An introduction to Logic
Programming
Chapter 6
Chapter topics
1. Introduction
2. Relational Logic Programming: specify
relations among entities
3. Logic Programming: data structures: lists,
binary trees, symbolic expressions, natural
numbers (church)
4. Prolog: arithmetic, cuts, negation
5. Meta-circular Interpreters for Logic
Programming
2
Logic Programming: Introduction
• Origin: automate the process of proving
logical statements
• Mind switch:
– Formula ⇔ procedure declaration
– Query ⇔ procedure call
– Proving ⇔ computation
3
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
4
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,...,Xk
H  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 definition
query  procedure application
interpretation  prove query via program
5
Logic programming model
Kowalski's interpretation:
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
6
Relational LP
• a computational model based on Kowalski's
interpretation
• The Prolog language ('70) - contains RLP +
additional programming features
7
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)
8
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.
9
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.
10
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.
11
% 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
12
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
13
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.
14
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.
15
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.
16
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.
17
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.
18
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.
19
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.
20
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.
21
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.
22
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.
23
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.
24
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)."
25
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.
26
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.
27
Relational LP Syntax - Rules
ancestor1(Ancestor, Descendant) :parent(Ancestor, Descendant).
ancestor1(Ancestor, Descendant) :Descendant),
parent(Ancestor, Person).
ancestor1(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.
28
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.
29
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> ’.’
30
Summary - RLP Semantic and syntax
parent(rina, moshe).
parent(rina, rachel).
parent(rachel, yossi).
parent(reuven, moshe).
ancestor(Ancestor, Descendant) :ancestor(Ancestor, Descendant) :-
parent(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).
31
Operational Semantics for LP
Input: a program P and a query Q
Interpreter of LP:
1. Unify - pattern matching between an atomic
formula from Q and a head of some rule/fact
from P.
2. 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.
32
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 }
33
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}
34
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
35
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
36
Combination of substitutions
s ◦ s'
1. s' is applied to the terms of s
2. A variable X for which s(X) is defined, is removed from the
domain of s'
3. The modified s' is added to s.
4. 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}.
37
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}.
38
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
39
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
40
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.
41
Operational Semantics for LP
Interpreter of LP:
1. Unify - pattern matching between an atomic
formula from Q and a head of some rule/fact
from P.
2. 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.
42
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.
43
answer-query Algorithm
1. PT := proof-tree(make_node(Q))
2. 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.
44
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)
45
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.
46
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).
47
?- son(S, haran).
48
?- anc(abraham, D).
What happens if rules 1 and 2 of
'anc' are switched?
49
What happens if bodies 1 and 2 in rule 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.
51
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).
52
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).
53
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)
54
6.2 Logic Programming
typeless atomic terms, type
safe
decidable
multi-directional
Relational LP
functors,
typeless composite terms,
type safe
partially decidable,
multi-directional
LP
Prolog
arithmetics,
uni-directional
dynamically typed,
not type safe,
system predicates (e.g. !)
55
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).
56
6.2.1 Atomic formula in FLP - examples
parent(rina, Child)
predicate
terms (a constant and a variable)
p(f(f(f(g(a,g(b,c))))))
predicate
a term (functors f,g combining constants a,b,c)
member(cube(red(X)), Lst)
predicate
terms (functors cube and red applied to variable X)
57
6.2.1.1 Formalizing the syntax extension
<term> -> <constant> | <variable> | <composite-term>
<composite-term> -> <functor> ’(’ (<term>’,’)* <term>’)’
<functor>
-> <constant>
58
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
- 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))}
59
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.
60
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).
le(s(X), s(Z)) :- le(X, Z).
%1
%2
• no data structure definition - definition via use.
61
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 ;
multi-directional definition X = s(zero) ;
Functionality depends on the
X = s(s(zero)) ;
locations of variables in the query.
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))) ...
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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}
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Summary
• Proof tree types
• LP with functors
• Unify + occur check error
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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.
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Examples
?- X=[].
X=[].
?- Y = [1,2,3].
Y = [1, 2, 3].
?- X=[a|[ ]].
X = [a].
?- Y=[1,2,3], X= [a,b| Y].
Y = [1, 2, 3],
X = [a, b, 1, 2, 3].
?- X = [a].
X = [a].
?- [a|[ ]] = [a].
true.
?- X = [a, b, c|[d,e,f]].
X = [a,b,c,d,e,f].
?- X=[1|t]. /* not a list */
X = [1|t].
?- X= [a | [ b | [] ]].
X = [a, b].
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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).
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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]).
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append(Xs, [a,d], [b,c,a,d])
2
{Xs=[X1|Xs1],
Ys1=[a,d],
X1=b
Zs1=[c,a,d]}
append(Xs1, [a,d], [c,a,d])
2
{Xs1=[X2|Xs2],
Ys2=[a,d],
X2=c
Zs2=[a,d]}
append(Xs2, [a,d], [a,d])
1
{Xs2=[],
Xs3=[a,d]}
true
2
{Xs2=[X3|Xs3],
Ys3=[a,d],
X3=a
Zs3=[d]}
append(Xs3, [a,d], [d])
2
{Xs3=[X4|Xs4],
Ys4=[a,d],
X4=d
Zs4=[]}
append(Xs4, [a,d], [])
fail
append([], Xs, Xs).
append([X|Xs], Ys, [X|Zs] ) :- append(Xs, Ys, Zs).
%2
%1
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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).
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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.
71
72
reverse(RList,[a,b,c])
2
{Rlist=[H1|T1],
R1=[a,b,c]}
1
{T1=[], S1=[]}
append([], [H1], [a,b,c])
1
{T2=[],
S2=[]}
fail
reverse(T1,S1),
append(S1, [H1], [a,b,c])
2
{T1=[H2|T2],
R2=S1}
reverse(T2, S2)
append(S2, [H2], S1)
append(S1, [H1], [a,b,c])
append([], [H2], S1)
append(S1, [H1], [a,b,c])
1
{Xs1=[H2], S1=[H2]}
append([H2], [H1], [a,b,c])
fail
2
{T2=[H3|T3],
R3=S2}
reverse(T3, S3)
append(S3, [H3], S2)
append(S2, [H2], S1)
append(S1, [H1], [a,b,c])
1
{T3=[],S3=[]}
append([], [H3], S2)
append(S2, [H2], S1)
append(S1, [H1], [a,b,c])
...
1
{Xs1=[H3], S2=[H3]}
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
reverse([], []).
reverse([H|T], R) :- reverse(T, S), append(S, [H], R).
append([], Xs, Xs).
append([X|Xs], Y, [X|Zs] ) :- append(Xs, Y, Zs).
%1
%2
%1
%2
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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
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6.3 Prolog
typeless atomic terms, type
safe
decidable
multi-directional definitions
RLP
functors,
typeless composite terms,
type safe
partially decidable,
multi-directional definitions
LP
Prolog
arithmetics,
uni-directional procedures
dynamically typed,
not type safe,
system predicates (e.g. !)
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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
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6.3.2 The cut operator
For a node v, in which rule H :- B1, ...Bi, !, Bi+1, ..., Bn
is 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
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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
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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
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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).
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Example: trimming unwanted answers
Eliminates wrong answers
color(P, red) :- red(P),!.
color(P, black) :- black(P),!.
color(P, unknown).
red(a).
black(b).
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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.
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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).
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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).
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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.
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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.
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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).
87
Comments
(A,B) is a tuple.
?- X=(Y,T).
X = (Y, T).
?- (X,Y)=(A,B,C,D).
X = A,
Y = (B, C, D).
88
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)).
89
Meta-interpreter - version 3
Pre-processing – Program transformation.
Every rule A :- B1, B2, ..., Bn in 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).
90
Meta-interpreter - version 3
Pre-processing – Program transformation.
Every rule A :- B1, B2, ..., Bn in program P,
is written in program P' as a fact rule(A,[B1, B2, ..., Bn]).
Program P':
rule(
rule(
rule(
rule(
rule(
rule(
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)]).
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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).
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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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