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### An introduction to Logic Programming

Chapter 6

Chapter topics

- Introduction
- Relational Logic Programming: specify relations among entities
- Logic Programming: data structures: lists, binary trees, symbolic expressions, natural numbers (church)
- Prolog: arithmetic, cuts, negation
- Meta-circular Interpreters for Logic Programming

Logic Programming: Introduction

- Origin: automate the process of proving logical statements
- Mind switch:
- Formula ⇔ procedure declaration
- Query ⇔procedure call
- Proving ⇔ computation

Logic Programming: Introduction

Every programming language has:

- Syntax: set of formulas (facts and rules)
- Semantics (set of answers)
- Operational Semantics (how to get an answer):
- Unification
- Backtracking

Logic Programming: Introduction

- The building blocks of LP expressions are formulas: formulas defined relations. For example, formula p(X,Y) - a relation between X and Y called p.
- axioms - are the knowledge base. axioms have the form:

X1,X2,...,XkH B1 and B2 ... and Bn

- A query - claim that the computation is attempting to proof.

A query has the form X1,X2,...,Xk Q1 and Q2 ... and Qm

A typical logic programming answer "X1=..., ... Xk=... also X1 =..., ... Xk=... "

- Operational semantics: set of axioms procedure definitionquery procedure application

interpretation prove query via program

Logic programming model

Kowalski's interpretation: An axiom H B1 and B2 ... and Bnrepresents a procedure

- H is the procedure’s head and the Bi’s are its body
- To solve (execute) H, we recursively solve B1 ... Bn

Relational LP

- a computational model based on Kowalski's interpretation
- The Prolog language ('70) - contains RLP + additional programming features

Relational LP - Syntax

- atomic formula:

predicate_symbol(term1, ..., termn)

- predicate symbols start with lowercase
- terms:
- symbols (representing individual constants)
- variables (start with uppercase or _ for anonymous)

Relational LP - Semantics

- Values:
- all atomic values are symbols (typeless)
- Very few primitive predicates, such as = (unification), \=, true, false
- Computation output:
- an answer to a query.
- An answer to a query is a (possibly partial) substitution (assignment) of query variables.

Relational LP Syntax - formulas

- atomic formula:

Syntax: predicate_symbol(term1,...,termn)

Examples:

male(moshe)

color(red)

parent(reuven, moshe)

parent(moshe, rina)

parent(Parent, Child)

ancestor(A,D)

address(_City, hertzel, 20)

- The only difference between predicates and individual constant symbols are their context/location.

Relational LP Syntax - Procedures

- A fact is an assertion of an atomic formula.

Syntax: H. where H is an atomic formula.

Examples:

parent(rina, moshe).

color(red).

ancestor(A,A).

- Variables in facts are universally quantified. "for all A, it holds that ancestor(A,A)".
- Procedures are an ordered collection of axioms (facts and rules)sharing the same predicate name and arity.

% Signature: parent(Parent, Child)/2

% Purpose: Parent is a parent of Child

parent(rina, moshe).

parent(rina, rachel).

parent(rachel, yossi).

parent(reuven, moshe).

% Signature: female(Person)/1

% Purpose: Person is a female.

female(rina).

female(rachel).

- Predicates have arity(no. of parameters). specified in /n in the comment above the procedure.
- not necessarily unique

Relational LP Syntax - Queries

A query has the syntax:

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

where the Qi are atomic formulas.

Meaning:

Assuming the program axioms, do Q1 and ... and Qn hold?

',' means conjunction.

For example,

?- parent(rina, moshe).

"Is rina a parent of moshe?”

A computation is a proof of a query, returns:

true ;

false. user requests another answer

Relational LP Syntax - Queries

A query has the syntax:

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

where the Qi are atomic formulas.

Meaning:

Assuming the program axioms, do Q1 and ... and Qn hold as well?

',' means conjunction.

For example,

?- parent(rina,X).

"Does there exist an X which is a child of rina?"

X = moshe ;

X = rachel.

- Variables in queries are existentially quantified.

Relational LP Syntax - Queries

"Is there an X which is a child of rina, and is also a parent of some Y?"

?- parent(rina,X),parent(X,Y).

X = rachel,

Y = yossi.

"Find two parents of moshe?":

?- parent(X,moshe),parent(Y,moshe).

X = rina,

Y = rina ;

X = rina,

Y = reuven ;

X = reuven,

Y = rina ;

X = reuven,

Y = reuven.

Relational LP Syntax - Queries

"Find two different parents of moshe?":

?- parent(X,moshe),parent(Y,moshe),X \= Y.

X = rina,

Y = reuven ;

X = reuven,

Y = rina ;

false.

?- parent(X,moshe), X \= Y, parent(Y,moshe).

false.

?- X=3.

X = 3.

?- X\=3.

false.

?- 4\=3.

true.

Relational LP - Syntax

% Signature: loves(Someone, Somebody)/2

% Purpose: Someone loves Somebody

loves(rina,Y). % rina loves everybody.

loves(moshe, rachel).

loves(moshe, rina).

loves(Y,Y). % everybody loves himself

- Variables in axioms are universally quantified.

"for all Y loves(rina,Y)"

can be renamed

"for all X loves(rina,X)"

- Using a variable in a fact is defining it. The scope is the fact itself.

Relational LP - Syntax

% Signature: loves(Someone, Somebody)/2

% Purpose: Someone loves Somebody

loves(rina,Y). % rina loves everybody.

loves(moshe, rachel).

loves(moshe, rina).

loves(Y,Y). % everybody loves himself

Queries:

?- loves(rina,moshe).

true ;

false.

Relational LP - Syntax

% Signature: loves(Someone, Somebody)/2

% Purpose: Someone loves Somebody

loves(rina,Y). % rina loves everybody.

loves(moshe, rachel).

loves(moshe, rina).

loves(Y,Y). % everybody loves himself

Queries:

?- loves(rina,X).

true ;

X = rina.

Relational LP - Syntax

% Signature: loves(Someone, Somebody)/2

% Purpose: Someone loves Somebody

loves(rina,Y). % rina loves everybody.

loves(moshe, rachel).

loves(moshe, rina).

loves(Y,Y). % everybody loves himself

Queries:

?- loves(X,rina).

X = rina ;

X = moshe ;

X = rina.

Relational LP - Syntax

% Signature: loves(Someone, Somebody)/2

% Purpose: Someone loves Somebody

loves(rina,Y). % rina loves everybody.

loves(moshe, rachel).

loves(moshe, rina).

loves(Y,Y). % everybody loves himself

Queries:

?- loves(X,X).

X = rina ;

true.

this query has two answers.

Relational LP Syntax - Rules

- Syntax:H :−B1, . . . , Bn.

is an assertion of an implication statement.

The conjunction of B1, .., Bn implies the head H.

Bi's and H are atomic formulas.

% Signature: mother(Mum, Child),

% Purpose: Mum is a mother of Child

mother(Mum, Child) :- parent(Mum, Child), female(Mum).

- Variables occurring in rule heads are universally quantified. The lexical scope of the variable is the rule.
- A variable can multiple times in the head.
- variables are bound within a rule.

Relational LP Syntax - Rules

% Signature: mother(Mum, Child),

% Purpose: Mum is a mother of Child

mother(Mum, Child) :- parent(Mum, Child), female(Mum).

?- mother(M,C).

M = rina,

C = moshe ;

M = rina,

C = rachel ;

M = rachel,

C = yossi ;

false.

Relational LP Syntax - Rules

% Signature: mother(Mum, Child),

% Purpose: Mum is a mother of Child

mother(Mum, Child) :- parent(Mum, Child), female(Mum).

“Find a two-different-kids mother”

?- mother(M,C1),mother(M,C2),C1\=C2.

M = rina,

C1 = moshe,

C2 = rachel ;

M = rina,

C1 = rachel,

C2 = moshe ;

false.

Relational LP Syntax - Rules

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

% Signature: ancestor(Ancestor, Descendant)/2

% Purpose: Ancestor is an ancestor of Descendant.

ancestor(Ancestor, Descendant) :-

parent(Ancestor, Descendant).

ancestor(Ancestor, Descendant) :-

parent(Ancestor, Person),

ancestor(Person, Descendant).

- Variables occurring in the rule body and not in the head are existentially quantified.

"for all Ancestor and for all Descendant, ancestor(Ancestor, Descendant) if there exists some Person such that parent(Ancestor, Person) and ancestor(Person, Descendant)."

Relational LP Syntax - Rules

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

% Signature: ancestor(Ancestor, Descendant)/2

% Purpose: Ancestor is an ancestor of Descendant.

ancestor(Ancestor, Descendant) :-

parent(Ancestor, Descendant).

ancestor(Ancestor, Descendant) :-

parent(Ancestor, Person),

ancestor(Person, Descendant).

?- ancestor(rina,D).

D = moshe ;

D = rachel ;

D = yossi ;

false.

Relational LP Syntax - Rules

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

% Signature: ancestor(Ancestor, Descendant)/2

% Purpose: Ancestor is an ancestor of Descendant.

ancestor(Ancestor, Descendant) :-

parent(Ancestor, Descendant).

ancestor(Ancestor, Descendant) :-

parent(Ancestor, Person),

ancestor(Person, Descendant).

?- ancestor(A,yossi).

A = rachel ;

A = rina ;

false.

- The reported result/functionality depends on the variables and their location in the query.

Relational LP Syntax - Rules

ancestor1(Ancestor, Descendant) :-

parent(Ancestor, Descendant).

ancestor1(Ancestor, Descendant) :- ancestor1(Person, Descendant),

parent(Ancestor, Person).

?- ancestor1(A,yossi).

A = rachel ;

A = rina ;

ERROR: Out of local stack

?- ancestor1(rina,yossi).

true ;

ERROR: Out of local stack

- This procedure is not tail recursive.
- Since this query cannot be answered using the base case, new similar queries are infinitely created.

Note

Facts can be considered as rules with an empty body.

For example,

parent(rina, moshe).

parent(rina, moshe):- true.

have equivalent meaning.

true - is the zero-arity predicate.

Concrete syntax of Relational Logic Programming

<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> ’.’

Summary - RLP Semantic and syntax

parent(rina, moshe).

parent(rina, rachel).

parent(rachel, yossi).

parent(reuven, moshe).

ancestor(Ancestor, Descendant) :- parent(Ancestor, Descendant).

ancestor(Ancestor, Descendant) :- parent(Ancestor, Person),

ancestor(Person, Descendant).

?- ancestor(A,yossi).

A = rachel ;

A = rina ;

false.

Concepts:

- predicate symbol, individual constant symbol, axioms (facts/rules), query.

Semantics:

- individual constant symbols - entities, axioms - relations of entities.
- Quantification of variables (universal/existential)
- Answers are partial substitutions to query variables (or true/false indications).

Operational Semantics for LP

Input: a program P and a query Q

Interpreter of LP:

- Unify - pattern matching between an atomic formula from Q and a head of some rule/fact from P.
- Answer-query (proof-tree)- Create a proof tree. Back track from a leaf if it is a "dead end" fail leaf, or if it is a success leaf and there may be additional answers to the query.

Unification

The unification operation:

two atomic formulas ==> substitution

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 finite mapping, s, from variables to terms, such that s(X)≠X.

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

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

Not substitutions:

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

Application of Substitution

atomic formula ◦ substitution ==> atomic formula'

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)

- A unifier of atomic formulas A and B is a substitution s, such that A◦s = B◦s.
- The unification operation returns a unifier.
- Goal of Unify(A,B): find the most general unifier.

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

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

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

- Less general unifiers {X = 4, Z = 3, Y = 4, W = 5}, {X = 4, Z = 3, Y = 4, W = 0}

Instantiation and Generalization

- An atomic formula A’ is an instance of an atomic formula A if there is a substitution s such that A◦s = A’
- A is more general than A’ if A’ is an instance of A

Most General Unifier (MGU)

- 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

Combination of substitutions

s ◦ s'

- s' is applied to the terms of s
- A variable X for which s(X) is defined, is removed from the domain of s'
- The modified s' is added to s.
- Identity bindings are removed.

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

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

Disagreement Set

- The disagreement set of atomic formulas is the set of left most symbols on which the formulas disagree.

disagreement-set(p(X, 3, X, W ), p(Y, Z, 4, W )) = {X, Y }.

disagreement-set(p(5, 3, X, W ), p(5, 3, 4, W )) = {X, 4}.

Unify - A unification algorithm

Signature: unify(A, B)

Type: atomic-formula*atomic-formula -> a substitution or FAIL

Post-condition: result = mgu(A, B) if A and B are uniﬁable or FAIL, otherwise

Unify - examples

1. unify[ p(X, 3, X, W), p(Y, Y, Z, Z) ] ==>

help[ {} ] ==>

D = {X, Y}

help[ {X = Y } ] ==>

D = {Y, 3}

help[ {X = 3, Y = 3 } ] ==>

D = {Z, 3} ]

help[ {X = 3, Y = 3, Z = 3 } ] ==>

D = {W, 3}

help[ {X = 3, Y = 3, Z = 3, W = 3 } ] ==>

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

2. unify[ p(X, 3, X, 5), p(Y, Y, Z, Z) ] ==>

FAIL

Properties of unify(A, B) algorithm:

- The algorithm always terminates.
- Unification is a generalization of Pattern matching
- If B does not include variables and A does not include repeated variable occurrences, the time complexity can be linear.

Operational Semantics for LP

Interpreter of LP:

- Unify - pattern matching between an atomic formula from Q and a head of some rule/fact from P.
- Answer-query (proof-tree)- Create a proof tree. Back track from a leaf if it is a "dead end" fail leaf, or if it is a success leaf and there may be additional answers to the query.

answer-query: an interpretation algorithm for LP

Input:

A query: Q = ?- Q1, ..., Qn. Each component is called goal

A program P , with numbered rules (denoted by number(R))

A goal selection policy Gsel

A rule selection policy Rsel

Output:

A set of (possibly partial) substitutions for variables of Q.

General Idea:

Repeated effort to select a goal using roles.

answer-query Algorithm

- PT := proof-tree(make_node(Q))
- Return {s | s ∈ labels(Success(P T ))/Q }, where Success(PT) is the set of Success nodes of P T , and labels/Q is the restriction of the substitutions in labels to the variables of Q.

proof-tree(current_node)

If label(current_node) is ?- true, ..., true.

1. Mark current_node as a Success node

2. If the path from the tree root to current_node is labeled with the substitutions s1, . . . , sn, label current_node with the substitution s1 ◦ s2 ◦ . . . ◦ sn

Else 1. Select a goal G true in label(current_node) according to Gsel.

2. Rename variables in every rule and fact of P .

3. While has-next?(iterator(current_node)):

(a) Advance iterator: next(iterator(current_node))

(b) Rule selection: Starting from iterator(current_node), and according to Rsel, select a rule R = [A :- B1, ..., Bm.] such that Unify(A, G) = s' succeeds with the unifying substitution s'.

(c) Rule application: if a rule R is selected then

i. Construct a new query node by removing G, adding the body of R, and applying s' to the resulting query:

new_node = make_node([label(current_node)−G+B1,...,Bm]◦s')

ii. Add a child node and start a new proof:

add_child(current_node, < s', number(R) >, (new_node))

proof-tree(new_node)

Comments about answer-query

- Variable renaming according to depth in tree. Xi at depth i.
- Unify(A,G), where G is the selected goal and A the head of the selected rule.
- Let XG is a variable from G and XA a variable of A
- Selecting XA=XG or XG=XA does not change query results.
- Selecting XA=XG leaves the query variables in the tree.
- The goal and rule selection decisions can affect the performance of the interpreter.

Example 6.5

% Signature: father(F,C)/2

parent(abraham,isaac). %1

parent(isaac, jacob). %2

parent(haran,lot). %3

parent(haran,yiscah). %4

parent(haran,milcah). %5

% Signature: male(P)/1

male(isaac). %1

male(lot). %2

% Signature: son(C, P)/2

son(X, Y) - parent(Y, X), male(X). %1

% Signature: ancestor(Ancestor, Descendant)/2

anc(Anc, Des) :- parent(Anc, Des). %1

anc(Anc, Des) :- parent(Anc, Person), %2

anc(Person, Des).

What happens if rules 1 and 2 of 'anc' are switched?

Significant kinds of proof trees:

- Finite success proof tree: A ﬁnite tree with a successful path.
- Finite failure proof tree: A ﬁnite tree with no successful path.
- Inﬁnite success proof tree: An inﬁnite tree with a successful path. In this case it is important not to explore an inﬁnite path. For Prolog: Tail recursion is safe.
- Inﬁnite failure proof tree: An inﬁnite tree with no successful path.

6.1.6 Relational logic programming and Structured Query Language (SQL) operations

- RLP naturally represents structured databases (tables with static columns).
- A procedure consisting of facts can represent a table in the database.
- Often databases are access via elementary SQL operations such as: select, project, union, Cartesian product and join.
- Select (rows from table r, fitting some criteria):

r1(X1, X2, X3) :- r(X1, X2, X3),

X2 \= X3.

- Project (some columns from table r):

r1(X1, X3) :- r(X1, X2, X3).

6.1.6 Relational logic programming and Structured Query Language (SQL) operations

- Union (unite tables r and s, with identical columns):

r_union_s(X1, ..., Xn) :- r(X1, ..., Xn).

r_union_s(X1, ..., Xn) :- s(X1, ..., Xn).

- Cartesian product (all combinations of rows from r and s):

r_X_s(X1, ..., Xn, Y1, ..., Ym) :-

r(X1, ..., Xn ),

s(Y1, ..., Ym).

- Natural Join (join tables r and s, with mutual column X):

r_join_s(X1, ..., Xn, X, Y1, ..., Ym) :-

r(X1, ..., Xn, X ),

s(X, Y1, ..., Ym).

6.1.5.3 Halting problem, RLP decidability

LRLP = {(P,Q) | P|- Q in RLP syntax}

Claim: Given a program P and a query Q, the problem "Is Q provable from P ", denoted P|- Q, is decidable.

Proof:

- The number of terms(constants/variables) and predicates appearing in P and Q is finite.
- Thus, the number of possible atomic formula (i.e. goals appearing in a node of the proof tree) is finite (except for renaming).
- Let N(P,Q) be that number.
- Then, any path that is longer than N(P,Q) is infinite. QED
- Most programming languages are only partially decidable (recursively enumerable/TM recognizable)

6.2 Logic Programming

typeless atomic terms, type safe

decidable

multi-directional

Prolog

LP

Relational LP

functors,

typeless composite terms, type safe

partially decidable,

multi-directional

arithmetics,

uni-directional dynamically typed,

not type safe,

system predicates (e.g. !)

6.2.1 Logic Programming

A functor symbol is added to the syntax, to represent data structures.

Terms (definition):

- constant individual symbols
- variables
- f(t1, . . . , tn) for terms t1, . . . , tn and a functor f.

Implications

- additional expressiveness (composite data structures)
- the LP language is partially decidable (recursively enumerable/TM recognizable) (in the same rank as other programming languages).

6.2.1 Atomic formula in FLP - examples

parent(rina, Child)

p(f(f(f(g(a,g(b,c))))))

member(cube(red(X)), Lst)

terms (a constant and a variable)

predicate

predicate

a term (functorsf,g combining constants a,b,c)

predicate

terms (functors cube and red applied to variable X)

6.2.1.1 Formalizing the syntax extension

<term> -> <constant> | <variable> | <composite-term>

<composite-term> -> <functor> ’(’ (<term>’,’)* <term>’)’

<functor> -> <constant>

6.2.2.1 Uniﬁcation for terms that include functors

- A substitution s is a ﬁnite mapping from variables to terms, such that s(X) does not include X.
- Unify remains the same, except for two points:
- Disagreement set can happen within a term

- unify(member(X,tree(X,Left,Right)) ,

member(Y,tree(9,void,tree(3,void,void))))

==> {Y=9, X=9, Left=void, Right=tree(3,void,void)}

− unify(t(X, f(a),X),

t(g(U),U,W))

==> {X=g(f(a)), U=f(a), W=g(f(a))}

6.2.2.1 Uniﬁcation for terms that include functors

- A substitution s is a ﬁnite mapping from variables to terms, such that s(X) does not include X.
- Unify remains the same, except for two points:

1. Disagreement set can happen within a term

2. Validation of occur check error (i.e. s(X) includes X).

− unify(t(X,f(X),X),

t(g(U),U,W))

==> fails due to occur check error! Expansion is infinite

- Unify algorithm for LP is modified so that it fails if occur check error is found in the {X=t} substitution at the disagreement-set.

6.2.3.2 Natural number arithmetic

Natural numbers can be represented by Church numerals: The constant zero denotes the number 0, s(0) - denotes 1, s(...s(s(0))...), n times - denotes natural number n, where s is a functor.

% Signature: natural_number(N)/1

% Purpose: N is a natural number.

natural_number(zero). %1

natural_number(s(X)) :- natural_number(X). %2

% Signature: le(X,Y)/2

% Purpose: X is less or equal Y.

le(zero, X) :- natural_number(X). %1

le(s(X), s(Z)) :- le(X, Z). %2

- no data structure definition - definition via use.

6.2.3.2 Natural number arithmetic

?- le(s(s(zero)), s(s(s(s(zero))))).

true.

?- le(s(s(X)), s(s(s(s(zero))))).

X = zero ;

X = s(zero) ;

X = s(s(zero)) ;

false.

?- le(s(s(X)), s(s(s(s(Y))))).

X = Y, Y = zero ;

X = zero,

Y = s(zero) ;

X = zero,

Y = s(s(zero)) ;

X = zero,

Y = s(s(s(zero))) ...

multi-directional definition - Functionality depends on the locations of variables in the query.

substitution for first leaf to the left

{X1=zero, N=s(Z1)}{X2=Z1} {Z1=zero}= {X1=zero, N=s(zero), X2=zero, Z1=zero}

substitution for second leaf:

{X1=zero, N=s(Z1)}{X2=Z1} {Z1=s(X3)} {X3=zero}=

{X1=zero, N=s(s(zero)), X2=s(zero), Z1=s(zero), X3=zero}

Summary

- Proof tree types
- LP with functors
- Unify + occur check error

6.2.3.3 Lists in LP

- Lists are a primitive composite data structure.
- Unlike non-primitive data-structure prefix notation f(t1, . . . , tn)list functor appears in infix notation.
- Syntax

[ ] - a 0-arity functor representing the empty list.

[Head|Tail] - a 2-arity functor representing a list that is constructed from its head and its tail, where the tail is also a list.

Examples

- ?- Y = [1,2,3].
- Y = [1, 2, 3].

?- Y=[1,2,3], X= [a,b| Y].

Y = [1, 2, 3],

X = [a, b, 1, 2, 3].

?- X = [a, b, c|[d,e,f]].

X = [a,b,c,d,e,f].

?- X=[1|t]. /* not a list */

X = [1|t].

?- X=[].

X=[].

?- X=[a|[ ]].

X = [a].

?- X = [a].

X = [a].

?- [a|[ ]] = [a].

true.

?- X= [a | [ b | [] ]].

X = [a, b].

6.2.3.3 LP lists - List membership

% Signature: member(X, List)/2

% Purpose: X is a member of List.

member(X, [X|Xs]).

member(X, [Y|Ys]) :- member(X, Ys).

% checks membership

?- member(a, [b,c,a,d]).

% takes an element from a list

?- member(X, [b,c,a,d]).

% generates a list containing b

?- member(b, Z).

6.2.3.3 LP lists - List concatenation

% Signature: append(List1, List2, List3)/3

% Purpose: List3 is the concatenation of List1 and List2.

append([], Xs, Xs).

append([X|Xs], Ys, [X|Zs]) :- append(Xs, Ys, Zs).

/* addition of two lists */

?- append([a,b], [c], X).

/* finds a difference between lists */

?- append(Xs, [a,d], [b,c,a,d]).

/* divides a list into two lists */

?- append(Xs, Ys, [a,b,c,d]).

2

{Xs=[X1|Xs1], Ys1=[a,d],

X1=b

Zs1=[c,a,d]}

append(Xs1, [a,d], [c,a,d])

append([], Xs, Xs). %1

append([X|Xs], Ys, [X|Zs] ) :- append(Xs, Ys, Zs). %2

2

{Xs1=[X2|Xs2], Ys2=[a,d],

X2=c

Zs2=[a,d]}

append(Xs2, [a,d], [a,d])

2

{Xs2=[X3|Xs3], Ys3=[a,d],

X3=a

Zs3=[d]}

1

{Xs2=[], Xs3=[a,d]}

true

append(Xs3, [a,d], [d])

2

{Xs3=[X4|Xs4], Ys4=[a,d],

X4=d

Zs4=[]}

append(Xs4, [a,d], [])

fail

6.2.3.3 LP lists - List concatenation

% Signature: append(List1, List2, List3)/3

% Purpose: List3 is the concatenation of List1 and List2.

append([], Xs, Xs).

append([X|Xs], Ys, [X|Zs]) :- append(Xs, Ys, Zs).

- List preﬁx and suﬃx:

prefix(Xs, Ys) :- append(Xs, Zs, Ys).

suffix(Xs, Ys) :- append(Zs, Xs, Ys).

- Redeﬁne member:

member(X, Ys) :- append(Zs, [X|Xs], Ys).

- Adjacent list elements:

adjacent(X, Y, Zs) :- append(Ws, [X,Y|Ys], Zs).

- Last element of a list:

last(X, Ys) :- append(Xs, [X], Ys).

6.2.3.3 LP lists - List Reverse

% Signature: reverse(List1, List2)/2

% Purpose: List2 is the reverse of List1.

reverse([], []).

reverse([H|T], R) :- reverse(T, S),

append(S, [H], R).

?- reverse([a,b,c,d],R).

R=[d,c,b,a]

?- reverse(R,[a,b,c,d]).

- Rule body ordering impacts the performance in various directions.

2

{Rlist=[H1|T1],

R1=[a,b,c]}

- reverse(T1,S1), append(S1, [H1], [a,b,c])

2

{T1=[H2|T2], R2=S1}

1

{T1=[], S1=[]}

- reverse(T2, S2)append(S2, [H2], S1)append(S1, [H1], [a,b,c])

- append([], [H1], [a,b,c])

reverse([], []). %1

reverse([H|T], R) :- reverse(T, S), append(S, [H], R). %2

append([], Xs, Xs). %1

append([X|Xs], Y, [X|Zs] ) :- append(Xs, Y, Zs). %2

2

{T2=[H3|T3], R3=S2}

1

{T2=[], S2=[]}

fail

- reverse(T3, S3)append(S3, [H3], S2) append(S2, [H2], S1)append(S1, [H1], [a,b,c])

- append([], [H2], S1) append(S1, [H1], [a,b,c])

1

{T3=[],S3=[]}

1

{Xs1=[H2], S1=[H2]}

...

- append([], [H3], S2) append(S2, [H2], S1)append(S1, [H1], [a,b,c])

- append([H2], [H1], [a,b,c])

1

{Xs1=[H3], S2=[H3]}

fail

- append([H3], [H2], S1)append(S1, [H1], [a,b,c])

...

S1=[H3,H2]

- append([H3,H2], [H1], [a,b,c])

H3=a, H2=b, H1=c

Rlist=[c,b,a]

...

true

6.2.3.3 LP lists - List Reverse

- An iterative version: uses the unification mechanism to accumulate the result in the second parameter which is returned in the base case.
- The help procedure is global. In Prolog all procedures are global.

% Signature: reverse(List1, List2)/2

reverse(Xs, Ys):- reverse_help(Xs,[],Ys).

% Signature: reverse_help(List1, AccReversed, RevList)/2

reverse_help([ ],Ys,Ys ).

reverse_help([X|Xs], Acc, Ys ) :-

reverse_help(Xs,[X|Acc],Ys).

?- reverse([a,b,c],R).

R=[c,b,a]

?- reverse(R,[a,b,c]).

ERROR: Out of local stack

6.3 Prolog

typeless atomic terms, type safe

decidable

multi-directional definitions

Prolog

LP

RLP

functors,

typeless composite terms, type safe

partially decidable,

multi-directional definitions

arithmetics,

uni-directional procedures dynamically typed,

not type safe,

system predicates (e.g. !)

6.3.2 The cut operator - pruning trees

The cut system predicate, denoted !, is a Prolog built-in predicate, for pruning proof trees.

- avoiding traversing failed sub-trees.
- eliminates wrong answers or infinite branches

6.3.2 The cut operator

For a node v, in which rule H :- B1, ...Bi, !, Bi+1, ..., Bnis applied, and having a branch to a node u, in which the current goal is !, all alternative branches splitting from nodes in the path between v (including) and node u are trimmed.

v

u

Example: trimming unwanted answers

p(X) :- a(X).

p(X) :- b(X),c(X),d(X),e(X).

p(X) :- f(X).

a(1).

b(1).

b(2).

c(1).

c(2).

d(2).

e(2).

f(3).

?- p(X).

X = 1 ;

X = 2 ;

X = 3 ;

fail

Example: trimming unwanted answers

p(X) :- a(X).

p(X) :- b(X),c(X),!,d(X),e(X).

p(X) :- f(X).

a(1).

b(1).

b(2).

c(1).

c(2).

d(2).

e(2).

f(3).

?- p(X).

X = 1 ;

fail

Some answers were eliminated

Example: trimming unwanted answers

Problem domain: colored pieces, each piece has one color.

color(P, red) :- red(P).

color(P, black) :- black(P).

color(P, unknown).

red(a).

black(b).

The queries to return a single solution

?- color(a, Color).

Example: trimming unwanted answers

Eliminates wrong answers

color(P, red) :- red(P),!.

color(P, black) :- black(P),!.

color(P, unknown).

red(a).

black(b).

Example: avoiding unnecessary searches (duplicate answers)

member(X,[X|Ys]).

member(X,[Y|Zs]) :- member(X, Zs).

Adding cut:

member(X,[X|Ys]) :- !.

member(X,[Y|Zs]) :- member(X, Zs).

?- member(5, [5, 9, 24, 17, 5, 2])).

% After one application the proof tree is complete.

?- member(9, [5, 9, 24, 17, 5, 2])).

% After 2 applications the proof tree is complete.

?- member(X, [5, 9, 24, 17, 5, 2]).

% Only one answer will be returned.

Example: cut implements negation

% Signature: not_member(Element, List)/2

% Purpose: the relation in which Element is not a member of List

not_member(_, []).

not_member(X, [Y|Ys]) :- X \= Y, not_member(X, Ys).

Alternatively,

not_member(X,Xs):- member(X,Xs),!,false.

not_member(X,Xs).

6.3.3 Negation in Logic Programming

- not(X) is primitive operator for negation in Prolog.
- Negation by failure: not(X) does not mean that X is false, it means that X can't be proven.
- For example, with the program:

man('Adam').

woman('Eve').

?- not(man('Abel')).

true

- We can implement not, as follows:

not(Goal) :- Goal, !, false.

not(Goal).

6.4 Meta-circular interpreters for LP

- Based on unification and backtracking.
- Two points of selection:

(a) Goal selection - leftmost for Prolog.

(b) Rule selection - top-to-bottom for Prolog, with backtracking to the following rules, in case of a failure.

Meta-Interpreter - version 1

% Signature: solve(Goal)/1

% Purpose: Goal is true if it is true when posed to the original program P.

solve( A ) :- A.

Meta-interpreter - version 2

% Signature: solve(Goal)/1

% Purpose: Goal is true if it is true when posed to the original program P.

solve(true) :- !.

solve( (A, B) ) :- !, solve(A), solve(B).

solve(A) :- clause(A, B), solve(B).

Comments

The Prolog system predicate clause, which for a query ?- clause(A,B).

selects the first program rule whose head unifies with A, and unifies B with

the rule body.

append([ ],Xs,Xs).

append([X|Xs],Y,[X|Zs]) :- append(Xs,Y,Zs).

reverse([], []).

reverse([H|T], R) :- reverse(T, S), append(S, [H], R).

?- clause(append(X,[1,2], Z), Body).

X = [], Z = [1,2], Body = true;

X = [X1|Xs1 ], Z = [X1|Zs1], Body = append(Xs1, [1,2], Zs1).

?- clause(reverse(List,RList), Body).

List = [], RList = [], Body = true ;

List = [_G513|_G514],

Body = (reverse(_G514, _G517),append(_G517, [_G513], RList)).

Meta-interpreter - version 3

Pre-processing – Program transformation.

Every rule A :- B1, B2, ..., Bnin program P,

is written in program P' as a fact rule(A,[B1, B2, ..., Bn]).

Program P:

member(X,[X|Xa]).

member(X,[Y|Ys]) :- member(X, Ys).

append([ ], Xs, Xs).

append([X|Xs],Ys,[X|Zs]) :- append(Xs,Ys,Zs).

reverse([], []).

reverse([H|T], R) :- reverse(T, S), append(S, [H], R).

Meta-interpreter - version 3

Pre-processing – Program transformation.

Every rule A :- B1, B2, ..., Bnin program P,

is written in program P' as a fact rule(A,[B1, B2, ..., Bn]).

Program P':

rule( member(X,[X|Xa]), [ ]).

rule( member(X,[Y|Ys]), [member(X,Ys)]).

rule( append([ ],Xs,Xs), [ ]).

rule( append([X|Xs],Ys,[X|Zs]), [append(Xs,Ys,Zs)]).

rule( reverse([], []), []).

rule( reverse([H|T], R), [reverse(T, S), append(S, [H], R)]).

Meta-interpreter - version 3

% Signature: solve(Goal)/1

% Purpose:

% Goal is true if it is true when posed to the program P.

solve(Goal) :- solve(Goal, []).

% Signature: solve(Goal, Rest_of_goals)/2

solve([ ], [ ] ).

solve([ ], [G| Goals] ) :- solve(G, Goals).

solve([A|B], Goals):- append(B,Goals,Goals1),

solve(A,Goals1).

solve(A, Goals) :- rule(A, B),

solve(B, Goals).

Logic Programming Summary

Pure(relational) LP:

- typeless
- atomic terms, atomic formula
- program axioms, queries
- lexical scoping, global definitions
- unification, build proof tree (backtracking)
- decidability

FLP:

- functors, composite terms
- lists

Prolog:

- cut!
- meta-circular interpreters (clause, tuples/list)

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