The Course
Logic Programming ID2213
Thomas Sjöland
[email protected]
SCS, Software and Computer Systems
ICT - School of Information and Communication Technology
KTH, The Royal Institute of Technology
Outline of lectures
W35:
F1: Theory, Elementary Programs, unification
Theory, Model Theory of LP, proof trees and search trees
W36:
F2: Programming Style, recursion, equality primitives, representation
Advanced Recursive Techniques, accumulators, diff-structures, ADT
W37:
F3: Search Based Programming, cut and negation
Concurrency, Search Based Programming, state space, puzzles, games
W38:
F4: Logic programming and Grammars, parsing with DCG
W39:
F5: Program Transformation. Higher-order programming.
Metaprogramming, Expert Systems
W40:
F6: Case study: A compiler for a simple stack machine
W41:
F7: Case study: Support for reasoning about electronic circuits
Red1: Project presentation 4 hours
W42: Written Examination
F1: Theory and simple programs
Sterling and Shapiro ch. 1,2,4,5,6
Nilsson and Maluszynski ch.1,2,3,6
Theory for Logic Programming
Outline
Informal introduction to logic programming theory
Data in logic programs: Individual constants, term
constructors, logical variables, compound terms, trees and
lists
Equality theory, Unification
Logic Programs: Definite (Horn) Clauses
Model theory (least Herbrand model, term interpretation)
Proof theory and operational semantics of Prolog
(SLD-resolution, proof trees)
Simple databases
Recursive rules
Logic Programming
Using proofs to compute
To each proof you can order a computation
To each computation you can order a proof
Representation of
knowledge and computations
- algorithms
- functions
- relations
Data in Logic Programs
Programs express facts about a world of objects
Constants
Functors
Numbers
Compounded structures (normally finite)
Lists
Trees
Objects in Logic Programs
Individual constants
a
b
foo
4711
-37
Functors
structure names of trees and graphs
same syntax as non-numerical constants
Arity (number of arguments):
term/4, a/0
Syntax example:
term(a,b,-4711,other(b,a))
34.5
Logical Variables - Syntax
Syntax: begin with a capital letter (or '_')
X
Y
Z
Foo
_Bar
_
Variables can occur wherever constants or
structures occur
Range over logical objects
_ is "anonymous" or "void"
Programs are Theories
sets of relations (predicates) over objects
The classical form of a definition is as a clausal form where
a positive literal P has exactly one occurrence:
P or not Q1 or ... or not Qn
This can be written as P if Q1 & ... & ... Qn.
If all goals Qi are true the clause becomes P.
Program = Definitions + Query
The general form of a relation definition is
P if (Q11 & ... & Q1n)
or ...or
(Qm1 & ... & Qmn).
1..m and 1..n are index sets large enough
to cover all goal atoms, Qij
Program = Definitions + Query
Elementary literals (atoms)
true, false, X=Y
cannot be redefined
(only used in queries and definition bodies)
Defined literals (p above)
Definite Clauses: Facts
Facts: statements of form P :- true.
Also written simply as P.
Example:
brother(nils,karl).
Means that the binary relation
brother holds between individual
constants nils and karl.
Definite Clauses: Rules
Rules:
conditional formulae of the form
P :- Q1,....,Qn.
P is called the head and Q1,...,Qn the body of
the clause and P, Q1,...,Qn are atomic
formulas (relation symbols). Some of the Qi
may be predefined relation symbols (=, <)
":-" is read as "if", "," is read as "and"
Definite Clauses: Rules, example
Example of a rule:
grandfather(X,Y) :father(X,Z), father(Z,Y).
The binary relation grandfather holds
between two individuals represented by
variables X and Y if the relation father
holds between X and some Z and between
that Z and Y.
Clause Syntax
Example :
p(17).
p(X) :- X<8, q(X).
p(X) :- q(X), X=s(Y), p(Y).
In english the above example could be stated as follows:
- The property p holds for the constant 17.
- The property p holds for a term denoted by the
variable X if X<8 and q holds for X.
- The property p holds for X if q holds for X, X equals
a term s(Y) and p holds for Y.
Programs are Theories
Definitions are collections of facts and rules
- sets of relations (predicates) over the objects
e.g. (for predicate p/2 using q/2 and r/2)
p(foo,bar).
p(Foo,Bar) :- q(Foo,Baz), r(Baz,Bar).
Functions are special cases of relations
(deterministic)
Query, Goal
formula to be verified (or falsified)
Questions posed to the system are of the form
?- Q1,...,Qn.
for example
?- q(Foo,Baz), r(Baz,Bar).
If the system succeeds to prove the formula, the values of the
variables (the bindings) that are the result are shown,
otherwise the proof attempt has failed or possibly loops.
Note that more than one solution is possible.
How Prolog works
A user query ?- p(Args).
is proven using resolution
- look for all definition clauses of p
- pick one, save others as alternatives
- match the arguments in Args with the
terms in the head of the clause, create
necessary variable bindings
- if the matching unification fails, try next
alternative
- else prove the goal clauses of the body
from left to right
- if all proofs are done, the bindings are
presented
Database for family relationships
parent(Parent, Child), male(Person) and female(Person)
parent(erik, jonas).
male(erik).
parent(erik, eva).
male(jonas).
parent(lena, jonas). female(lena).
?- parent(lena, jonas).
Yes
?-parent(X,Y).
X=erik, Y=jonas;
X=erik, Y=eva;
X= lena, Y= jonas
?- parent(X, jonas).
X=erik;
X=lena
Logical variables - semantics
Variables can occur wherever constants or
structures occur.
Range over logical objects.
Bound to structures and to other variables.
The semantics is "single-assignment"
- starts "unbound"
- once bound, stays the same in the whole proof
Example cont.: rules
father(Dad, Child):- parent(Dad, Child), male(Dad).
mother(Mum, Child):- parent(Mum, Child), female(Mum).
?- father(X,Y).
X=erik, Y=jonas
X=erik, Y=eva
?- mother(erik, jonas).
No
?- mother(Erik, jonas). Yes. Why?
%sibling(Person1, Person2) :- ...
sibling(X,Y) :- parent(Z,X), parent(Z,Y).
%cousin(Person1, Person2):- ...
cousin(X,Y) :- parent(Z,X), parent(U,Y), sibling(Z,U).
syntactic sugar ';'
The symbol ';' can be used to avoid defining auxiliary
predicates or to reduce the number of clauses.
';' is read as "or".
A clause of the form
P :- Q1, (Q2 ; Q3), Q4.
is the same as
P :- Q1, Q, Q4.
Q :- Q2.
Q :- Q3.
Equality theory - Substitutions
X equals Y iff
X is an unbound variable or Y is an unbound variable
or
X and Y are (bound to) the same constant
or
X and Y are terms with the same functor and arity
e.g. X is term(Z1,..,Zn) and Y is term(U1,...,Un)
and for all arguments 1=<i=<n: Zi equals Ui.
Substitutions
A substitution is a function Subst: Var -> Term
Substitutions can be applied to terms or substitutions
and also to formulas
We may represent a substitution as a conjunction of
simple equalities v=t where a variable v occurs on
the left hand side at most once
or as a set {v/t | v=t} meaning a function that
replaces v with t for each v/t in the set
Unifier
A unifier is a substitution s such that ssts
(applying s to s and to t creates identical terms)
Most general unifier
A unifier s is more general than a unifier d
iff
there exists another unifier w such that s w  d
A unifier s is the most general unifier of two terms
iff
s is more general than any other unifier of the two
terms
Most general unifier, example
Example :
t(X,Y,Z) and t(U,V,W)
are unified by
{X/a,Y/b,Z/c, U/a,V/b,W/c}
consider for instance the mgus in this case
{X/U,Y/V,Z/W} and {U/X,V/Y,W/Z}
Unification procedure
An algorithm that constructs most general unifiers for two
terms in an environment is a unification procedure.
Since the most general unifier is unique (modulo renaming of
variables), unification can also be understood as a function
unify : Subst x Term x Term -> Subst
Theory and simple programs
(cont) Operational Semantics,
SLD
Sterling and Shapiro ch. 1,2,4,5,6
Nilsson and Maluszynski ch.1,2,3,6
Example cont.: structured data
Use compound (not atomic) terms for the description of persons.
parent(erik, jonas).
parent(erik, eva).
parent(lena, jonas).
male(person(erik,50,180cm)).
male(person(jonas,25,_)).
father(Dad,Child) :- Dad = person(DadName,_,_),
Child=person(ChildName,_,_),
parent(DadName, ChildName),
male(Dad).
?-father(person(_,50,_), person(X,_,_)).
X=jonas
(second solution: X = eva)
NB: how does the unification algorithm work here?
Logical Variables in Programs
Variables and parameters are implicitly quantified
syntax: variables start with capital letter
p(X,Y) :- q(X,Z), r(Z,Y).
is understood as
forall X,Y:(p(X,Y) <exists Z:(q(X,Z), r(Z,Y)))
Parameters (X,Y) are often confusingly named "global
variables" as opposed to "local variables" (Z)
but if X is global and Y is local, what is Y, if X=Y occurs
in program?
Example cont.: recursive rules
ancestor(Ancestor, Descendant) :parent(Ancestor, Descendant).
ancestor(Ancestor, Descendant) :parent(Ancestor, Person),
ancestor(Person, Descendant).
parent(ulf, erik). …
?- ancestor(X, Y).
Declarative vs procedural
A logic program can be understood in either of two ways:
it can be seen as a set of Horn clauses specifying facts about
data (a theory). This is the declarative or modeltheoretical reading of a logic program. What?
it can be viewed as a program describing a particular
execution (how to find a proof). This is the procedural
or proof-theoretical reading of a logic program. How?
Modus Ponens
P
Q:-P
-------------Q
Proof methods with Horn clauses
Given a database:
p :- q,r.
q :- q1, q2.
q1.
q2.
r.
Proof methods to
prove p:
Forward chaining
- use modus ponens to
accumulate known truths,
starting from facts.
Backward chaining
- prove p by proving q
and then proving r etc.
(used in prolog)
Model Theory:
Herbrand interpretation
When reading a program as a specification we need to
determine the meaning of the symbols.
A term interpretation, or "Herbrand interpretation" is an
association of a unique function to each functor occurring
in the program and an association of sets of tuples of terms
to relations.
An interpretation is a model for a program if all statements in
the interpretation are true.
Model Theory:
Least Herbrand Model
The least Herbrand model is the least term interpretation such
that it is a model.
For definite clauses such a unique model always exists.
Least Herbrand Model computed
The model can be inductively built up from the relation
symbols and the terms built from constants and terms in
the program by constructing a fixpoint.
Use the monotone Tp-operator. (N&M p. 29 ch 2.4),
ground(P) is the set of all ground instances of clauses
in a program P (assume always at least one functor or
constant and only finite structures).
Tp(I) := {A0 | A0:-A1,...,Am in ground(P) &
{A1,...,Am} subset I }.
Start from the empty theory and determine the least fixpoint
for I=Tp(I) U I.
Note that the model does not contain variables.
Constructing a model with Tp
p :- q,r.
q :- q1, q2.
q1.
q2.
r.
s.
0: {}
1: {q1,q2,r,s}
2: {q1,q2,r,s,q}
3: {q1,q2,r,s,q,p}
4: {q1,q2,r,s,q,p}
Done
The fixpoint is the model
{q1,q2,r,s,q,p}
Constructing a model with Tp
p(X) :- q(X,Y),r(X).
q(X,Y) :- q1(X,Y), q2(Y,X).
q1(a,b).
q2(b,a).
r(a).
0: {}
1: {q1(a,b),q2(b,a),r(a)}
2: {q1(a,b),q2(b,a),r(a),q(a,b)}
3: {q1(a,b),q2(b,a),r(a),q(a,b),p(a)}
4: {q1(a,b),q2(b,a),r(a),q(a,b),p(a)}
Done
Herbrand universe: {a,b}
The fixpoint is the model:
{q1(a,b),q2(b,a),r(a),q(a,b),p(a)}
Infinite structures
Assuming that the least Herbrand Model defines the intended
meaning of the program, unification must preserve the
property that infinite (cyclic) terms are not allowed. This
requires an occurs-check in the unification algorithm
prohibiting for example X=f(X) from generating
X=f(f(f(f(f(f(f(........
This is very inefficient so occurs-check is the responsibility of
the programmer. In critical cases a special test must be
performed after the unification.
Note that SICStus Prolog uses rational trees in
X=f(X)
Theoretically sound unification:
unify_with_occurs_check/2
Proof theory:
Execution is search for a proof
or failure, generating an or-tree
restrictions on the variables are shown as bindings of the variables
Search trees and proof trees
proof
Search tree
Proof tree
SLD-resolution rule
<- A1,..,Ai-1,Ai,Ai+1,...,Am
B0 <- B1,...,Bn
-----------------------------------------<- (A1,...,Ai-1,B1,...,Bn,Ai+1,...,Am)s
Where P is a program,
A1,...,Am are atomic formulas (goals),
B0<- B1,...,Bn is a (renamed) clause in P
and s=mgu(Ai,B0)
Goal and clause selection
A goal selection function specifies which goal
Ai is selected by the SLD-rule.
The order in which clauses are chosen is
determined with a clause selection rule.
Soundness of SLD-resolution
Any query (goal) that is provable with SLDresolution is a logical consequence of the
program.
Completeness of SLD-resolution
Any query (goal) that is (true) in the least
Herbrand model is provable with SLDresolution.
In the case of an infinite SLD-tree, the selection
function has to be fair (as in breadth first
search). For finite SLD-trees left-first-withbacktracking as used in Prolog gives a complete
method.
Conclusion
LP can be used as a uniform language for
representing databases, e.g. data structure
and queries can be written in a single
language
LP extends traditional databases by having
recursive rules and structured data facilities
F2: Logic Programming Style
Sterling and Shapiro ch. 2,6,7,13 (except 2.4, 2.5, 3.6, 6.3, 7.6, 13.3, 13.4)
Nilsson and Maluszynski ch.7 (except 7.3)
Outline
Programming techniques
Arithmetic in Prolog
Different primitives for equality: =/2, ==/2
Recursive definitions
Procedural - declarative
Imperative - logical style
binary trees, lists
append/3 reverse/2 quicksort/2
Specifying the use of a procedure
For serious projects it is good programming practice to specify the intended use of important
procedures, such as the predicates intended to use in a library.
For instance this could be given as a comment of the following form:
% procedure foo(T1,T2,...Tn)
%
% Types: T1: type 1
%
T2: type 2
%
T3: type 3
%
...
%
Tn: type n
% Relation scheme:...
% Modes uf use: (input arguments T1,T2) (output arguments T3,...,Tn)
% Multiplicities of solution: deterministic (one solution only)
Built-in arithmetics
is/2
a built-in predicate for evaluation of arithmetical expressions
?- Value is Expression. - first, Expression is evaluated and,
second, unified with Value
For example,
??????-
X = 2, Y is 1+X*3. - Y = 7
X = 2, 4 is X*X.
- yes
Z is 1+x.
- instantiation error, x is a constant
Z is 1+X.
- instantiation error, X is not instantiated
2 is 1+X.
- instantiation error, X is not instantiated
X=1+2.
- Yes. X = 1+2.
Built-in arithmetics
is/2 evaluates expressions containing:
-
+ - * / // mod
plus, minus, multiplication, division, integer division, remainder
/\ \/ # \ << >>
bitwise conjunction, disjunction, exclusive or, negation,
shift to the left, shift to the right
abs(X), min(X), max(X), sin(X), cos(X), sqrt(X).
(for a complete list, see the SICStus manual)
Typical error: failing to unify floating point numbers.
NB: different "equals"
=
= = ( \== )
=:= ( =\= )
is/2
?????-
- unification
- equality (inequality) of terms
- arithmetic, boolean (not)equal
- evaluation and unification
X=2, X=Y.
X=2, X==Y.
X=:=2.
X=2, Y=2, X=:=Y.
X=2+3, Y is X.
- Yes. X=2, Y=2.
- No.
- instantiation error
- Yes. X=2, Y=2.
- Yes. X=‘+’(2,3), Y=5.
Elementary programs
Sterling and Shapiro ch. 1,2,4,5,6
Nilsson and Maluszynski ch.1,2,3,6
Composing recursive programs
think about declarative meaning of recursive
data type (a definition)
write down recursive clause and base clause
run simple examples - check different goals
check what is happening (do you get the
expected result?)
Composing recursive programs
Typical errors:
missing (or erroneously failing) base case
error in data structure representation
wrong arity of structures
mixing an element and a list
permuted arguments
Natural numbers
Unary syntax
For example,
0
s(0)
- denotes zero
- denotes 1
...
s(…s(s(0))…)
- denotes n
Defining the natural numbers
natural_number(0).
natural_number(s(X)) :natural_number(X).
Natural numbers
plus
plus(0, X, X) :- natural_number(X).
plus(s(X), Y, s(Z)) :- plus(X, Y, Z).
?- plus(s(0),0,s(0)). - checks 1+0=1
Yes.
?- plus(X,s(0),s(s(0)). - checks X+1=2, (e.g. compute X=2-1)
X=s(0).
?- plus(X, Y, s(s(0))). - checks X+Y=2, (e.g. generate all
pairs of natural numbers, whose sum equals 2)
X=0, Y=s(s(0));
X=s(0), Y=s(0);
X=s(s(0)), Y=0.
Natural numbers
less or equal
le(0, X) :- natural_number(X).
le(s(X), s(Z)) :- le(X, Z).
multiplication
times(0, X, 0) :- natural_number(X).
times(s(X), Y, Z) :plus(Y, Z1, Z), times(X, Y, Z1).
check how substitution works!
Recursive Arithmetic
sum([],0).
sum([H|T],S) :- sum(T,V), S is H+V.
sum0([],0).
sum0([H|T],S) :- sum0(T,V), S=H+V.
Binary trees
Syntax (not built-in, create own compound terms)
For example,
void
- denotes empty tree
tree(Element, Left, Right)
- denotes a tree, where Element is
root and Left, Right are subtrees
tree(5,tree(8,void,void),tree(9,void,tree(3,void,void)))
defining a tree
binary_tree(void).
binary_tree(tree(Element, Left, Right)) :binary_tree(Left), binary_tree(Right).
Binary trees
membership
tree_member(X,tree(X,_,_)).
tree_member(X,tree(Y,Left ,_)):- tree_member(X,Left).
tree_member(X,tree(Y,_,Right)):- tree_member(X,Right).
NB: X might be equal to Y in clauses 2 and 3!
Lists
Syntax
[Head|Tail] cons cell
Head is an element, Tail is a list '.'(Head,Tail)
[]
empty list
simpler syntax
[a | [] ] = [a]
[a | [ b | [] ] ] = [a, b]
[erik], [person(erik,_,_),jonas|[lena, eva]]
defining a list
list([]).
list([X|Xs]) :- list(Xs).
- defines the basis
- defines the recursion
Lists
checking membership
member(X, [X|Xs]).
member(X, [Y|Ys]) :- member(X, Ys).
?- member(a, [b,c,a,d]).
- checks membership
?- member(X, [b,c,a,d]).
- takes an element from a list
?- member(b, Z).
- generates a list containing b
Lists
concatenation of lists
append([], Xs, Xs).
append([X|Xs], Y, [X|Zs]) :- append(Xs, Y, Zs).
?- append([a,b], [c], X).
- addition of two lists
?- append(Xs, [a,d], [b,c,a,d]).
- finds a difference between lists
?- append(Xs, Ys, [a,b,c,d]).
- divides a list into two lists
Check SLD-tree!
Typical error: wrong "assembly" of a resulting list
Lists
reversing lists
reverse([], []).
reverse([H|T],R) :- reverse(T,S), append(S,[H],R).
?- reverse([a,b,c,d],R).
- gives R=[d,c,b,a]
Check SLD-tree!
Typical error: wrong "assembly" of resulting list
wrong_reverse([H|T],R):reverse(T,S), append(S,H,R).
Lists
sorting
quicksort([X|Xs], Ys) :partition(Xs, X, Littles, Bigs),
quicksort(Littles, Ls),
quicksort(Bigs, Bs),
append(Ls, [X|Bs], Ys).
quicksort([], []).
partition([Y|Ys], X, [Y|Ls], Bs) :X>Y, partition(X, Ys, Ls, Bs).
partition([Y|Ys], X, Ls, [Y|Bs]) :X=<Y, partition(X, Ys, Ls, Bs).
partition([], _, [], []).
Dictionaries
Finding and adding a value in a dictionary (an (ordered) binary tree)
%lookup(+,?,?)
lookup(Key, tree(Key,Value, Left, Right), Value):- !.
lookup(Key, tree(Key1, Value1, Left, Right), Value) :Key < Key1, lookup(Key, Left, Value).
lookup(Key, tree(Key1, Value1, Left, Right), Value) :Key > Key1, lookup(Key, Right, Value).
?- lookup(1, D, fifi),lookup(2, D, mumu),lookup(1,D, X).
D=tree(1, fifi, _C, tree(2, mumu, _B, _A)), X=fifi.
NB: for finding a key of a value, the traversal of tree should be implemented.
Composing recursive programs
Final example
Define a predicate unsort_deg(Xs, D) that, given a
list of numbers, finds its unsort degree D.
The unsort degree of a list is the number of pairs of
element positions in the list such that the first
position precedes the second in the list, but the
number occupying the first position is greater than
the number occupying the second position.
Some examples:
the unsort degree of the list [1, 2, 3, 4] is 0
the unsort degree of the list [2, 1, 4, 3] is 2
the unsort degree of the list [4, 3, 2, 1] is 6
Representing sets
Sets can be represented by the existing datatypes in a convenient way by
enforcing an order on a structure used to store the set. For instance using
an ordered list (or tree) where each element has a unique occurrence and
where all operations are assumed to take ordered unique lists as input
and produce ordered unique lists.
If the sets are allowed to contain uninstantiated elements, however, we may
have some problems with enforcing the requirement that the lists are
ordered and unique, since the requirement may be violated in a later
stage.
Consider for instance[X,Y,Z] as a representation of a set with three
uninstantiated elements. Of course if X=Y is executed, the list no
longer contains unique elements. Perhaps even more obvious is that we
cannot ensure that the order of the elements is the one intended until the
elements are at least partially known.
invariants are the responsibility of the programmer
Syntactic support
Using op/3 properties (priority, prefix,infix,postfix,
associativity) of operators can be defined and then used.
(see manual for details)
Predicates defined by the user are written with the same
syntax as structures, for instance
:- op(950, xfy, [in]).
foo(Y) :- X in Y, baz(Y in U,Z).
Advanced recursive techniques
Sterling and Shapiro ch. 7,8,13.3, 15
Nilsson and Maluszynski ch. 7.3.
Outline
Programming with accumulating parameters
Programming with difference-structures
Queues with difference-structures
Abstract data types, separation of data
definitions (types) and the program's logic
Accumulating parameters
reverse lists
a) naive reverse (using append in each recursion step)
reverse([], []).
reverse([X|Xs], Ys) :reverse(Xs, Zs), append(Zs, [X], Ys).
b) reverse-accumulate
reverse(Xs, Ys) :reverse(Xs, [], Ys).
reverse([], Acc, Acc).
reverse([X|Xs], Acc, Ys) :reverse(Xs, [X|Acc], Ys).
advice: draw
simple SLD-tree and check substitutions!
Built-in arithmetics
Example
Define the predicate for computing the factorial of a given integer.
a) recursion
factorial(0, 1).
factorial(N, F) :N > 0,
N1 is N -1,
factorial(N1, F1),
F is N*F1.
Built-in arithmetics
Example
Define the predicate for computing the factorial of a given integer.
b) recursion with an accumulator
factorial(N, F) :- factorial(N, F, 1).
factorial(0, F, F).
factorial(N, F, F1) :N > 0,
N1 is N -1,
F2 is N*F1,
factorial(N1, F, F2).
Built-in arithmetics
Example
Define the predicate for computing the sum of members of integer-list.
a) recursion
sumlist([], 0).
sumlist([I|Is], Sum) :sumlist(Is, Sum1),
Sum is Sum1 + I.
b) iteration (with accumulator)
sumlist(List, Sum) :- sumlist(List, 0, Sum).
sumlist([], Sum, Sum).
sumlist([I|Is], Sum1, Sum) :Sum2 is Sum1 + I,
sumlist(Is, Sum2, Sum).
Using Abstract Data Types
Separation of data definitions (types) and the
program's logic.
Specify a set of objects
Specify set of operations (relations, functions)
on the objects
Allow access to objects only through defined
operations
Using Abstract Data Types
Assume the "data type predicates" specifying
operations on a given representation of lists:
cons(H,T,[H|T]).
nil([]).
equal(X,X).
Using this method allows change in representation
without change in the code of the algorithm.
Abstract form of append/3
append(A,B,C) :nil(A),
equal(B,C).
append(A,B,C) :cons(H,T,A),
cons(H,R,C),
append(T,B,R).
Changing Representation
Note that the representation of lists can be
changed without changing the algorithmic
code defining append/3
by replacing these "datatype predicates":
cons(H,T,foo(T,H)).
nil(bar).
Difference-lists
syntax
D1-D2,
D-D
where D1 is a list, which ends with D2
- empty list
diff-lists representing [a,b,c]
[a, b, c| R] – R
[a, b, c, d, e] - [d,e]
[a, b, c]-[]
adding two lists by unification only
[a, b, c]
[d]
[a,b,c,d]
use D1=[a,b,c|R1]-R1=D0-R1
use D2=[d|R2]-R2
use D=D0-R2, assuming that R1=[d|R2]
Difference-lists
concatenation
append_dl(D0-D1, D1-D2, D0-D2).
reverse
reverse(X, Y) :- reverse_dl(X, Y-[]).
reverse_dl([], Xs-Xs).
reverse_dl([X|Xs], Ys-Zs) :reverse_dl(Xs, Ys-[X|Zs]).
Difference-lists
sorting
quicksort(Xs, Ys) :- quicksort_dl(Xs, Ys-[]).
quicksort_dl([X|Xs], Ys-Zs) :partition(X, Xs, Littles, Bigs),
quicksort_dl(Littles, Ys-[X|Z1]),
quicksort_dl(Bigs, Z1-Zs).
quicksort_dl([], Xs-Xs).
partition(X,[Y|Ys],[Y|Ls],Bs) :X > Y, partition(X,Ys,Ls,Bs).
partition(X,[Y|Ys],Ls,[Y|Bs]) :X =< Y, partition(X,Ys,Ls,Bs).
partition(_,[],[],[]).
Queues
A queue may be implemented as a difference list
enqueue and dequeue
- enqueue(Element, OldQueue, NewQueue)
enqueue(X, Qh-[X|Qt], Qh-Qt).
- dequeue(Element, OldQueue, NewQueue)
dequeue(X,[X|Qh]-Qt, Qh-Qt).
Queues (cont)
S=[in(5), in(9), in(10), out(X1), out(X2), in(4)]
- an input list for queueing
queue(S) :- queue(S, Q-Q).
queue([], Q).
queue([in(X)|Xs], Q) :- enqueue(X, Q, Q1),
format("In ~d ~n", X),
queue(Xs, Q1).
queue([out(X)|Xs], Q) :- dequeue(X, Q, Q1),
format("Out ~d ~n", X),
queue(Xs, Q1).
F3: Programming with search
Sterling and Shapiro ch. 6,7,11,14,20,21
Nilsson and Maluszynski ch.4,5,11,12, A.3
Outline - Search and control
Proof trees
Cut
Negation, SLDNF
Outline
Cut
execution of a program with cuts
insertion of cuts in your own program:
"green" and "red" cuts
implementation of if-then-else
Negation
basic concepts
execution of programs with negation
implementation of negation
Controlling search
Sometimes when a solution to a subproblem has been
found, no other solutions to it or to earlier proved
subgoals of the current goal need to be considered.
By using the non-logical primitive predicate !,
named 'cut', you remove alternative branches
to subgoals and to the clause that is currently being
proved. The alternatives on a 'higher' level, that is
to the clause which the current goal is a part of are,
though, kept. This can decrease the amount of
unnecessary computation.
Cut
syntax
!, can be placed in the body of a clause or a goal as one of its atoms
to cut branches of an SLD-tree
p(X) :- q(X), !, r(X).
effects
divides the body into two parts: when "!" is reached, it is evaluated to
true and all backtracking of the left-side of the body is disallowed.
The execution of the right-side of the clause body continues as
usual.
new matches of the head of the clause are disallowed
e.g. backtracking is stopped one level up in the SLD-tree
Cut performs two operations
P :- Q, !, R.
P :- ...
removes alternatives to Q that haven't been
tried when passing the cut
removes alternatives to P that haven't been
tried when passing the cut
Cut
example: execution of a program with cuts
Consider the following program:
top(X,Y):- p(X,Y).
top(X,X) :- s(X).
p(X,Y) :- true(1), q(X), true(2), r(Y).
p(X, Y) :- s(X), r(Y).
q(a). q(b).
r(c). r(d).
s(e).
true(X).
?- top(X,Y).
¤ in the given program
¤ when true(1) is replaced by !
¤ when true(2) is replaced by !
(seven answers)
(five answers)
(three answers)
Cut
inserting cuts in your own program
in order to increase efficiency
"green" cut:
does not change the semantics of a program (cuts away
only failing branches in an SLD-tree)
"red" cut:
changes the semantics of a program (cuts also away
success branches in an SLD-tree)
in general, the red cuts are considered harmful
Cut
"Green cut": an example
Given two sorted integer-lists Xs and Ys, construct a sorted
integer-list Zs, containg elements from Xs and Ys.
merge([], Ys, Ys).
merge(Xs, [], Xs).
merge([X|Xs], [Y|Ys], [X|Zs]) :X < Y, merge(Xs, [Y|Ys], Zs).
merge([X|Xs], [Y|Ys], [X, Y|Zs]) :X = Y, merge(Xs, Ys, Zs).
merge([X|Xs], [Y|Ys], [Y|Zs]) :X > Y, merge([X|Xs], Ys, Zs).
Cut
"Green cut": an example (cont.)
merge([], Ys, Ys):- !.
merge(Xs, [], Xs):- !.
merge([X|Xs], [Y|Ys], [X|Zs]) :X < Y, !, merge(Xs, [Y|Ys], Zs).
merge([X|Xs], [Y|Ys], [X, Y|Zs]) :X = Y, !, merge(Xs, Ys, Zs).
merge([X|Xs], [Y|Ys], [Y|Zs]) :X > Y, !, merge([X|Xs], Ys, Zs).
Cut
"Red cut": an example
Find the minimum of two integers.
Try:
minimum(X, Y, X) :- X =< Y, !.
minimum(X, Y, Y).
?- minimum(4,5,Z).
?- minimum(5,4,Z).
?- minimum(4,5,5).
- Yes, Z = 4.
- Yes, Z = 4.
- Yes.
Correction:
minimum(X, Y, Z) :- X =< Y, !, Z=X.
minimum(X, Y, Y).
Cut
"Red cut": an example
Checking membership in a list.
member(X, [X|Xs]) :- !.
member(X, [Y|Ys]) :- member(X, Ys).
?- member(a, [b,c,a,d]).
?- member(X, [b,c,a,d]).
?- member(b, Z).
- checks membership,
OK
- takes elements from a list,
takes only the first element
- generates lists containing b
generates only one list
OBS: Check the example lookup/3 from the previous lecture!
Cut
implementation of if-then-else
P :- Condition, !, TruePart.
P :- ElsePart.
or
(Condition -> TruePart; ElsePart)
For example,
minimum(X, Y, Z) :- (X =< Y -> Z = X; Z = Y).
Negation: how to use and prove
negative information?
A negated query 'not_p(x)' should succeed if the proof of the
statement 'p(x)' fails and it should fail if the proof of the
statement 'p(x)' succeeds.
%not_p(++) (++ stands for a ground term)
not_p(X) :- p(X), !, false.
not_p(_).
Closed world assumption
That which is not stated explicitly is false
animal(cow).
not_animal(X) :- animal(X), !, false.
not_animal(_).
?- not_animal(X), X=house.
A house is not an animal so the query should succeed
Unfortunately it fails. Why?
Closed world assumption
animal(cow).
not_animal(X) :- animal(X),!, false.
not_animal(_).
not_not_animal(X) :- not_animal(X),!, false.
not_not_animal(_).
?- not_not_animal(X), X=house.
Unfortunately this succeeds. Why?
Negation
how to use and prove negative information?
to apply closed world assumption (cwa):
the statement \+ A is derivable if A is a formula which
cannot be derived by SLD-resolution.
- Problem with infinite SLD-trees
implementation: negation as failure (naf):
the statement \+ A is derivable if the goal A has a
finitely failed SLD-tree.
The problem when A has variables remains.
unmarried_student(X) :- \+ married(X), student(X).
student(erik).
married(jonas).
Negation
SLDNF-resolution
the combination of SLD-resolution to resolve positive
literals and negation as failure to resolve negative
literals
foundation(X)
on_ground(X)
off_ground(X)
above(X, Y)
above(X, Y)
on(c, b).
on(b, a).
:::::-
on(Y, X), on_ground(X).
\+ off_ground(X).
on(X, Y).
on(X, Y).
on(X, Z), above(Z, Y).
Negation
Four kinds of SLDNF-derivations:
refutations (that end with success branches);
infinite derivations;
(finitely) failed derivations;
stuck derivations (if none of the previous apply).
NB: check examples in N&M, pp. 71-73.
Negation
implementation of \+
\+ Goal :\+ Goal.
call(Goal), !, fail.
example
p(a).
?- p(X).
?- \+ \+ p(X).
Yes, X = a.
true, X is not instantiated
Search based programming
Sterling and Shapiro ch. 6,7,11,14,20,21
Nilsson and Maluszynski ch.4,5,11,12, A.3
Outline- Search based programming
State space programming
generate-and-test
searching in a state-space
Graph theoretical examples
Euler paths, Hamilton paths
Puzzle-solving, game-playing
Sterling and Shapiro ch. 14,20
Nilsson and Maluszynski ch. 11
Generate-and-test
A technique in algorithm design, which defines
two processes
the first generates the set of candidate solutions
the second tests the candidates
In PROLOG:
find(X):- generate(X), test(X).
Important optimisation:
to "push" the tester inside the generator
as "deep" as possible
Generate-and-test
important optimisation: to "push" the tester inside
the generator as "deep" as possible
find(X):generate1(X),
generate2(X),
generate3(X),
generate4(X),
test1(X),
test2(X),
test3(X),
test4(X).
Generate-and-test
Example 1
Finding parts of speech in a sentence:
verb(Sentence,Word) :
member(Word,Sentence),verb(Word).
noun(Sentence,Word) :
member(Word,Sentence),noun(Word).
article(Sentence,Word):
member(Word,Sentence),article(Word).
noun(man).
noun(woman).
article(a).
verb(loves).
?- noun([a, man, loves, a woman], N).
N=man; N=woman
NB. member/2 should not contain cuts. Why?
Generate-and-test
Example 2
Place N queens on a NxN chess-board in such a way that any two queens
are not attacking each other.
a) Naive generate and test places N queens and then test whether they
are not attacking each other.
The answer is a list of queens' positions, for example [3, 1, 4, 2].
queens(N, Qs) :range(1, N, Ns),
% Ns is the list of integers in 1..N
permutation(Ns, Qs), % Qs is a permutation of Ns
safe(Qs).
% true, if the placement Qs is safe
range(M, N, [M|Ns]) :M < N, M1 is M +1, range(M1, N, Ns).
range(N, N, [N]).
Generate-and-test
Example 2 (cont.)
permutation(Xs, [Z|Zs]) :select(Z, Xs, Ys),
permutation(Ys, Zs).
permutation([], []).
select(X, [X|Xs], Xs).
select(X, [Y|Ys], [Y|Zs]) :- select(X, Ys, Zs).
safe([Q|Qs]) :- safe(Qs), \+ attack(Q, Qs).
safe([]).
attack(X, Xs) :- attack(X, 1, Xs).
attack(X, N, [Y|Ys]) :- X is Y+N; X is Y-N.
attack(X, N, [Y|Ys]) :- N1 is N+1, attack(X, N1, Ys).
Generate-and-test
Example 2 (cont.)
b) When generating a position of a queen, test whether it is permitted
queens(N, Qs) :range(1, N, Ns),
queens(Ns, [], Qs).
queens(UnplacedQs, SafeQs, Qs) :select(Q, UnplacedQs, UnplacedQs1),
\+ attack(Q, SafeQs),
queens(UnplacedQs1, [Q|SafeQs], Qs).
queens([], Qs, Qs).
select/3, attack/2 are the same as in a).
Searching in a State-space
- loop-avoidance in searching for a path
- efficiency issues
- different search strategies
Searching in a State-space
Many problems in computer science can be formulated as
follows:
Given some start-state S0 and a set of goal-states
determine whether there exists a sequence
S0 ---> S1, S1 ---> S2, …, Sn-1 ---> Sn,
such that Sn belongs to a set of goals.
States can be seen as nodes in a graph whose edges
represent the pairs in the transition-relation, then the
problem reduces to that of finding a path from the startstate to one of the goal-states.
Searching in a state-space
Finding a path
path(X,X).
path(X,Z) :- edge(X,Y), path(Y,Z).
edge(X,Y) :- % define construction/finding of the next node
For example
edge(a,b). edge(b,c). edge(c,d).
?- path(a,d).
?- path(a,X).
?- path(X,d).
- Yes.
- Yes. X=a; X=b; X=c; X=d.
- Yes. X=d; X=a; X=b; X=c.
Searching in a state-space
Loop detection
path(X, Y) :path(X, Y, [X]).
path(X, X, Visited).
path(X, Z, Visited):edge(X, Y),
\+ member(Y, Visited),
path(Y, Z, [Y|Visited]).
member(X, [X|Y]) :- !.
member(X, [Y|Z]) :- member(X, Z).
Searching in a state-space
Returning the path as an answer to the goal
path(X, Y, Path) :path(X, Y, [X], Path).
path(X, X, Visited, Visited).
path(X, Z, Visited, Path):edge(X, Y),
\+ member(Y, Visited),
path(Y, Z, [Y|Visited], Path).
member(X, [X|Y]) :- !.
member(X, [Y|Z]) :- member(X, Z).
Searching in a state-space
Puzzle: Missionaries and cannibals
Three missionaries and three cannibals must cross a river, but the
only available boat will hold only two people at a time. There is no
bridge, the river cannot be swum, and the boat cannot cross the
river without someone in it. The cannibals will eat any
missionaries they outnumber on either bank of the river.
The problem is to get everyone across the river with all the
missionaries uneaten.
Searching in a state-space
puzzle( Moves ) :path( state(3, 3, left), state(3, 3, right), Moves).
path(InitNode, FinalNode, Path) :path(InitNode, FinalNode, [InitNode], Path).
path(InitNode, FinalNode, _, []) :- InitNode = FinalNode, !.
path(Node0, FinalNode, VisitedNodes, [Arc|Path]):edge(Node0, Arc, Node1),
\+ member(Node1, VisitedNodes),
path(Node1, FinalNode, [Node1|VisitedNodes], Path).
Searching in a state-space
Example (cont.)
edge( state(M0, C0, L0), move( M, C, D), state (M1, C1, L1) ):member(M, [0, 1, 2]),
member(C, [0, 1, 2]),
M + C >= 1,
M + C =< 2,
M0 >= M,
C0 >= C,
M1 is 3 - (M0 - M),
C1 is 3 - (C0 - C),
( M1 =:= 0 ; M1 =:= 3; M1 = := C1),
(L0 = left -> ( D = leftRight, L1 = right);
( D = rightLeft, L1 = left) ).
Searching in a state-space
Better representation – simpler algorithm
Store the whole graph as one fact
graph([edge(a,b),edge(c,d),edge(b,c)]).
path(X, Y, Path) :- graph(G),
path(X, Y, Path, G, []).
path(X, X, [X], G, G).
path(X, Z, Path, G0, G1):deleteedge(X, Y, G0, Gt),
path(Y, Z, [Y|Path], Gt,G1).
deleteedge(X, Y, [edge(X,Y)|T], T) :- !.
deleteedge(X, Y, [A|T], [A|R]) :deleteedge(X, Y, T, R).
Searching in a state-space
Basic search methods
depth-first - interprets current nodes as a stack
breadth-first - interprets current nodes as a queue
bounded-depth - "controls" the depth of the search
iterative-deepening -"controls" the depth of the search
heuristic methods – using domain specific knowledge
Searching in a state-space
Heuristic search
to solve larger problem, some domain-specific
knowledge must be added to improve search
efficiency
the term heuristic is used for any advice that is
often effective, but isn't guaranteed to work in
every case
a heuristic evaluation function estimates the cost
of a shortest path between a pair of states
Many search strategies
Bottom-up - inductively generate facts from known
facts
Top-down - recursively find supporting rules for query
Serial - alternatives one at a time, backtrack for more
Or- parallel - to prove A or B try A and B in parallel
collect all solutions or choose one (first found, best
etc.)
Concurrent - to prove A & B try A and B concurrently
AND-parallel - to prove A & B where A and B have nothing
in common do A and B in parallel, then combine results
and go on
Iterative deepening – search all solutions down to a
maximum depth, then increase max
Tabled execution – keep ongoing calls in a table to
avoid redundant work
Parallelism and Concurrency
Concurrent logic programming models
Parallel execution on multiprocessors
Sterling and Shapiro ch. 14.2
Nilsson and Maluszynski ch.12,14, A.3
Concurrent Logic Programming
essense
goals in the body of a clause can execute as interleaving
processes
p(X) :- q(X), r(X).
Synchronisation on variable unification
"X? = Y" means "X must not be bound by unification"
effects
divides the body into two parts and switches execution
between the goals.
deadlock is possible
Parallel logic programming
Computation can be distributed over several
computers possibly sharing memory.
OR-parallelism
copying of backtrack stack
sharing of binding environment
AND-parallelism
independent AND (CIAO-Prolog)
stream-and ("Penny", parallel AKL)
Game playing
Game trees
a game tree is an explicit representation of all possible
plays of the game. The root node is the initial position of
the game, its successors are the positions which the first
layer can reach in one move, their successors are the
positions resulting from the second player's replies and so
on.
the trees representing the games contain two types of node:
max at even levels from the root, and min nodes at odd
levels of the root.
a search procedure combines an evaluation function, a
depth-first search and the minimax backing-up
procedure.
Game playing
4
Maximise
4
8
4
4
Minimise
2
8
5
2
5
7
2
7
3
Maximise
5
F4: Logic and grammars
Sterling and Shapiro ch. 19 (except 19.2),24
Nilsson and Maluszynski ch.10
SICStus Prolog Manual.
Outline
grammars
context free grammars
context dependent grammars
Context Free Grammars
A context free grammar is a 4-tuple
< N, T, P, S >
where N and T are finite, disjoint sets of
nonterminal and terminal symbols respectively,
(N U T)* denotes the set of all strings (sequences)
of terminals and non-terminals
P is a finite subset of N x (N U T)* ,
S is a nonterminal symbol called the start symbol.
Empty string is denoted by e and elements of P are
usually written in the form of production rules:
A ::= B1, …, Bn
(n > 0)
A ::= e
(n = 0)
Grammars
Example 1
<sentence> ::= <noun-phrase><verb-phrase>
<noun-phrase> ::= the <noun>
<verb-phrase> ::= runs
<noun> ::= engine
<noun> ::= rabbit
<sentence> derives the strings the rabbit runs, the engine runs.
How?
Grammars
Example 2
prod_rule(sentence, [noun_phrase, verb_phrase]).
prod_rule(noun_phrase, [the, noun]).
prod_rule(verb_phrase, [runs]).
prod_rule(noun, [rabbit]).
prod_rule(noun, [engine]).
An interpreter for production rules:
derives_directly(X, Y) :append(Left, [Lhs|Right], X),
prod_rule(Lhs, Rhs),
append(Left, Rhs, Temp),
append(Temp, Right, Y) .
?- derives([sentence], X).
derives(X, X).
derives(X, Z) :derives_directly(X, Y),
derives(Y, Z).
Yes. What is X unified to?
How can you force all productions to be considered before terminating?
Grammars
Example 3
sentence(Z) :- append(X, Y, Z), noun_phrase(X), verb_phrase(Y).
noun_phrase([the|X]) :- noun(X).
verb_phrase([runs]).
noun([rabbit]).
noun([engine]).
append([], X, X).
append([X|Xs], Y, [X|Zs]) :- append(Xs, Y, Zs).
?- sentence([the, rabbit, runs]).
?- sentence([the, X, runs]).
?- sentence(X).
Yes.
X = rabbit, X = engine
Grammars
Example 4
Usage of difference-lists:
sentence(X0-X2) :noun_phrase(X0-X1), verb_phrase(X1-X2).
noun_phrase([the|X]-X2) :- noun(X-X2).
verb_phrase([runs|X]-X).
noun([rabbit|X]-X).
noun([engine|X]-X).
????-
sentence([the, rabbit|X] - X).
sentence([the, rabbit, runs] - []).
sentence([the, rabbit, runs, quickly] - [quickly]).
sentence(X).
Grammars
Example 5
"Collecting" the result:
sentence(s(N, V), X0-X2) :noun_phrase(N, X0-X1), verb_phrase(V, X1-X2).
noun_phrase(np(the, N), [the|X]-X2) :- noun(N, X-X2).
verb_phrase(verb(runs), [runs|X]-X).
noun(noun(rabbit), [rabbit|X]-X).
noun(noun(engine), [engine|X]-X).
?- sentence(S, [the, rabbit, runs] - []).
Yes.
S=s(np(the, noun(rabbit)), verb(runs)).
Grammars
Context dependent grammars
<sentence> ::= <noun-phrase>(X) <verb>(X)
<noun-phrase>(X) ::= <pronoun>(X)
<noun-phrase>(X) ::= the <noun>(X)
<verb>(singular) ::= runs
<verb>(plural) ::= run
<noun>(singular) ::= rabbit
<noun>(plural) ::= rabbits
<pronoun>(singular) ::= it
<pronoun>(plural) ::= they
Grammars
Example 6
sentence(X0-X2) :- noun_phrase(Y, X0-X1), verb(Y, X1-X2).
noun_phrase(Y, X - X2) :- pronoun(Y, X-X2).
noun_phrase(Y, [the|X]-X2) :- noun(Y, X-X2).
verb(singular, [runs|X]-X).
verb(plural, [run|X]-X).
noun(singular, [rabbit|X]-X).
noun(plural, [rabbits|X]-X).
pronoun(singular, [it|X]-X).
pronoun(plural, [they|X]-X).
Recognizers for languages:
Lexers and parsers
A program that recognizes a string in a formal language is often
divided into two distinct parts:
Lexer: translation from lists of character codes to lists of 'tokens'
Parser: the translation from lists of 'tokens' to parse trees
Concrete syntax (describes a string in a language, a list of tokens)
<A> ::= <B>< C>
[foo,bar]
Abstract syntax (describes a syntax tree, a term)
<A> :: <B>< C>
‘A’(B, C)
Logical Input
char_infile(FileName,Offset,List) :open(FileName,read,S),
skipchars(S,Offset),
readchars(S,List),
close(S),
!.
skipchars(_S,0) :- !.
skipchars(S,I) :- I>0, get0(S,_), J is I-1,
skipchars(S,J).
readchars(S,L) :- get0(S,C), readchars0(S,L,C).
readchars0(_,L,-1) :- !, L=[].
readchars0(S,L,C) :- L=[C|R], readchars(S,R).
A recognizer for s-expressions
An s-expression is a general representation form for data
used in LISP and in some Prolog dialects, especially
those embedded in a LISP or SCHEME environment.
We will show how a program identifying s-expressions
looks in Prolog.
Backus-Naur grammar
for a legal s-expression
<s-expr> ::= <s-atom> | '(' <s-exprs> ['.' <s-expr>] ')'
<s-exprs> ::= <s-expr> [<s-exprs>]
<s-atom> ::= [<blanks>] <non_blanks> [<blanks>]
<non_blanks> ::= <non_blank> [<non_blanks>]
<blanks> ::= <blank> [<blanks>]
A non-blank is a character with a character code greater than 32 (decimal). A blank is a
character with a character code less than or equal to 32.
The input to the parser is represented as a list of characters (or character codes).
A recognizer for the s-expression
grammar
The predicate sExpr(L1,L2,S) succeeds if the
beginning of the list L1 is a list of characters
representing an s-expression. As a result of the
computation the structure S contains a structure (a
list) representing the s-expression. The rest of the
input text is stored in the list L2.
p :- sExpr("(A.B)",Out,S),
write(Out), write(S).
sExpr(In,Out,S) :(blanks(In,I0); In=I0),
(sAtom(I0,Out,S)
;
lpar(I0,I1),
sExprs(I1,I2,S,Last),
(dot(I2,I3),
sExpr(I3,I4,Last)
;
I4=I2,
Last=[]),
rpar(I4,Out)).
sExprs(In,Out,[H|T],Last) :sExpr(In,I1,H),
(sExprs(I1,Out,T,Last)
;
Out=I1, T=Last).
sAtom(In,Out,A) :- nonBlanks(In,Out,A).
nonBlanks(In,Out,[H|T]) :nonBlank(In,I1,H),
(nonBlanks(I1,Out,T)
;
I1=Out,
T=[]).
blanks(In,Out) :- blank(In,I1), (blanks(I1,Out) ; I1=Out).
blank(In,Out) :- In=[C|Out], C<=32.
nonBlank(In,Out,C) :- In=[C|Out], C>=48.
lpar(In,Out) :- In=[#(|Out].
rpar(In,Out) :- (blanks(In,I1); In=I1), I1=[#)|Out].
dot(In,Out) :- (blanks(In,I1); In=I1), I1=[#.|Out].
A Grammar For Horn Clauses
<clause> ::= <head> [ ':-' <body> ]
<head> ::= <goal>
<body> ::= <goal> [ ',' <body> ]
<goal> ::= <name> '(' <termlist> ')' | <cut> | <false>
<termlist> ::= <term> [ ',' <termlist> ]
<term> ::= <variable>
| <dataconstructor> [ '(' termlist ')' ]
| <constant>
| <list>
| <integer>
| <expression>
<list> ::= '[' <term> [ <moreterms> ] [ '|' <term> ] ']'
<moreterms> ::= ',' <term> [ <moreterms> ]
<expression> ::= <term> <operator> <term> | <operator> <term>
<operator> ::= '=' | '<' | '=<' | '>=' | '>' | '+' | '-' | '*' | '!'
<cut> ::= '!'
<false> ::= 'false'
Outline - Definite Clause Grammars
Definite clause grammars
Parsing
Translating grammars into logic programs
Sterling and Shapiro ch. 19 (except 19.2)
Nilsson and Maluszynski ch 10.4,10.5
SICStus Prolog Manual
Definite Clause Grammars
(DCGs) special syntax for language specifications.
The system automatically compiles DCG into a Prolog
clause.
DCGs are a generalisation of CFGs.
Definite Clause Grammars
Formalism
<N, T, P>
N - a possibly infinite set of atoms (non-terminals)
T - a possibly infinite set or terms (terminals)
P is in N x (N U T)* - a finite set of production rules
N and T are disjoint
Definite Clause Grammars
Syntax
terminals are enclosed by list-brackets;
nonterminals are written as ordinary compound terms or
constants;
',' separates terminals and nonterminals in the right-hand
side;
'-->' separates nonterminal to the left from terminals and
nonterminals in the right-hand side;
extra conditions, in the form of Prolog procedure calls,
may be included in the right-hand side of a grammar
rule. Such procedure calls are written enclosed in '{ }'
brackets;
the empty string is denoted by the empty list []
Definite Clause Grammars
Example
sentence --> noun_phrase, verb_phrase.
noun_phrase --> [the], noun.
verb_phrase --> [runs].
noun --> [rabbit].
noun --> [engine].
?- sentence(X,A).
Yes
X=[the, rabbit, runs], A=[]
X=[the, engine, runs], A=[]
WHY? see the next page.
Definite Clause Grammars
Compilation of DCG's into Prolog
p(A1, …, An) --> T1, .., Tm
is translated into the clause:
p(A1, …, An, X0, Xm) :- T1', …, Tm',
where each Ti' is of the form:
q(S1,…Sn, Xi-1, Xi) if Ti is of the form q(S1, …, Sn)
'C'(Xi-1, X, Xi)
if Ti is of the form [X]
T, Xi-1 = Xi
if Ti is of the form {T}
Xi-1 = Xi
if Ti is of the form []
and X1,…Xm are distinct variables
Definite Clause Grammars
Example
translated to:
expr(X) -->
term(Y),
[+],
expr(Z),
{X is Y + Z}.
NB:
sentence --> noun, verb.
expr(X, X0, X4) :term(Y, X0, X1),
'C'(X1, +, X2),
expr(Z, X2, X3),
X is Y+Z, X3 = X4.
sentence(B,C) :noun(B,A),
verb(A,C).
Definite Clause Grammars
Example grammar
<expr> ::= <term> + <expr>
<expr> ::= <term> - <expr>
<expr> ::= <term>
<term> ::= <factor> * <term>
<term> ::= <factor> / <term>
<term> ::= <factor>
<factor> ::= 0|1|2|….
Definite Clause Grammars
Example grammar as DCG (cont.)
expr(X) --> term(Y), [+], expr(Z), { X is Y + Z}.
expr(X) --> term(Y), [-], expr(Z), { X is Y-Z}.
expr(X) --> term(X).
term(X) --> factor(Y), [*], term(Z), { X is Y*Z}.
term(X) --> factor(Y), [/], term(Z), { X is Y/Z}.
term(X) --> factor(X).
factor(X) --> [X], {integer(X)}.
?- expr(X, [2, *, 2, +, 4, *, 4], []).
X = 20.
NB: avoid rules, which lead to "left hand recursion"
expr --> expr, [+], expr.
Definite Clause Grammars
Example
Write a DCG which accepts strings in the language an bm cn (n, m >= 0).
a) if n and m are fixed: n=1, m=2.
abc --> a, b, c.
a --> [a].
b --> [bb].
c --> [c].
test(String) :- abc(String, []).
?- test(X).
Yes. X=[a,bb,c]
?- test([a,bb,c]).
Yes.
Definite Clause Grammars
Example (cont.)
Write a DCG which accepts strings in an bm cn dm (n, m >= 0).
b)general case:
abcd(N,M) --> lit(a,N), lit(b,M), lit(c,N), lit(d,M).
lit(L, 0) --> [].
lit(L, I) --> [L], lit(L, I1), {I is I1+1}.
test(String) :- abcd(N, M, String, []).
?- test([a, b, b, c, d, d]).
Yes.
Interpreter as transition relation
An interpreter can be formulated as a transition relation p/2 from states to new states
S0 ---> S1 ---> S2 ---> ... ---> Sn-1 ---> Sn
p(S0,S1), p(S1,S2), … p(Sn-1, Sn)
Assignment as predicate
For instance
X:=E
would be modeled as a predicate
assign(X,E,StateIn,StateOut) :- ...
imperative program
state transition relation in a procedure in an imperative language
proc {Prog
Z := X
X := Y
Y := Z
}
Program as transition relation
prog(In,Out) :assign(Z,X,In,T1),
assign(X,Y,T1,T2),
assign(Y,Z,T2,Out).
Using grammar notation
The state transformation could be expressed using the grammar notation
prog --> assign(z,x),
assign(x,y),
assign(y,z).
Transition relations declaratively
State transition predicates are expressed in a perfectly declarative way
They still look very similar to the usage of assignment in an imperative language
It expresses the change of a thread of state objects logically
Ö4: Case study:
A compiler for three model
computers
Clocksin, ch. 9
F7: Program Transformation
Sterling and Shapiro ch. 13,16,18
Outline
Transformations of programs
fold/unfold - partial evaluation
Higher order programming
defining and using apply, map
Transformation rules for
programs
Since logic programs are defined as axioms, it is often
possible to define logically based transformation rules,
usable to improve programs.
Here are some examples of such rules:
Equality reordering
P :- … e1…q…e2…
-------------------------P :- e1,e2…q …
Motivation:
In a (pure) logic program the search will be made more efficient if information
about equalities is known as soon as possible to the search procedure.
Equality removal
P :- …,e1,…,e1,…,e1,…
-------------------------------P :- …,e1,…
Motivation:
In a (pure) logic program statements about equalities need not be repeated.
Unification is idempotent, that is the bindings of variables achived by applying a
unification Term1 = Term2 will not be extended by a second application of the
same unification.
Clause level transformation
A clause (j) is a specialized version of a clause (i)
if (Pi :- Q1,…,Qm) s = Pj :- Q1',…,Qm' in a program:
.
.
(i) Pi :- Q1,…,Qm
.
.
(j) Pj :- Q1',…,Qm'
.
.
When clause (j) is a specialized version of clause (i) it is redundant (in a pure logic program).
But: specialization can increase efficiency and also make an otherwise looping program
terminate. That is it might be advantageous to generate different versions for different forms of
the arguments and then apply other transformations to the resulting clauses.
Removal of failing clauses
Suppose that
(Pi :- Q1,…,Qm) s = Pj :- false
This specialized clause may be removed, since it can never contribute to a
solution.
Moreover, clauses Ck containing a goal that only matches the head Pj can
be converted into
Ck :- false
and the process can be repeated.
Removal of repeated goals
P :- …,Qn,…,Qn,…
------------------------P :- …,Qn,…
Motivation:
The reasoning concerning equalities above holds for goals in general in a pure logic
program. Note that we consider programs equivalent with respect to the Herbrand
model. The set of solutions found in proofs may not coincide exactly after such a
transformation, but the set of true consequences of a program indicated by the model
do coincide given an appropriate interpretation of the free variables in the proofs.
Reordering of goals
P :- …,Qi,…,Qk,…
-------------------------P :- …,Qk,…,Qi,…
Motivation:
This rule can for instance be used to push recursive calls towards the end of a clause.
Fold/Unfold
P :- Q.
Q :- B1.
Q :- B2.
|
|
|
P :- (B1 ; B2).
-----------> (unfolding)
<---------- (folding)
Applying the rules above in combination with rules like those given earlier can create more
efficient programs (note that variables must be renamed and the introduction of explicit
equalities might be necessary to keep the semantics of P).
Partial deduction/evaluation
Partial deduction - logical semantics is kept
Partial evaluation - operational semantics is kept (same number of
solutions, presented in the same order)
Partial evaluation is much harder than partial deduction since it should work even in the
presence of non-logical goals such as cut (!), I/O etc.
Stepwise enhancement
identify program skeletons that indicate the control flow.
enhance by adding operations (S&S chap 13)
list([X|Xs]) :- list(Xs).
list([]).
sumlist([X|Xs],S) :sumlist(Xs,S0), S is S0+X.
sumlist([],0).
Stepwise enhancement (cont)
length([_|Xs],L) :length(Xs,L0), L is L0+1.
length([],0).
sum_length_list([X|Xs],S,L) :sum_length_list(Xs,S0,L0),
S is S0+X, L is L0+1.
sum_length_list([],0,0).
Functions
Functions are deterministic relations.
There is one unique value in the output
domain for each input tuple.
A function f: Term* -> Term can for instance
be encoded as a definition in a logic program
as f(X1,…,Xn,Y) with a unique output Y
for each input tuple X1,…,Xn.
The relation f/n corresponding to a function
with n-1 arguments is deterministic, that
is, when all arguments (except possibly the
one corresponding to the output value of the
function) are fully instantiated.
Higher order
Higher order functions are not directly
expressible since functions are not objects in
the first-order logical model of a program.
We will see later how such programming
techniques can be encoded using
metaprogramming techniques.
Higher order programming:apply
general form
apply(foo,X1...Xn) :- foo(X1...Xn).
Higher order programming: map
mapping a predicate Predname(In,Out)
to each element of a list
map_list([X|Xs],Predname,[Y|Ys]) :apply(Predname,X,Y),
map_list(Xs,Predname,Ys).
map_list([],_,[]).
All-solutions predicates
It might be useful to collect several solutions in a list. Prolog gives support
for this through some "higher-order" predicates.
father(sven,olle). father(sven,lisa).
father(bengt,lisa). father(bengt,sven).
children(X,Kids) :findall(Kid, father(X,Kid),Kids).
The query
?- children(bengt,Kids).
gives
Kids=[lisa,sven]
All-solutions predicates (cont)
father(sven,olle). father(sven,lisa).
father(bengt,lisa). father(bengt,sven).
the query
?- findall(F, father(F,Kid),Fathers).
gives
Fathers=[sven,sven,bengt,bengt]
All-solutions predicates (cont)
Instead of a single solution collecting all fathers to some child we might
want a separate solution for each child. There is another set precidate for
this.
father(sven,olle). father(sven,lisa).
father(bengt,lisa). father(bengt,sven).
?- bagof(F, father(F,Kid),Fathers).
Kid=lisa
Fathers=[sven,bengt]
Kid=sven
Fathers=[bengt]
Kid=olle
Fathers=[sven]
All-solutions predicates (cont)
It is often sensible to present sorted lists of unique solutions. This is
achieved by setof.
father(sven,olle). father(sven,lisa).
father(bengt,lisa). father(bengt,sven).
?- setof(F, father(F,Kid),Fathers).
Kid=lisa
Fathers=[bengt,sven]
Kid=sven
Fathers=[bengt]
Kid=olle
Fathers=[sven]
Ö5: Metaprogramming,
Expert Systems
Sterling and Shapiro ch. 10,17 (not 17.5),19.2,22
Nilsson and Maluszynski ch. 8,9
Outline - Metaprogramming
What is a metastatement?
Metalogic predicates (built in)
solve, augmenting solve
Iterative deepening
Mixing object and metalevel programming
Support for dynamically changing knowledge bases
What is a metastatement?
A metastatement is a statement about statements
Stockholm is a nine-letter word.
'X+1-Y' has the size five.
'X+1-Y' contains two variables.
This statement is true.
This statement is false.
'P :- Q1...Qn.' is a clause.
This is a metastatement.
Meta-logic
Metalogic refers to reasoning about a formalization of some (other) logical
system.
If the metalogic deals with itself it is called circular or metacircular.
In logic programming metalogic has two meanings:
1.using logical inference rules expressed as axioms with a meta-interpreter.
2.expressing properties of the proof procedure.
For the latter case the term "meta-logical predicate" is used, for instance in the
SICStus manual.
Ground representation of facts
and rules
The logical approach to metaprogramming requires a clear division between object
level and meta-level.
Formulas are represented as ground facts, where in particular variables on the
object level are represented as constants on the meta-level
Each constant of the object language is represented by a unique constant of the
meta-language
Each variable of the object language is represented by a unique constant of the
meta-language
Each n-ary functor of the object language is represented by a unique n-ary functor
of the meta-language
Each n-ary predicate symbol of the object language is represented by a unique nary functor of the meta-language
Each connective of the object language is represented by a unique functor of the
meta-language (with corresponding arity)
Ground representation
clause(if(list(x),equals(x,[]))).
clause(if(list(x),and(equals(x,cons(x,h,t)),list(t)))).
clause(if(p(x),true)).
clause(if(p(x),and(q(x,a),p(b)))).
SLD-resolution rule
The SLD-resolution rule (p. 43 Nilsson & Maluszynski)
<- A1,..,A(i-1),Ai,A(i+1),...,Am B0 <- B1,...,Bn
----------------------------------------------------------
<- (A1,...,A(i-1),B1,...,Bn,A(i+1),...,Am)s
The SLD-rule encoded as a relation
step(Goal,NewGoal) :select(Goal,Left,Selected,Right),
clause(C),
rename(C,Goal,Head,Body),
unify(Head,Selected,Mgu),
combine(Left,Body,Right,TmpGoal),
apply(Mgu,TmpGoal,NewGoal).
The SLD-rule encoded as a relation (cont)
select/4 describes the relation between a goal and the selected subgoals
clause/1 describes the property of being a clause in the object language
rename/4 describes the relation between four formulas such that two are
uniquely renamed variants of the other two
unify/3 describes the relation between two atoms and their mgu
combine/4 describes the relation between a goal and three conjunctions
apply/3 describes the relation between a substitution and two goals
The SLD-rule encoded as a relation (cont)
derivation(G,G).
derivation(G0,G2) :step(G0,G1),
derivation(G1,G2).
Why self-interpreters?
flexibility
alternative search strategies
debugging
programs that change during their execution
collecting the actual proof of a satisfied goal (explanation)
non-standard logics: fuzzy logic, non-monotonic logic, modal logic
program transformation, program verification, program synthesis
Non-ground representation
Efficiency of the previous representation is low
-> Use object language variables for meta-level also
Seems straightforward, but mixing object level and meta-level has important
semantic consequences.
Consider this representation.
(with 'if' and 'and' as infix operators):
for facts: ax(Fact if true).
for rules: ax(Head if Body).
Metainterpreter for pure Prolog
using the above representation
solve(true).
solve(P) :- ax(P if Q), solve(Q).
solve(P and Q) :- solve(P), solve(Q).
In order to extend the meta-interpreter to handle also non-logical features of Prolog a different
interpreter must be written. For instance ! (cut) is hard to handle.
Metainterpreter generating a
proof
Augmented metainterpreter
solve(true,true).
solve(P,if(P,ax(P if Q),Qt)) :- ax(P if Q), solve(Q,Qt).
solve(P and Q,and(Pt,Qt)) :- solve(P,Pt), solve(Q,Qt).
This metainterpreter generates a proof .
Depth-bounded pure Prolog
An augmented meta-interpreter for depth-bounded search pure Prolog
solve(true,N) :- N>=0.
solve(P,N) :- N>0, ax(P if Q), N1 is N-1, solve(Q,N1).
solve(P and Q,N) :- solve(P,N), solve(Q,N).
This meta-interpreter finds a proof with a search tree depth of at most N levels.
depth(0).
depth(N) :- depth(N1), N is N1+1.
solve(G) :- depth(N), solve(G,N).
solve/1 tries at gradually deeper levels of the tree, iterative deepening.
Meta-logic
built-in predicates that perform operations that
require reasoning about:
the current instantiation of terms;
decomposing terms into their constituents.
Instantiation checking:
var(X)
nonvar(X)
ground(X)
- checks that X is uninstantiated variable (not a
structure)
- opposite to var/1.
- checks that X is completely instantiated
Meta-logic
Example.
?- var(X), X=1
?- X=1, var(X).
?- nonvar(father(X,Y)).
-Yes, X=1.
-No.
-Yes.
Define the predicate plus/3 which uses built-in arithmetics and
performs plus and minus.
plus(X, Y, Z):nonvar(X), nonvar(Y), Z is X+Y.
plus(X, Y, Z):nonvar(X), nonvar(Z), Y is Z-X.
plus(X, Y, Z):nonvar(Y), nonvar(Z), X is Z-Y.
Meta-logic
Type checking:
integer(X)
- X is instantiated to an integer
float(X)
number(X)
atom(X)
- X is instantiated to a float
- X is instantiated to a number
- X is instantiated to an atom (non-variable
term of arity 0, other than a number)
- X is instantiated atom or number
- X is uninstantiated or instantiated to an
atom or number
- X is instantiated to a term of arity > 0, i.e.
a list or a structure
atomic(X)
simple(X)
compound(X)
Meta-logic, decomposing terms
functor(Term, FunctorName, Arity)
Term has functor FunctorName and arity Arity
?- functor(father(erik, jonas), father, 2).
- Yes
?- functor(father(erik, jonas), F, A).
- F=father, A=2
?- functor(Term, father, 2).
- Term=father(_A, _B)
?- functor(Term, father, N).
- instantiation error
Meta-logic, decomposing terms
arg(N, Term, Argument)
the Nth argument of a compound term Term is Argument
?- arg(1, father(erik, jonas), Arg).
?- arg(2, father(erik, X), jonas).
- Arg=erik.
- X = jonas.
?- arg(N, father(erik, jonas), jonas).
- instantiation error.
?- arg(2, Y, jonas).
- instantiation error.
?- arg(1, father(X, Y), Z).
- Z = X.
?- arg(3, father(X, Y), Z).
- No.
Meta-logic, decomposing terms
Term =.. List
(=.. is called univ)
List is a list whose head is the atom corresponding to the principal functor
of Term, and whose tail is a list of the arguments of Term.
?- father(person(erik,A,b),person(jonas,X,Y)) =.. List.
- List=[father, person(erik,A,B), person(jonas, X,Y)].
?- Term =.. [father, erik, X].
- Term=father(erik,X).
?- father(erik, jonas) =.. [father, erik, jonas].
- Yes.
Meta-logic
Define the predicate subterm(Sub,Term), for checking if Sub is subterm
of Term
subterm(T, T).
subterm(S, T) :compound(T),
functor(T, F, N),
subterm(N, S, T).
subterm(N, S, T) :N > 1, N1 is N-1,
subterm(N1, S, T).
subterm(N, S, T) :N > 0, arg(N, T, Arg),
subterm(S, Arg).
Meta-logic
Define the predicate subterm(Sub,Term) for checking if Sub is
subterm of Term
subterm(T, T).
subterm(S, T) :compound(T),
T =.. [F|Args],
subtermList(S, Args).
subtermList(S, [Arg|Args]) :subterm(S, Arg).
subtermList(S, [Arg|Args]) :subtermList(S, Args).
Meta-logic
define =.. using functor/3 and arg/3
a) Term T is given
T =.. [F|Args] :functor(T, F, N),
args(T, Args, 0, N).
args(_, [], N, N).
args(T, [Arg|Args], I, N) :I < N,
I1 is I+1,
arg(I1, T, Arg),
args(T, Args, I1, N).
Meta-logic
define =.. using functor/3 and arg/3
b) List [F|Args] is given
T =.. [F|Args] :length(Args, N),
functor(T, F, N),
args(Args, T, 1).
args([], _, _).
args([Arg|Args], T, N) :arg(N, T, Arg),
N1 is N+1,
args(Args, T, N1).
Support for dynamic knowledge bases
assert(Clause)
clause(Head, Body)
retract(Clause)
- Clause is added to the program
- clause with head Head and body Body
- Clause is erased from the program
For example,
member(X, [X|Ys]).
member(X, [Y|Ys]) :- member(X,Ys).
?- clause(member(H1, H2), B).
H1=X, H2=[X|Ys], B= true;
H1=X, H2=[Y|Ys], B=member(X,Ys).
Expert systems
Expert System =
Knowledge-base (KB)
+ Inference engine (IE)
+ User Interface (UI)
Expert System Shell = (IE) + (UI)
Expert systems:
Production rules
knowledge base often consist of a set
of production rules of the form
IF A1 AND(OR) A2 ….
THEN C1 AND C2 …
Expert systems:
forward/backward chaining
forward-chaining: start from assumptions
(axioms) and find conclusions
backward-chaining: start from conclusions
(hypothesis) and look for supporting
assumptions
Expert systems in Prolog
in PROLOG
knowledge base = set of clauses
inference engine = SLD-resolution
SLD-resolution is a backward chaining proof procedure
Uncertainty
Expert systems might contain rules and/or
facts which hold with some degree of
uncertainty
IF it is summer and there are no clouds
THEN the temperature is above 20 degrees C
CERTAINTY 80%
Expert systems, forward/backward
Example
Develop an expert system, which finds a name of a fruit when some fruit characteristics
such as shape, diameter, surface , colour and the number of seeds are given. Gonzalez:
"The engineering of knowledge-based systems." (p91).
Rule 1:
IF Shape=long and Colour=green or yellow
THEN Fruit=banana
fruit(Name, Shape, Diameter, Surface, Colour, FruitClass, SeedCount, SeedClass):Shape == long,
(Colour == green; Colour == yellow),
Name = banana.
Rule 2:
IF Shape=round or oblong and Diameter > 4 inches
THEN FruitClass=vine
fruit(Name, Shape, Diameter, Surface, Colour, FruitClass, SeedCount, SeedClass):var(FruitClass),
(Shape == round; Shape ==oblong),
integer(Diameter),
Diameter>4,
FruitClass= vine,
fruit(Name, Shape, Diameter, Surface, Colour, FruitClass, SeedCount, SeedClass).
Expert systems, forward/backward
Rule 4:
IF SeedCount=1
THEN SeedClass=stonefruit
fruit(Name, Shape, Diameter, Surface, Colour, FruitClass, SeedCount, SeedClass)):var(SeedClass),
integer(SeedCount), SeedCount =:= 1,
SeedClass = stonefruit,
fruit(Name, Shape, Diameter, Surface, Colour, FruitClass, SeedCount, SeedClass).
Rule 5:
IF Seedcount>1
THEN SeedClass=multiple
fruit(Name, Shape, Diameter, Surface, Colour, FruitClass, SeedCount, SeedClass):var(SeedClass),
integer(SeedCount), SeedCount > 1,
SeedClass = multiple,
fruit(Name, Shape, Diameter, Surface, Colour, FruitClass, SeedCount, SeedClass).
Rule 11:
IF FruitClass=tree and Colour=red and SeedClass=stonefruit
THEN Fruit=cherry
fruit(Name, Shape, Diameter, Surface, Colour, FruitClass, SeedCount, SeedClass):FruitClass == tree,
Colour == red,
SeedClass == stonefruit,
Name = cherry.
?- fruit(Name, round, 3, Surface, red, FruitClass, 1, SeedClass). Yes. Name = cherry.
User interface - dialogue
An expert system is often not automatic
=> user interaction guides the search
graphical user interface
dialogue
Explanation facilities
Expert systems should be able to explain its
conclusions to different people (experts,
programmers, users)
- How did you come to your conclusion?
- Why does A follow from B ?
Knowledge acquisition
An expert system should allow incremental
updates of the knowledge base and rule
base
knowledge can be aquired through dialogue
with an expert, or through analysis of the
system
Rule base as Prolog clauses
IF A1 AND (A2 OR A3)
THEN C1 AND C2
C1 :- A1, (A2 ; A3).
C2 :- A1, (A2 ; A3).
Taxonomy of a car-engine
car
fuel system
fuel pump
fuel
electric
system
ignition
starting
motor
spark
plugs
fuse
battery
needs(car,fuel_system). ....needs(electric_system,fuse).
Expert System: diagnosis
IF Y is a necessary component for X and Y is
malfunctioning THEN X is also
malfunctioning
IF X exhibits a fault-symptom Z THEN either
X is malfunctioning or there exists another
malfunctioning component which is
necessary for X
Expert system: diagnosis
X has an indirect fault if there exists a
component which is necessary for X and
which malfunctions
malfunctions(X) :needs(X,Y), malfunctions(Y).
malfunctions(X) :symptom(Y,X), not indirect(X).
indirect(X) :needs(X,Y), malfunctions(Y).
Expert systems: abduction
The knowledge base is usually incomplete
symptoms are not known, but need to be
established by asking questions in a
dialogue
KB + cause |- symptom
Finding cause is named abduction
self-interpreter generating a proof
solve(true,true).
solve(P,proof(P,Qt)) :kb(P if Q), solve(Q,Qt).
solve(P and Q,and(Pt,Qt)) :solve(P,Pt), solve(Q,Qt).
This meta-interpreter generates a proof .
query-the-user
solve(true).
solve(P and Q) :- solve(P), solve(Q).
solve(symptom(X,Y)) :- confirm(X,Y).
solve(P) :- kb(P if Q), solve(Q).
confirm(X,Y) :write('Is the '),
write(Y), tab(1),write(X), write('? '),
read(yes).
knowledge base
kb(malfunctions(X) if
possible_fault(Y,X) and symptom(Y,X)).
kb(possible_fault(flat,tyre) if true).
:- solve(malfunctions(X)).
Is the tyre flat?
>yes
X=tyre
(anything but 'yes' as answer makes the query fail)
F8: Case study:
Support for reasoning about
electronic circuits
Clocksin, ch. 7,8
Ö6: Constraint Logic Programming
SICStus Prolog Manual
Nilsson and Maluszynski ch.14
Course compendium
This lecture is given by Christian Schulte.
The written material is based on slides from
Per Kreuger SICS and Christian Schulte KTH/ICT/ECS
Significance
Constraint programming identified as a
strategic direction in computer science
research
[ACM Computing Surveys, December 1996]
This Talk…
…concerned with constraints for
solving combinatorial problems
designed as basic tutorial
Application Areas
Timetabling
Scheduling
Crew rostering
Resource allocation
Workflow planning and optimization
Gate allocation at airports
Sports-event scheduling
Railroad: track allocation, train allocation, schedules
Automatic composition of music
Genome sequencing
Frequency allocation
…
Techniques
Artificial intelligence
Operations research
Algorithms
Programming languages
Overview
Basics
modeling: variables and constraints
solving: propagation, distribution, search
Solving realistic problems: scheduling
modeling
solving
Constraint programming
research
systems
Case studies
instruction scheduling for compiler
bus scheduling for real-time system
Conclusion
Constraint Programming (CP)
based on the idea of an abstract space of
statements or conditions, a
constraint space
xy
y>3
x{4,5} y{4,5}
Constraint Programming (CP)
Some such statements can be regarded as
completely determined:
E.g.
“ A given train trip will leave from Avesta
15.05 on Thursday, October 10th 2002.”
Constraint Programming (CP)
Other statements are less exact with respect to
e.g. the resources some task will need:
E.g.
“OVAKO Steel needs to transport between
320 and 280 kilotons of steel from Hofors
and Hellefors to Malmö next year.”
Constraint Programming (CP)
Such statements can be represented as
constraints in a constraint programming
system.
These constitute conditions on the values that
variables may take (i.e. a form of type).
Constraint Programming (CP)
It is nontrivial to determine e.g. how the two
above mentioned propositions would be
related in some model of a planning
problem.
Sometimes it is possible to determine some
form of “consistency” of such propositions.
Constraint Programming (CP)
Enumeration
It is also possible (in principle) to compute
one or more witnesses (a consistent
assignment of values to all the variables) of
such a system of conditions.
This is called to enumerate the constraint
space and generally involves search.
Constraint Programming (CP)
If more than one witness is computed they can
be compared with respect to various cost
measures.
To determine the (in some sense) best
assignment is modeled as an optimization
problem in the constraint system.
Constraint propagation
Much of the search in the enumeration of a
constraint space for a given problem can
generally be eliminated by a technique
called constraint propagation.
x in {2,…,5} & y in {3,…,9}
2x+3=y  x in {2,3} & y in {7,…,9}
2x+3=y & y<9  x<3  x=2  y=7
Constraint propagation
Much of the search in the enumeration of a
constraint space for a given problem can generally
be eliminated by a technique called constraint
propagation.
Each (non primitive constraint) can be regarded as a
temporarily suspended computation (of the values
of the involved variables) that can be made to
interact with other such suspended/reinvoked
computations.
Constraint propagation
Computations are suspended when
information needed to determine a value is
missing but is rescheduled as soon as that
information becomes available.
One can view the constraint programming
system as a pool of concurrent processes
that communicate, interact and synchronize
through shared variables.
Constraint propagation
Propagation is normally not sufficient to
completely determine the values of the
involved variables.
To achieve this we need to combine
propagation with enumeration in such a
way that values chosen during search
trigger further propagation which in turn
guides the continued search, etc.
Constraint Modelling (CP)
The computational complexity of the task to
find a solution to a given problem depends
to a large extent on the expressive power of
the language used to formalize the problem.
To formulate a mathematical model of some
real process is generally difficult. It requires
a thorough understanding of both the
problem domain and the methods employed
to solve the problem.
Constraint Programming (CP)
To some extent this is still more of a craft than
a science.
A large body of typical problems with
standard models have been identified.
Early attempts to develop a methodology has
started to give results.
Finite Domains
The constraint programming systems that have
been most actively developed the last ten
years are those that built on finite domains.
In such a system each variable can take on
values from a finite set of discrete values.
This type of variable is natural to use to model
discrete entities such as the number of engines
or staff that have been allocated to a given
task.
Continuous Domains
Finite domain constraints are, however,
unnecessarily restrictive when the modeling
concerns values that can be assumed to vary
over continuous domains (with an infinite
number of possible values), for instance
time.
Comparison with Operations
Research ---OR
Techniques from Operations Research, e.g. linear
programming (LP) and integer programming
(IP) efficiently handles models where:
1. Most of the variables are continuous.
2. The model can be relatively directly expressed
as a set of simple (linear) equations and
inequalities.
3. A simple (linear) and well defined cost function
captures well the “goodness” of different
solutions to a given problem.
Comparison with Operations
Research (2)
The techniques that have been developed in
constraint programming, using finite domains,
work well also when:
1. A majority of the variables model naturally
discrete entities.
2. The cost function is hard to determine.
3. The model contains complicated (for instance
non-linear) conditions.
In this way these two classes of techniques can be
said to complement each other.
Global constraints
The first type of constraint that was studied
in constraint programming was constraints
that limits one or relate two variables, e.g.
constraints like:
<
>
=<
>=
=
/=
Global constraints
In contrast to these simple binary constraints the
focus has in recent years more and more been on
complex constraints between an unlimited
number of variables.
E.g. constraints:
- relating variables with the value of a linear sum
- maintaining pairwise disequality of an
arbitrarily large set of variables
- implementing various scheduling, matching,
packing and/or placement mechanisms
Global constraints (2)
The expression “global constraints” for this
type of constraints was introduced in
{BC94} and refers to arguments that can be
made over a multitude of variables related
with a non-binary condition.
Global constraints (2)
Global constraints can in principle often be
encoded in terms of a set of simpler binary
constraints which semantically have the
same meaning.
This is rarely practical, however, since an
efficient solution can seldom be achieved
by only considering the variables pairwise.
Global constraints (2)
Global constraints constitute abstractions of
more complicated properties of problems
and enables computations on a more
detailed model
Algorithms as constraint
abstractions
Often methods from operations analysis,
matching theory or graph algorithms can be
integrated into a constraint programming
system as global constraints.
This is an active and very promising
research area in constraint
programming{bel00}.
An example: Getting Started
A toy problem…
Modeling
Solving: propagation and search
Send More Money (SMM)
Find distinct digits for letters, such that
SEND
+ MORE
= MONEY
Constraint Model for SMM
Variables:
S,E,N,D,M,O,R,Y  {0,…,9}
Constraints:
distinct(S,E,N,D,M,O,R,Y)
1000×S+100×E+10×N+D
+
1000×M+100×O+10×R+E
= 10000×M+1000×O+100×N+10×E+Y
S0
M0
Solution for SMM
Find values for variables such that
all constraints satisfied
Finding a Solution
Enumerate assignments: poor!
Constraint programming
compute with possible values
prune inconsistent values
constraint propagation
search
distribute:
explore:
define search tree
explore for solution
Some Concepts
Constraint store
Basic constraint
Propagator
Non-basic constraint
Constraint propagation
Constraint Store
finite domain constraints
x{3,4,5} y{3,4,5}
Stores basic constraints
map variables to possible values
Domains: finite sets, real intervals, trees, …
Propagators
Implement non-basic constraints
distinct(x1,…,xn)
x + 2×y = z
Propagators
xy
y>3
x{3,4,5} y{3,4,5}
Amplify store by constraint propagation
Propagators
xy
y>3
x{3,4,5} y{4,5}
Amplify store by constraint propagation
Propagators
xy
y>3
x{3,4,5} y{4,5}
Amplify store by constraint propagation
Propagators
xy
y>3
x{4,5} y{4,5}
Amplify store by constraint propagation
Propagators
xy
x{4,5} y{4,5}
Amplify store by constraint propagation
Disappear when entailed
no more propagation possible
Constraint Space
xy
y>3
x{4,5} y{4,5}
Store with connected propagators
Propagation for SMM
Results in store
S=9
M=1
E{4,…,7} N{5,…,8} D{2,…,8}
O=0
R{2,…,8} Y{2,…,8}
Propagation alone not sufficient!
create simpler sub-problems
distribution
Distribution
xy
x{4,5} y{4,5}
x=4
xy
y>3
y>3
x{4} y{4}
x4
xy
y>3
x{5} y{5}
Yields spaces with additional constraints
Enables further constraint propagation
Distribution Strategy
Pick variable x with at least two values
Pick value n from domain of x
Distribute with
x=n
and
xn
Part of model
Search
Iterate propagation and distribution
Orthogonal: distribution  exploration
Nodes:
 Distributable  Failed
 Succeeded
SMM: Solution
SEND
+ MORE
= MONEY
9567
+ 1085
= 10652
Heuristics for Distribution
(CLP jargong: Labelling)
Which variable?
least possible values (first-fail)
application dependent heuristic
Which value?
minimum, median, maximum
x=m
or
split with median m
x<m
or
xm
xm
In general: application specific
SMM: Solution With First-fail
SEND
+ MORE
= MONEY
9567
+ 1085
= 10652
Send Most Money (SMM++)
Find distinct digits for letters, such that
SEND
+ MOST
= MONEY
and MONEY maximal
Best Solution Search
Naïve approach:
compute all solutions
choose best
Branch-and-bound approach:
compute first solution
add “betterness” constraint to open nodes
next solution will be “better”
prunes search space
Also possible: restart strategy
Branch-and-bound Search
Find first solution
Branch-and-bound Search
Explore with additional constraint
Branch-and-bound Search
Explore with additional constraint
Branch-and-bound Search
Guarantees better solutions
Branch-and-bound Search
Guarantees better solutions
Branch-and-bound Search
Last solution best
Branch-and-bound Search
Proof of optimality
Modelling SMM++
Constraints and distribution as before
Order among solutions with constraints
so-far-best solution S,E,N,D,M,O,T,Y
current node
S,E,N,D,M,O,T,Y
constraint added
10000×M+1000×O+100×N+10×E+Y
<
10000×M+1000×O+100×N+10×E+Y
SMM++: Branch-and-bound
SEND
+ MOST
= MONEY
9782
+ 1094
= 10876
SMM++: All Solution Search
SEND
+ MOST
= MONEY
9782
+ 1094
= 10876
Summary
Modeling
variables with domain
constraints to state relations
distribution strategy
solution ordering
Solving
constraint propagation
constraint distribution
search tree exploration
The Art of Modeling
Avoid search, avoid search, avoid…
Techniques
increase propagation strength
stronger propagators
redundant propagators
remove symmetrical solutions
good distribution heuristics
smart search engines
Distribution  Exploration
Distribution
defines shape of search tree
Exploration
left-most depth-first
interactive, graphical
parallel
branch-and-bound [prunes tree]
Scheduling
Modeling
Propagation
Global constraints
Scheduling
Among the examples of global reasoning
that have successfully been introduced into
constraint programming systems are a
number of fundamental scheduling
mechanisms.
Scheduling
A scheduling problem consists of a number
of tasks with restrictions on start times, stop
times and task duration.
Often the tasks are partially ordered into
totally ordered sequences.
Such a totally ordered subset of tasks is
often called a job.
Scheduling
Each task uses one or more resources
during certain time intervals.
The so called job shop scheduling problem
is a classic and well studied case.
Scheduling
Resources can in general model widely
different type of entities.
For instance:
1. processing equipment in a production
process.
2. staff or vehicles in a transport net.
3. network resources such as routers and
transport links with limited capacity.
Scheduling
To arrange the tasks so that no limitations in
resources are violated is called to schedule
the tasks and it is in general a very difficult
(NP-complete) computational problem.
Nevertheless the many practical
applications for methods in this area make it
fairly well studied.
Scheduling: Given
Tasks
duration
resource
Precedence constraints
determine order among two tasks
Resource constraints
at most one task per resource
[disjunctive, non-preemptive scheduling]
Scheduling: Bridge Example
Scheduling: Solution
Start time for each task
All constraints satisfied
Earliest completion time
minimal make-span
Scheduling: Model
Variable for start-time of task a
start(a)
Precedence constraint: a before b
start(a) + dur(a)  start(b)
Propagating Precedence
a before b
a
b
start(a){0,…,7}
start(b){0,…,5}
Propagating Precedence
a before b
a
a
b
start(a){0,…,7}
start(b){0,…,5}
b
start(a){0,…,2}
start(b){3,…,5}
Scheduling: Model
Variable for start-time of task a
start(a)
Precedence constraint: a before b
start(a) + dur(a)  start(b)
Resource constraint:
a before b
or
b before a
Scheduling: Model
Variable for start-time of task a
start(a)
Precedence constraint: a before b
start(a) + dur(a)  start(b)
Resource constraint:
start(a) + dur(a)  start(b)
or
b before a
Scheduling: Model
Variable for start-time of task a
start(a)
Precedence constraint: a before b
start(a) + dur(a)  start(b)
Resource constraint:
start(a) + dur(a)  start(b)
or
start(b) + dur(b)  start(a)
Reified Constraints
Use control variable b{0,1}
c
 b=1
Propagate
c entailed 
c entailed 
b=1 entailed 
b=0 entailed 
propagate b=1
propagate b=0
propagate c
propagate c
not easy!
Reification for Disjunction
Reify each precedence
[start(a) + dur(a)  start(b)]  b0=1
and
[start(b) + dur(b)  start(a)]  b1=1
Model disjunction
b0 + b1  1
Model Is Too Naïve
Local view
individual task pairs
O(n2) propagators for n tasks
Global view
all tasks on resource
single propagator
smarter algorithms possible
Edge Finding
Find ordering among tasks (“edges”)
For each subset of tasks {a}B
assume: a before B
deduce information for
assume: B before a
deduce information for
join computed information
can be done in O(n2)
a and B
a and B
Scheduling Architecture
specification
compiler
Constraint programming
natural modeling
model
Scheduling Architecture
specification
compiler
Constraint programming
expressive modeling
model
Summary
Modeling
easy but not always efficient
constraint combinators (reification)
global constraints
smart heuristics
More on constraint-based scheduling
Baptiste, Le Pape, Nuijten. Constraint-based
Scheduling, Kluwer, 2001.
Why Does CP Matter?
Middleware for combining smart
algorithmic components
scheduling
graphs
flows
…
plus
essential extra constraints
Research in Constraints
Propagation algorithms
Search methods
Innovative applications
Programming and modeling languages
Hybrid methods
linear programming
local search
Constraints in Sweden
Swedish constraint network
SweConsNet
[founded May 2002]
http://www.dis.uu.se/~pierref/astra/SweConsNet/
CP Systems
Commercial
ILOG Solver
OPL Studio
SICStus Prolog
Eclipse
C++
Modeling
Prolog-based
Prolog-based
GNU Prolog
Mozart
CHOCO
Prolog-based
Oz
Claire
Free
Oz and Mozart
Constraint-based programming system
concurrent and distributed programming
combinatorial problem solving
and combinations: intelligent agents, …
Mozart implements Oz
concurrent constraint programming language
with: objects, functions, threads, …
Developed by Mozart Consortium
Saarland University, Germany
SICS/KTH, Sweden
Université catholique de Louvain, Belgium
Mozart Fact Sheet
Freely available at
www.mozart-oz.org
Many platforms supported
Unix
Windows
Mac OS
Active user community
Comes with extensive documentation
Many applications
Constraints in Mozart
Rich set of constraints
finite domains scheduling
finite sets
records
New propagators via C++ API
Search and combinators [unique]
programmable
concurrency-compatible
fully compositional
Book: Schulte, Programming Constraint Services. LNAI,
Springer 2002.g
Some Research Issues
Search methods
Architecture and implementation
Automatic selection of good propagator
domain versus bound
Challenging applications
Case Study
Instruction scheduling
Bus scheduling
Instruction Scheduling
Optimized object code by compiler
Minimum length instruction schedule
precedence
latency
resources
per basic block
Best paper CP 2001, Peter van Beek and Kent Wilken, Fast
Optimal Scheduling for Single-issue Processors with
Arbitrary Latencies, 2001.
Model
All issue times must be distinct
use single distinct constraint (as in SMM)
is resource constraint or unit duration
Latency constraints
precedence constraints (as before)
difference: duration  latency
Making It Work
Only propagate bounds information
relevant for distinct
Add redundant constraints
regions: additional structure in DAG
successor and predecessor constraints
[special case of edge-finding]
Results
Tested with gcc
SPEC95 FP
Large basic blocks
up to 1000 instructions
Optimally solved
less than 0.6% compile time increase
limited static improvement (< 0.1%)
better dynamic impact (loops)
Far better than ILP approach
Off-Line Scheduling of RealTime System
System with global clock
time-triggered real-time
processes
message exchange over shared bus
Infinite, periodic schedule
map to single fixed time window
repeat
Klaus Schild, Jörg Würtz. Scheduling of TimeTriggered Real-Time Systems, Constraints 5(4),
2000.
Model
Single resource: data bus
Maximal latencies:
messages valid for at most n time units
Infinite schedule
repeat finite schedule of given length
repetition does not violate constraints
Performance
Problem size
6,000,000 time units
3500 processes and messages
Model size
up to 10 million constraints
Run time
from 10 min to 2 hrs (200 MHz PPro)
Summary
Useful for
small components in software systems
large offline optimization
Widely applicable
in your area?
hardware design?
Conclusion
Constraint programming useful
easy modeling
open to new techniques
Constraints for Concurrency
Constraints
describe data structures
used for control
[as opposed to Prolog]
Logic variables as dataflow variables
unconstrained

constrained

synchronization is automatic
suspension
resumption
Well established idea
resumption condition is logical entailment
[Maher,87], ccp [Saraswat,90]
Problem Solving
Constraint domains
tree constraints (records, feature)
finite domains
finite sets
Programmable search and combinators
based on computation spaces
makes search compatible with concurrency
book:
Christian Schulte, Programming Constraint Services
LNAI 2302, Springer-Verlag, 2002
Descargar

Logic Programming 2G1530 5p