success leaf, we can determine it is a success tree,

otherwise we don’t know…

Operational Semantics for LPMethod: Build a node N for the given query Q

From current node N:

If it is a success leaf (N = true),

o extract answer and return it

o Backtrack to obtain more answers

If it is a failure leaf (N =false)

o Backtrack

If it is not a leaf node and N = (G1,…,Gn)

o select a goal Gi and

o select a rule H :- B1, …, Bk

If H can be unified with Gi via substitution Substhen create a new node N’ = (G1,…Gi-1, B1, …, Bk , Gi+1,…Gn) Subs

Otherwise, create a new node N’ = false.

o Make N’ the current node.

Input: a program P and a query Q

Output: partial subs to query variables (or true/false)

Algorithm uses procedures:• Gsel - for selecting a goal• Grule - for selecting a rule• Unify – for trying to unify

rule head and goal.

Unification

The unification operation:

Given two atomic formulas A1, A2

Return a substitution s such that A1s = A2s.

Examples:

p(3, X), p(Y, 4) ==> {X=4, Y=3}

p(X, 3, X), p(Y, Z, 4) ==> {X=4, Z=3, Y=4}

Substitution

A substitution is a finite mapping, s, from variables to terms, such that s(X)≠X.

Examples: {X=4, Y=4, Z=3}

{X=4, Z=3, U=X}

{X=4, Z=3, U=V}

{X=4, Z=3, Y=Y}

{X=4, Z=3, X=Y}

Application of substiutionatomic formula ◦ substitution ==> atomic formula‘

The application of substitution s to an atomic formula A, denoted A◦s, simultaneously replaces the terms for their variables.

Examples:

p(X, 3, X, W) ◦ {X = 4, Y = 4} =

p(4, 3, 4, W)

p(X, 3, X, W) ◦ {X = 4, W = 5} =

p(4, 3, 4, 5)

p(X, 3, X, W) ◦ {X = W, W = X} =

p(W, 3, W, X)

Instantiation and Generalization

An atomic formula A’ is an instance of an atomic formula A if there exists a substitution s such that A◦s = A’

A is more general than A’ if A’ is an instance of A

Examples:

p(X,3,X,W) is more general than p(4,3,4,W), which is more general than p(4,3,4,5).

p(X,3,X,W) is more general than p(W,3,W,W), which is more general than p(5,3,5,5).

p(X,3,X,W) is more general than p(W,3,W,X), which is more general than p(X,3,X,W).

Unifier

A unifier of atomic formulas A and B is a substitution s, such that A◦s = B◦s.

For example, the following substitutions are unifiers of p(X,3,X,W) and p(Y,Z,4,W):

{X=4, Z=3, Y=4}

{X=4, Z=3, Y=4, W=5}

{X=4, Z=3, Y=4, W=0}

The mgu of atomic formulas A and B is a unifier s of A and B such that A◦s = B◦s is more general than all other instances of A and B obtained by applying a unifier

Composition of Substitutions

s ◦ s'

1. A variable X for which s(X) is defined, is removedfrom the domain of s‘

2. s' is applied to the terms of s

3. The modified s' is added to s.

4. Identity bindings are removed.

Example:

{X=Y, Z=3, U=V} ◦

{Y=4, W=5, V=U, Z=X} =

{X=4, Z=3, Y=4, W=5, V=U}