1 history automatic theorem proving robinson 1965 “programming in logic” kowalski, 1974...

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History

Automatic Theorem Proving Robinson 1965

“PROgramming in LOGic” Kowalski, 1974 Deduction as computation

“Prolog”, Colmerauer, 1973

Edinburgh Prolog, 1977 “Prolog 10”, David Warren Warren Abstract Machine

Clocksin&Mellish, 1981

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What kind of a thing is Prolog?

Very different from C, C++, Java, and other Procedural languages

Symbolic Programming Language

Logic Programming Language

Declarative Programming Language

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Prolog program

A Prolog program is defined by a set of Predicates

Each Predicate has a unique Functor and Arityparent(Parent, Child)

a predicate whose functor is “parent” with arity 2.

Each Predicate is defined by a sequence of Clauses

A Clause is either a Fact or a Rule

fact(…).

head(…) :- goal1(…), … , goalN(…).

Each argument to a predicate is a Term

A Term is either: Atomic, Variable, or Compound

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Atomic terms

Atomic terms can be either atoms or numbers

Atoms can be a sequence of alphanumeric (including ‘_’) characters starting with a lower case letter, or a sequence of any characters embedded in single quote marks:johnDowding symbsys139p ‘quoted atom’ non_quoted_atom

Numbers can be integers or floating point

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Variables

Variables start with a capital letter or _ character X Parent _Sibling _

Variables can stand in for arbitrary Prolog terms (including other variables)

Variables that occur with the same name in the same clause represent the same variable, except for

The anonymous variable “_”, which always represents a unique variable.

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Compound Terms

A compound term is a structure with a functor and N arguments.

functor(Arg1, Arg2, …, ArgN)

Each argument can be an arbitrary Prolog term

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Example: Successor Arithmetic

Simple arithmetic using compound terms to represent integers:0 0

1 s(0)

2 s(s(0))

3 s(s(s(0)))

4 s(s(s(s(0))))

.

.

.

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Structural Induction and Recursion

Inductive proofs have a base (or terminating) case and an inductive step.

Mathematical induction – over the space of integers

Structural induction – over subsets of the space of Prolog Terms.

Many Prolog predicates have this structure:is_number(0).

is_number(s(Number)) :- is_number(Number).

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BUILT-IN Predicate: consult/1

consult(FileName)

Load the Prolog program in FileName into memory.

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Factorial

factorial(+N, ?Factorial) fact(1) = 1 fact(N+1)= (N+1) * fact(N)

%factorial(+Number, ?Factorial)

factorial(s(0), s(0)).

factorial(s(Number), Factorial):-

factorial(Number, PartialFactorial),

mult(s(Number), PartialFactorial, Factorial).

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Prolog Execution Model/Prolog Debugger

CALL EXIT

FAIL REDOparent(james, john).parent(james, alan).parent(florence, john).parent(florence, alan).parent(alan, elizabeth).parent(alan, emily).

Goal = parent(P, john) parent(james, john)parent(florence, john)

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Execution Model (conjunctions)

parent(james, john).parent(james, alan).parent(florence, john).parent(florence, alan).parent(alan, elizabeth).parent(alan, emily).

female(emily).female(florence).female(elizabeth).

parent(Mother, john) female(Mother)parent(james, john) female(james)parent(florence, john) female(florence)

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Execution Model – mult(s(s(s(0))), s(s(0)), Result)

mult(0, Term, 0):- is_number(Term).mult(s(Term1), Term2, Product):- mult(Term1, Term2, Partial), add(Term2, Partial, Product).

add(0, Sum, Sum):- is_number(Sum).add(s(Addend1), Addend2, s(Sum)):- add(Addend1, Addend2, Sum).

mult(s(s(0)),s(s(0)), Prod)

mult(s(0), s(s(0)), Partial) add(s(s(0)), Partial, Prod)

add(s(s(0)), Prod, Result)

mult(0, Term, 0):- is_number(Term).mult(s(Term1), Term2, Product):- mult(Term1, Term2, Partial), add(Term2, Partial, Product).

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Prolog Debugger Demo

Built-In predicates: trace/0. notrace/0.

Debugger actions: ‘c’ or <cr> - creep ‘s’ – skip ‘l’ – leap ‘r’ – retry ‘f’ – fail ‘a’ - abort

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And-Or Trees

or

andand and

ororor

ororor or

andandandandand and and

orororor

oror

or

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And-Or Tree (cont.)

or

and and

mult(0, Term, 0):- is_number(Term).

mult(s(Term1), Term2, Product):- mult(Term1, Term2, Partial), add(Term2, Partial, Product).

or or or

is_number(Term)mult(Term1, Term2, Partial)

add(Term2,Partial,Product)

and and and and and and

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Linked Lists

Prolog allows a special syntax for lists: [a,b,c] is a list of 3 elements [] is a special atom indicating a list with 0 elements

Internally, Prolog lists are regular Prolog terms with the functor ‘.’ (so called “dotted pairs”)[a,b,c] = ‘.’(a, ‘.’(b, ‘.’(c, []))).The symbol | in a list indicates “rest of list”, or the term that is the 2nd argument of a dotted pair.[a,b,c] = [a|[b,c]].[Head|Tail] is a common expression for dividing a list into its first element (Head) and the rest of the list (Tail).

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Example: list/1

% list(?List)

list([]).

list([_Head|Tail]):-

list(Tail).

Since Prolog is untyped, we don’t have to know anything about Head except that it is a term.

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Example: member/2

% member(?Element, ?List)

member(Element, [Element|_Tail]).

member(Element, [_Head|Tail]):-

member(Element, Tail).

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Example: delete/3

% delete(+Element, +List, -NewList)

% delete/3 succeeds if NewList results from removing

% one occurrence of Element from List.

delete(Element, [Element|Tail], Tail).

delete(Element, [Head|Tail], [Head|NewTail]):-

delete(Element, Tail, NewTail).

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Example: append/3

% append(?List1, ?List2, ?List3)

% append/3 succeds if List3 contains all the elements of

% List1, followed by all the elements of List2.

append([], List2, List2).

append([Head|List1], List2, [Head|List3]):-

append(List1, List2, List3).

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Example: “naïve” reverse

% "naive reverse": nreverse(?List, ?ReversedList).

nreverse([], []).

nreverse([Head|Tail], ReversedList):-

nreverse(Tail, ReversedTail),

append(ReversedTail, [Head], ReversedList).

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“Pure Prolog” and non-logical built-ins

All the examples so far have been “pure Prolog” Contain no built-ins with non-logical side-effects

Prolog has many built-in predicates: Type checking of terms Arithmetic Control execution Input and output Modify the program during execution Perform aggregation operations

Use of non-logical built-in predicates usually effects the reversability of your program.

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Type-checking Built-in Predicates

var(X) – is true when X is an uninstantiated variable.

nonvar(X) – is true when X is not a variable.

atom(X) – is true when X is a symbolic constant.

number(X) – is true when X is a number

atomic(X) – is true when atom(X) or number(X).

compound(X) – is true when X is a compound term.

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Term constructor/selectors: functor/3, arg/3

functor(+Term, ?Functor, ?Arity) Find the Functor and Arity of Term

functor(?Term, +Functor, +Arity) Constructs a new Term with Functor and Arity

arg(+N, +Term, ?SubTerm) Unifies SubTerm with the Nth argument of Term

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Arithmetic: Built-In is/2

Arithmetic expressions are not normally evaluated in Prolog.Built-In infix operator is/2 evaluates it’s 2nd argument, and unifies the result with it’s 1st argument.| ?- X = 5 + 2.X = 5+2?yes| ?- X is 5 + 2.X = 7 ?yes

Any variables in the right-hand side of is/2 must be instantiated when it is evaluated.

Revisit operator and arithmetic at a later time

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Cut (!)

The ! Symbol (pronounced “cut”) modifies the execution of your program by committing to certain choices. That is, it removes choice points.

Easy to describe what it does, more difficult to get used to using it properly.

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Cut (cont.)

Head1 :- Goal1, Goal2, …, GoalN, !, …Head2 :- …Head3 :- ……HeadN :- …

Removes the choice point that allows Head2 …HeadNRemoves any choice points that may have been introduced in Goal1 … GoalN.We will discuss Cut in more detail later on.

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Example: delete_first/3

% delete_first(+Element, +List, -NewList)

% removes the 1st occurrence of Element in List.

delete(Element, [Element|Tail], Tail):-

!.

delete(Element, [Head|Tail], [Head|NewTail]):-

delete(Element, Tail, NewTail).

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Built-Ins: Term Comparison Operators

Unifies with Term1 = Term2 iff unifies Term1 and Term2 if they are unifiable

Does not unify with Term1 \= Term2, Term1 and Term2 are unchanged

Identical Term1 == Term2 iff are the same terms

Not identical Term1 \== Term2

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Built-ins: true/0 and fail/0.

true. Always succeeds.

fail. Always fails.

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Meta-Predicates: call/1, \+/1

Meta-predicates take a Goal as an argument, and execute it.

call/1 executes it’s one argument.

\+(Goal) succeeds if call(Goal) fails.

\+/1 could also be defined using cut and fail.

\+(Goal) :- call(Goal), !, fail.

\+(_Goal).

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More about cut!

Common to distinguish between red cuts and green cuts Red cuts change the solutions of a predicate Green cuts do not change the solutions, but effect the efficiency

Most of the cuts we have used so far are all red cuts

%delete_all(+Element, +List, -NewList)delete_all(_Element, [], []).delete_all(Element, [Element|List], NewList) :- !, delete_all(Element, List, NewList).delete_all(Element, [Head|List], [Head|NewList]) :- delete_all(Element, List, NewList).

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Green cuts

Green cuts can be used to avoid unproductive backtracking% identical(?Term1, ?Term2)identical(Var1, Var2):-

var(Var1), var(Var2),!, Var1 == Var2.

identical(Atomic1,Atomic2):- atomic(Atomic1), atomic(Atomic2),!, Atomic1 == Atomic2.

identical(Term1, Term2):-compound(Term1),compound(Term2),functor(Term1, Functor, Arity),functor(Term2, Functor, Arity),identical_helper(Arity, Term1, Term2).

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Advice on cuts

Dangerous, easy to misuse

Rules of thumb: Use sparingly Use with as narrow scope as possible

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intersect/3

%intersect(+Set1, +Set2, ?Intersection)

intersect([], _Set2, []).

intersect([Element|RestSet1], Set2, [Element|Intersection]):-

member(Element, Set2),

!,

intersect(RestSet1, Set2, Intersection).

intersect([_Element|RestSet1], Set2, Intersection):-

intersect(RestSet1, Set2, Intersection).

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Example: Input/Output

repeat/0 is a built-in predicate that will always resucceed

% classifing terms

classify_term :-

repeat,

write('What term should I classify? '),

nl,

read(Term),

process_term(Term),

Term == end_of_file.

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I/O Example (cont)

process_term(Atomic):-

atomic(Atomic),

!,

write(Atomic), write(' is atomic.'), nl.

process_term(Variable):-

var(Variable),

!,

write(Variable), write(' is a variable.'), nl.

process_term(Term):-

compound(Term),

write(Term), write(' is a compound term.‘), nl.

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assert/1, asserta/1, and assertz/1

Asserting facts (most common)assert(Fact)

Asserting rulesassert( (Head :- Body) ).

asserta/1 adds the new clause at the front of the predicate

assertz/1 adds the new clause at the end of the predicate

assert/1 leaves the order unspecified

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Built-In: retract/1

retract(Goal) removes the first clause that matches Goal.

On REDO, it will remove the next matching clause, if any.

Retract facts:retract(Fact)

Retract rules:retract( (Head :- Body) ).

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Built-in: retractall/1

retractall(Head) removes all facts and rules whose head matches.

Could be implemented with retract/1 as:

retractall(Head) :-

retract(Head),

fail.

retract(Head):-

retract( (Head :- _Body) ),

fail.

retractall(_Head).

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Built-In: abolish(Predicate/Arity)

abolish(Predicate/Arity) is almost the same as

retract(Predicate(Arg1, …, ArgN))

except that abolish/1 removes all knowledge about the predicate, where retractall/1 only removes the clauses of the predicate.

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Aggregation: findall/3.

findall/3 is a meta-predicate that collects values from multiple solutions to a Goal:

findall(Value, Goal, Values)

findall(Child, parent(james, Child), Children)

Prolog has other aggregation predicates setof/3 and bagof/3, but we’ll ignore them for now.

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findall/3 and assert/1

findall/3 and assert/1 both let you preserve information across failure.:- dynamic solutions/1.findall(Value, Goal, Solutions):-

retractall(solutions/1),assert(solutions([])),call(Goal),retract(solutions(S)),append(S, [Value], NextSolutions),assert(solutions(NextSolutions)),fail.

findall(_Value, Goal, Solutions):-solutions(Solutions).

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Special Syntax III: Operators

Convenience in writing terms

We’ve seem them all over already:union([Element|RestSet1], Set2, [Element|SetUnion]):-

union(RestSet1, Set2, SetUnion),

\+ member(Element, SetUnion),

!.

This is just an easier way to write the term:‘:-’(union([Element|RestSet],Set2,[Element|SetUnion]),

‘,’(union(RestSet1,Set2,SetUnion),

‘,’(‘\+’(member(Element, SetUnion),

!)))

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Operators (cont)

Operators can come before their arguments (prefix) \+

Or between their arguments (infix) , + is <

Of after their arguments (postfix) Prolog doesn’t use any of these (yet)

The same Operator can be more than one type :-

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Precedence and Associativity

Operators also have precedence 5 * 2 + 3 = (5 * 2) + 3

Operators can be associative, or not,

Left associative or right associative

Explicit parenthesization can override defaults for associatiativity and precendence

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Built-in: current_op/3

current_op/3 gives the precedence and associativity of all current operators.

current_op(Precedence, Associativity, Operator)

where Precedence in an integer 1-1200

and Associativity is of fx or fy for prefix operators xf or yf for postfix operators xfx, xfy, yfx, yfy for infix operators

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Associativity

These atoms: fx, fy, xf, yf, xfx, xfy, yfx, yfy draw a “picture” of the associativity of the operator: The location of the f tells if the operator is prefix, infix,

or postfix. x means that the argument must be of lower precedence y means that the argument must be of equal or lower

precedence. A y on the left means the operator is left associative A y on the right means the operator is right associative

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Creating new operators

Built-in op/3 creates new operators

op(+Precedence, +Associativity, +Operator)

:- op(700, xfx, equals).

:- op(650, fx, $).

:- op(650, xf, cents).

$Dollars equals Cents cents :-

Cents is 100 * Dollars.