3. On Yacas programming


3.1 Example: implementing a formal grammar

To illustrate the use of rules, consider a theorem prover in a simple formal grammar. (The example is the "ABIN system" from the book: W. Robinson, Computers, minds and robots, Temple University Press, 1992. Warning: the book is about philosophy.)

Well-formed expressions consist of symbols A, B, I, N and are either

This defines a certain set of well-formed expressions (statements of the ABIN language); for example, NBII is a statement of the language but AB is not. The truth/falsehood interpretation of the ABIN language is the following. All well-formed expressions starting with NB are interpreted as true statements (they are "axioms" of the system). In addition, there is one deduction rule allowing one to prove "theorems":

Thus, NABIBI can be proved starting from the axiom NBI, but NANBB cannot be proved. The task at hand is to decide whether a given sequence of symbols is a provable statement of the ABIN language.

(The idea behind this interpretation is to assume that all B, BI etc. are some false statements that one could denote "B0", "B1" according to the number of "I" symbols; "N" is the logical Not and "A" is the logical And. Then the statement NABIB would mean "it is false that both B0 and B1 are true" and NANBB would mean "it is false that both B0 and negation of B0 are true". The NANBB statement is true in this interpretation but the deductive system of ABIN is too weak to obtain its proof.)


Implementation using predicates

The easiest way to model the ABIN language in Yacas is by using predicates. Our goal will be to define a predicate IsProvable(x) that will return True when x is a provable ABIN statement and False otherwise. We shall define IsProvable(x) recursively through several auxiliary predicates. Naturally, we would like to have a predicate to test well-formedness: IsExpr(x). It is necessary also to have predicates for B-expressions, N-expressions and A-expressions, as well as for axioms and theorems. We might implement expressions by lists of symbols, e.g. {"B", "I"} and begin to code by

IsExpr(x_IsList) <-- IsBExpr(x) Or
  IsNExpr(x) Or IsAExpr(x);
IsProvable(x_IsList) <-- IsAxiom(x) Or
  IsTheorem(x);
IsAxiom(x_IsList) <-- IsNExpr(x) And
  IsBExpr(Tail(x));

The definitions of IsBExpr(x) and IsNExpr(x) are simple recursion to express the rules 1 and 2 of the ABIN grammar. Note the use of Take to create a copy of a list (we'd better not modify the value of x in the body of the rule).

10 # IsBExpr({}) <-- False;
10 # IsBExpr({"B"}) <-- True;
20 # IsBExpr(x_IsList) <-- x[Length(x)]="I"
  And IsBExpr(Take(x, {1, Length(x)-1}));

10 # IsNExpr({}) <-- False;
20 # IsNExpr(x_IsList) <-- x[1] = "N" And
  IsExpr(Tail(x));

The predicate IsAExpr(x) is a little bit more complicated because our rule 3 requires to find two well-formed expressions that follow A. Also, for proving theorems we need to be able to extract the first of these expressions. With this in mind, we define another auxiliary function, FindTwoExprs(x), that returns the results of search for two well-formed expressions in the list x. The return value of this function will be a pair such as {True, 3} to indicate that two well-formed expressions were found, the first expression being of length 3. We shall use a For loop for this function:

FindTwoExprs(x_IsList) <-- [
 Local(iter, result);
 For( [ iter:=1; result:=False; ],
  iter < Length(x) And Not result,
    iter:=iter+1 )
   [
    result := IsExpr(Take(x, iter))
     And IsExpr(Take(x, {iter+1,
	   Length(x)}));
   ];
 {result, iter-1};
];

Now we can define the remaining predicates:

10 # IsAExpr(x_IsList)_(Length(x) <= 1)
  <-- False;
20 # IsAExpr(x_IsList) <-- x[1] = "A" And
  FindTwoExprs(Tail(x))[1];

IsTheorem(x_IsList) <-- If(IsNExpr(x) And
  IsAExpr(Tail(x)) And IsProvable(
    Concat({"N"}, Take(Tail(Tail(x)),
   FindTwoExprs(Tail(Tail(x)))[2]) ));

The ABIN language is now complete. Let us try some simple examples:

In> IsExpr({"A","B"});
Out> False;
In> IsExpr({"N","B","I"});
Out> True;
In> IsAxiom({"N","B","I"});
Out> True;
In> IsTheorem({"N","B","I"});
Out> False;
In> IsProvable({"N","B","I"});
Out> True;
In> IsProvable({"N","A","B","I","B"});
Out> True;

It is somewhat inconvenient to type long lists of characters. So we can create an interface function to convert atomic arguments to lists of characters, e.g. AtomToCharList(BII) will return {"B","I","I"} (provided that the symbol BII has not been given a definition). Then we define a function ABIN(x) to replace IsProvable.

AtomToCharList(x_IsAtom) <-- [
  Local(index, result);
  For( [ index:=Length(String(x));
     result:={}; ],
   index > 0, index:=index-1 )
   Push(result, StringMid'Get(index, 1,
     String(x)));
  result;
];
Holdarg(AtomToCharList, 1);
ABIN(x) := IsProvable(AtomToCharList(x));

In> AtomToCharList(NBII);
Out> {"N", "B","I","I"};
In> ABIN(NANBB);
Out> False;

It is easy to modify the predicates IsTheorem() and IsAxiom() so that they print the sequence of intermediate theorems and axioms used for deriving a particular theorem. The final version of the code is in the file examples/ABIN.ys. Now we can try to check a "complicated" theorem and see an outline of its proof:

In> ABIN(NAAABIIBIBNB);
Axiom {"NBII"}
Theorem NABIIBI derived
Theorem NAABIIBIB derived
Theorem NAAABIIBIBNB derived
Out> True;


3.2 Example: Using rules with special syntax operators creatively

Any Yacas function can be declared to have special syntax: in other words, it can be made into a prefix, infix, postfix, or bodied operator. In this section we shall see how prefix, infix, and postfix operators understood by Yacas can be adapted to a problem that seems to be far removed from algebra. Nevertheless it is instructive to understand how rewriting rules are used with special syntax operators.

Suppose we want to build a system that understands a simple set of English sentences and will be able to answer questions. For example, we would like to say "Tom had an apple and Jane gave 3 apples to Tom"; the system should understand that Tom has 4 apples now. In the usual LISP-based treatments of artificial intelligence, this problem would be illustrated with a cumbersome list syntax such as (had (Tom apple 1)) but we would like to use the power of the Yacas syntax and use plain English.

We shall create a set of rules that will "simplify" sentences to atoms such as True or False. As a side-effect, these "simplifications" will maintain a "knowledgebase" of information about all existing persons and objects.


The talking machine

The full source of this example is in the file examples/wordproblems.ys. In the next subsection we shall discuss the basic issues of the implementation. For now, here is an example session that shows what functionality we have in mind:

Unix>   yacas
True;
Numeric mode: "Internal"
To exit Yacas, enter  Exit(); or quit or Ctrl-c.
Type 'restart' to restart Yacas.
To see example commands, keep typing Example();
In> Load("wordproblems.ys")
Out> True;
In> Jitse and Ayal are persons;
OK, Jitse is a person.
OK,  Ayal is a person.
Out> {True,True};
In> apple is an object;
OK,  apple is an object.
Out> True;
In> there are many apples and pears;
Note: we already know that  apple  is an object
OK, we assume that the plural of " apple " is
  " apples ".
OK,  pear is an object.
OK, we assume that the plural of " pear " is
  " pears ".
Out> {True,True};
In> Serge had an apple;
OK,  Serge is a person.
OK,  Serge  has  1   apples  now.
Out> True;
In> Jitse had (10!) pears;
OK,  Jitse  has  3628800   pears  now.
Out> True;
In> Ayal had (2+3) apples and Serge had \
  2 pears;
OK,  Ayal  has  5   apples  now.
OK,  Serge  has  2   pears  now.
Out> {True,True};
In> Serge ate the apple;
OK,  Serge  has  no   apples  now.
Out> True;
In> Ayal ate a pear;// this should fail
Error:  Ayal  does not have enough
  pears  at this time.
Out> False;
In> Ayal gave an apple to Serge and \
  Serge gave a pear to Ayal;
OK,  Ayal  has  4   apples  now.
OK,  Serge  has  1   apples  now.
OK,  Serge  has  1   pears  now.
OK,  Ayal  has  1   pears  now.
Out> {True,True};
In> Ayal ate a pear;
OK,  Ayal  has  no   pears  now.
Out> True;
In> soup is an object and Ayal had \
  some soup;
OK,  soup is an object.
OK,  Ayal  has some  soup  now.
Out> {True,True};
In> Ayal gave soup to Serge and Serge \
  ate the soup;
OK,  Ayal  has  no   soup  now.
OK,  Serge  has some  soup  now.
OK,  Serge  has  no   soup  now.
Out> {True,True};
In> Serge has soup
Out> no;
In> Serge has apples
Out> 1;
In> Ayal has apples
Out> 4;
In> Serge has some soup
Out> False;
In> Serge has some apples
Out> True;
In> Ayal has some pears
Out> False;
In> Knowledge();
OK, this is what we know:
Persons: Jitse, Ayal, Serge
Object names: soup, pear, apple
Countable objects: pears, apples
Jitse has: 
 3628800  pears 

Ayal has: 
 4  apples 
 no  pears 
 no  soup 

Serge has: 
 1  apples 
 1  pears 
 no  soup 

Out> True;
In> software is an object
OK,  software is an object.
Out> True;
In> Ayal made some software
OK,  Ayal  has some  software  now.
Out> True;
In> Ayal gave some software to everyone
OK,  everyone is a person.
OK,  Ayal  still has some  software 
OK,  everyone  has some  software  now.
Out> True;
In> Ayal gave some software to Serge
OK,  Ayal  still has some  software 
OK,  Serge  has some  software  now.
Out> True;
In> Serge ate the software
OK,  Serge  has  no   software  now.
Out> True;

The string "OK" is printed when there is no error, "Note" when there is a warning, and "Error" on any inconsistencies in the described events. The special function Knowledge() prints everything the system currently knows.

Now we shall see how this system can be implemented in Yacas with very little difficulty.


Parsing sentences

A sentence such as "Mary had a lamb" should be parsed as a valid Yacas expression. Since this sentence contains more than one atom, it should be parsed as a function invocation, or else Yacas will simply give a syntax error when we type it in.

It is logical to declare "had" as an infix operator and "a" as a prefix operator quantifying lamb. In other words, "Mary had a lamb" should be parsed into had(Mary, a(lamb)). This is how we can do it:

In> [ Infix("had", 20); Prefix("a", 10); ]
Out> True;
In> FullForm(Mary had a lamb)
(had Mary (a lamb ))
Out> Mary had a lamb;
Now this sentence is parsed as a valid Yacas expression (although we have not yet defined any rules for the functions "a" and "had").

Note that we declared the precedence of the prefix operator "a" to be 10. We have in mind declaring another infix operator "and" and we would like quantifiers such as "a", "an", "the" to bind more tightly than other words.

Clearly, we need to plan the structure of all admissible sentences and declare all necessary auxiliary words as prefix, infix, or postfix operators. Here are the patterns for our admissible sentences:

"X is a person" -- this declares a person. Parsed: is(X, a(person))

"X and Y are persons" -- shorthand for the above. Parsed: are(and(X, Y), persons). "person" and "persons" are unevaluated atoms.

"A is an object" -- this tells the system that "A" can be manipulated. Parsed: is(A, an(object))

"there are many As" -- this tells the system that "A" can be counted (by default, objects are not considered countable entities, e.g. "milk" or "soup"). Parsed: are(there, many(As)). Here "As" is a single atom which will have to be stripped of the ending "s" to obtain its singular form.

"X ate N1 As", for example, Tom ate 3 apples -- parsed as ate(Tom, apples(3)). Since we cannot make the number 3 into an infix operator, we have to make apples into a postfix operator that will act on 3.

"X gave N As to Y" -- Here "N" is a number and "A" is the name of an object. Parsed as: gave(X, to(As(N), Y)). So to and gave are infix operators and to binds tighter than gave.

Sentences can be joined by "and", for example: "Tom gave Jane an apple and Jane ate 3 pears". This will be parsed as the infix operator "and" acting on both sentences which are parsed as above. So we need to make "and" of higher precedence than other operators, or else it would bind (apple and Jane) together.

"X made some A" -- note that if "A" is not countable, we cannot put a number so we need to write some which is again a prefix operator. made is an infix operator.

"X ate some A" -- the interpretation is that some A is still left after this, as opposed to "X ate the A" or "X ate A".

"X gave some A to Y" -- similarly, X still has some A left after this.

After each sentence, the system should know who has what at that time. Each sentence is parsed separately and should be completely interpreted, or "simplified".

All knowledge is maintained in the variable Knowledge which is an associative list with three entries:
Knowledge := {
	{"objects", {} },
	{"countable objects", {} },
	{"persons", {} }
};
The values under the keys "objects" and "countable objects" are lists of names of declared objects. The values of the "persons" key is a doubly nested associative list that specifies which objects each person has and how many. So, for example, Knowledge["persons"]["Tom"]["apples"] should give the number of apples Tom has now, or the atom Empty if he has none.


Declaring objects

Declaring persons is easy: we just create a new entry in the "persons" list. This can be done by an auxiliary routine DeclarePerson(). Note that after we have declared the words "is", "a" to be operators, we can just write the rule using them:

Infix("is", 20);
Prefix("a", 10);
_x is a person <-- DeclarePerson(x);
Here "person" will be left as an unevaluated atom and we shall never have any rules to replace it. Some other words such as "object", "objects" or "there" will also remain unevaluated atoms.

The operator "and" will group its operands into a list:

Infix("and", 30);
10 # x_IsList and _y <-- Concat(x, {y});
15 # _x and _y <-- Concat({x}, {y});
So expressions such as "Lisa and Anna and Maria" will be automatically transformed into {Lisa, Anna, Maria}. We shall adapt our rules to operate on lists of operands as well as on simple operands and that will automatically take care of sentences such as "there are many apples and ideas".

10 # there are many xs_IsList <--
  MapSingle("DeclareCountable", xs);
20 # there are many _xs <-- DeclareCountable(xs);

However, in the present system we cannot simultaneously parse "there are many apples and ideas" and "Isaac had an apple and an idea" because we have chosen had to bind tighter than and. We could in principle choose another set of precedences for these operators; this would allow some new sentences but at the same time disallow some sentences that are admissible with the current choice. Our purpose, however, is not to build a comprehensive system for parsing English sentences, but to illustrate the usage of syntax in Yacas.

Declaring objects is a little more tricky (the function DeclareCountable). For each countable object (introduced by the phrase "there are many ...s") we need to introduce a new postfix operator with a given name. This postfix operator will have to operate on a preceding number, so that a sentence such as "Mary had 3 lambs" will parse correctly.

If x were an unevaluated atom such as "lambs" which is passed to a function, how can we declare lambs to be a postfix operator within that function? The string representation of the new operator is String(x). But we cannot call Postfix(String(x)) because Postfix() does not evaluate its arguments (as of Yacas 1.0.49). Instead, we use the function UnList to build the expression Postfix(String(x)) with String(x) evaluated from a list {Postfix, String(x)}, and we use the function Eval() to evaluate the resulting expression (which would actually call Postfix()):
Eval(UnList({Postfix, String(x)} ));
We also need to declare a rulebase for the operator named String(x). We use MacroRuleBase for this:
MacroRuleBase(String(x), {n});

Finally, we would need to define a rule for "had" which can match expressions such as
_Person had n_IsNumber _Objects
where _Objects would be a pattern matcher for an unknown postfix operator such as lambs in our previous example. But we discover that it is impossible to write rules that match an unknown postfix operator. The syntax parser of Yacas cannot do this for us; so we should find a workaround. Let us define a rule for each object operator that will transform an expression such as {5 lambs} into a list {lambs, 5}. In this list, "lambs" will just remain an unevaluated atom.

Incidentally, the parser of Yacas does not allow to keep unevaluated atoms that are at the same time declared as prefix operators but it is okay to have infix or postfix operators.

A rule that we need for an operator named String(x) can be defined using MacroRule:
MacroRule(String(x), 1, 1, True) {x, n};
Now, after we declare "lambs" as an operator, the routine will define these rules, and anything on which "lambs" acts will be transformed into a list.
In> 5 lambs;
Out> {lambs, 5};
In> grilled lambs
Out> {lambs, grilled};

But what about the expression "many lambs"? In it, many is a prefix operator and lambs is a postfix operator. It turns out that for Yacas it is the prefix operator that is parsed first (and remember, we cannot have unevaluated atoms with the same name as a prefix operator!) so "many lambs" will be transformed into many(lambs) and not into an illegal expression {lambs, many}.


Implementing semantics

After implementing all the syntax, the semantics of these sentences is very easy to transform into rules. All sentences are either about how something exists, or about someone "having", "making", "eating", or "giving" certain objects. With the rules described so far, a complicated sentence such as
Ayal gave soup to Serge and Serge ate the soup
will be already parsed into function calls
{gave(Ayal, to(soup, Serge)), ate(Serge,
  {soup, 1}}
So now we only need to make sure that all this information is correctly entered into the knowledgebase and any inconsistencies (e.g. eating something you do not have) are flagged.

Here is the simplest rule: "giving" is implemented as a sequence of "eating" and "making".
10 # _x gave _obj to _y <-- [
	x ate obj;
	y made obj;
];

One more subtlety connected with the notion of "countable" vs. "uncountable" objects is that there are two different actions one can perform on an "uncountable" object such as "soup": one can eat (or give away) all of it or only some of it. This is implemented using the keyword "some" which is a prefix operator that turns its argument into a list,

some _obj <-- {obj, True};
This list looks like the result of another quantifier, e.g.
the _x  <-- {x, 1};
but in fact the special value True in it is used in the definition of "ate" so that when you "eat" "some" of the object, you still have "some" of it left.

To implement this, we have made a special rule for the pattern
_x had {obj, True} <-- ...
separately from the general rule
_x had {obj, n_IsNumber} <-- ...
and its shorthand
(_x had _obj )_(Not IsList(obj)) <--
  x had {obj, 1};

Admittedly, the module wordproblems.ys has not very much practical use but it is fun to play with and it illustrates the power of syntactic constructions from an unexpected angle.