Terms 0.1.0b6

Words

The main building block of Terms constructs are words.

To start with, there are a few predefined words: word, verb, noun, number, thing, and exist.

New words are defined relating them to existing words.

There are 2 relations that can be established among pairs of words.

As we shall see below, these relations are formally similar to the set theory relations “is an element of” and “is a subset of”.

In English, we express the first relation as “is of type”, and in Terms it is expressed as:

word1 is a word2.

So we would say that word1 is of type word2, defining word1 in terms of word2 (so word2 must have been defined before, or be predefined). The second relation is expressed in English as “is subtype of”, and in Terms:

a word1 is a word2.

So, we would say that word1 is a subtype of word2, also defining word1 in terms of word2. Among the predefined words, these relations are given:

word is a word.
verb is a word.
a verb is a word.
noun is a word.
a noun is a word.
thing is a noun.
a thing is a word.
exist is a verb.
a exist is a word.
number is a word.
a number is a word.

To define a new word, you put it in relation to an existing word. For example:

a person is a thing.
a man is a person.
a woman is a person.
john is a man.
sue is a woman.

These relations have consecuences, given by 2 implicit rules:

A is a B; a B is a C -> A is a C.
a A is a B; a B is a C -> a A is a C.

Therefore, from all the above, we have, for example, that:

thing is a word.
person is a word.
person is a noun.
john is a word.
a man is a thing.
john is a thing.
sue is a person.
...

With words, we can build facts. A fact consists of a verb and any number of (labelled) objects.

Verbs are special words, in that they determine the modifiers of the facts built with them. These modifiers are words, and are labeled. To define a new verb, you provide first an ancestor verb (or a series of ancestor verbs separated by colons), and then the types of words that can be modifiers for the verb in a fact, associated with their labels. For example:

to love is to exist, subj a person, who a person.

That can be read as: love is defined as a subtype of exist, and when used in facts it can take a subject of type person and an object labelled who also of type person.

The primitive verb is exist, that just defines a subj object of type thing. There are more predefined verbs, the use of which we shall see when we explain the treatment of time in Terms.

Facts

Facts are built with a verb and a number of objects. They are given in parenthesis. For example, we might have a fact such as:

(love john, who sue).

The subj object is special: all verbs have it, and in facts it is not labelled with subj, it just takes the place of the subject right after the verb.

Verbs inherit the object types of their ancestors. The primitive exist verb only takes one object, subj, of type word, inherited by all the rest of the verbs. So, if we define a verb:

to adore is to love.

It will have a who object of type person. If adore had provided a new object, it would have been added to the inherited ones. A new verb can override an inherited object type to provide a subtype of the original object type (like we have done above with subj; subj is predefined to be of type word.)

Facts are words, “first class citizens”, and can be used wherever a word can be used. Facts are words of type exist, and also of type <verb>, were <verb> is the verb used to build the fact. So our facts are actually synctactic sugar for (love john, who sue) is a love.

The objects in a fact can be of any type (a word, a verb, a noun, a thing, a number). In addition, they can also be facts (type exist). So, if we define a verb like:

to want is to exist, subj a person, what a exist.

We can then build facts like:

(want john, what (love sue, who john)).

And indeed:

(want john, what (want sue, what (love sue, who john))).

Rules

We can build rules, that function producing new facts out of existing (or newly added) ones. A rule has 2 sets of facts, the conditions (given first) and the consecuences. The facts in each set of facts are separated by semicolons (conjunctions), and the symbol -> (implication) separates the conditions from the consecuences. A simple rule might be:

(love john, who sue)
->
(love sue, who john).

The facts in the knowledge base are matched with the conditions of rules, and when all the conditions of a rule are matched by coherent facts, the consecuences are added to the knowledge base. The required coherence among matching facts concerns the variables in the conditions.

We can use variables in rules. They are logical variables, used only to match words, and with a scope limited to the rule were they are used. We build variables by capitalizing the name of the type of words that it can match, and appending any number of digits. So, for example, a variable Person1 would match any person, such as sue or john. With variables, we may build a rule like:

(love Person1, who Person2)
->
(love Person2, who Person1).

If we have this rule, and also that (love john, who sue), the system will conclude that (love sue, who john).

Variables can match whole facts. For example, with the verbs we have defined, we could build a rule such as:

(want john, what Exists1)
->
(Exists1).

With this, and (want john, what (love sue, who john))., the system would conclude that (love sue, who john).

Variables that match verbs (or nouns) have a special form, in that they are prefixed by the name of a verb (or a noun), so that they match verbs (or nouns) that are subtypes of the prefix verb (or noun). For example, with the words we have from above, we might make a rule like:

(LoveVerb1 john, who Person1)
->
(LoveVerb1 Person1, who john).

In this case, LoveVerb1 would match both love and adore, so both (love john, who sue) and (adore john, who sue) would produce the conclusion that (love sue, who john) or (adore sue, who john).

For a more elaborate example we can define a new verb:

to be-allowed is to exist, subj a person, to a verb.

and a rule:

(want Person1, what (LoveVerb1 Person1, who Person2));
(be-allowed Person1, to LoveVerb1)
->
(LoveVerb1 Person1, who Person2).

Then, (be-allowed john, to adore) would allow him to adore but not to love.

We can use word variables, e.g. Word1, that will match any word or fact.

In conditions, we may want to match a whole fact, and at the same time match some of its component words. To do this, we prepend the fact with the name of the fact variable, separated with a colon. With this, the above rule would become:

(want Person1, what Love1:(LoveVerb1 Person1, who Person2));
(be-allowed Person1, to LoveVerb1)
->
(Love1).

Integers

Integers are of type number. We don’t define numbers, we just use them. Any sequence of characters that can be cast as an integer type in Python are numbers in Terms, e.g.: 1.

Number variables are composed just with a capital letter and an integer, like N1, P3, or F122.

Pythonic conditions

In rules, we can add a section where we test conditions with Python, or where we produce new variables out of existing ones. This is primarily provided to test arithmetic conditions and to perform arithetic operations. This section is placed after the conditions, between the symbols <- and ->. The results of the tests are placed in a condition python variable, and if it evaluates to False, the rule is not fired.

To give an example, let’s imagine some new terms:

to aged is to exist, age a number.
a bar is a thing.
club-momentos is a bar.
to enters is to exist, where a bar.

Now, we can build a rule such as:

(aged Person1, age N1);
(want Person1, what (enters Person1, where Bar1))
<-
condition = N1 >= 18
->
(enters Person1, where Bar1).

If we have that:

(aged sue, age 17).
(aged john, age 19).
(want sue, what (enters sue, where club-momentos)).
(want john, what (enters john, where club-momentos)).

The system will (only) conclude that (enters john, where club-momentos).

Negation

We can use 2 kinds of negation in Terms, classical negation and negation by failure.

Classical negation

Any fact can be negated by prepending ! to its verb:

(!aged sue, age 17).

A negated fact is the same as a non-negated one. Only a negated fact can match a negated fact, and they can be asserted or used in rules. The only special thing about negation is that the system will not allow a fact and its negation in the same knowledge base: it will warn of a contradiction and will reject the offending fact.

Negation by failure

In pythonic conditions, we can use a function runtime.count with a single string argument, a Terms fact (possibly with variables), that will return the number of facts in the db matching the given one. We can use this to test for the absence of any given fact in the knowledge base, and thus have negation by failure.

Some care must be taken with the count function. If a fact is entered that might match a pythonic count condition, it will never by itself trigger any rule. Rules are activated by facts matching normal conditions; and pythonic conditions can only allow or abort those activations. In other words, when a fact is added, it is tested against all normal conditions in all rules, and if it activates any rule, the pythonic conditions are tested. An example of this behaviour can be seen here. If you examine the ontology in the previous link, you will see that it is obviously wrong; that’s the reason I say that care must be taken. Counting happens in time, and it is not advisable to use it without activating time.

Time

In the monotonic classical logic we have depicted so far, it is very simple to represent physical time: you only need to add a time object of type number to any temporal verb. However, to represent the present time, the now, i.e., a changing distinguished instant of time, this logic is not enough. We need to use some non-monotonic tricks for that, that are implemented in Terms as a kind of temporal logic. This temporal logic can be activated in the settings file:

[mykb]
dbms = postgresql://terms:terms@localhost
dbname = mykb
time = normal
instant_duration = 60

If it is activated, several things happen.

The first is that the system starts tracking the present time: It has an integer register whose value represents the current time. This register is updated every config['instant_duration'] seconds. There are 3 possible values for the mode setting for time: If the setting is none, nothing is done with time. If the setting is normal, the current time of the system is incremented by 1 when it is updated. If the setting is real, the current time of the system is updated with Python’s import time; int(time.time()).

The second thing that happens is that, rather than defining verbs extending exist, we use 2 new verbs, occur and endure, both subtypes of exist. These new verbs have special number objects: occur has an at_ object, and endure a since_ and a till_ objects.

The third is that the system starts keeping 2 different factsets, one for the present and one for the past. All reasoning occurs in the present factset. When we add a fact made with these verbs, the system automatically adds to occur an at_ object and to endure a since_ object, both with the value of its “present” register. The till_ object of endure facts is left undefined. We never explicitly set those objects. Each time the time is updated, all occur facts are removed from the present and added to the past factset, and thus stop producing consecuences. Queries for occur facts go to the past factset if we specify an at_ object in the query, and to the present if an at_ object is not provided. The same goes for endure facts, substituting at_ with since_. We might say that the endure facts in the present factset are in present continuous tense.

The fourth thing that happens when we activate the temporal logic is that we can use a new predicate in the consecuances of our rules: finish. This verb is defined like this:

to finish is to exist, subj a thing, what a exist.

And when a rule with such a consecuence is activated, it grabs the provided what fact from the present factset, adds a till_ object to it with the present time as value, removes it from the present factset, and adds it to the past factset.

There is also the temporal verb exclusive-endure, subverb of endure. The peculiarity of exclusive-endure is that whenever a fact with such verb is added to the knowledge base, any previous present facts with the same subject and verb are finish ed.

A further verb, happen, derived from occur, has the singularity that, when a fact is added as a consecuence of other facts, and is built with a verb derived from happen, is fed through the pipeline back to the user adding the facts that are producing consecuences.

Querying

Queries are sets of facts separated by semicolons, with or without variables. If the query contains no variables, the answer will be true for presence of the asked facts or false for their absence. To find out whether a fact is negated we must query its negation.

If we include variables in the query, we will obtain all the variable substitutions that would produce a true query, in the form of a json list of mappings of strings.

However, we can not add special constraints, like we can in rules with pythonic conditions.

Miscelaneous technical notes.

• I have shown several different kinds of variables, for things, for verbs, for numbers, for facts. But the logic behind Terms is first order, there is only one kind of individuals, and the proliferation of kinds of variables is just syntactic sugar. Person1 would be equivalent to something like “for all x, x is a person and x…”. LoveVerb1 would be equivalent to something like “for all x, a x is a verb, subverb of love, and x…”.

• The design of the system is such that both adding new facts (with their consecuences) and querying for facts should be independent of the size of the knowledge base. The only place where we depend on the size of the data is in arithmetic conditions, since at present number objects are not indexed as such.

• The Python section of the rules is exec``ed with a dict with the ``condition variable in locals and an empty dict as globals. We might add whatever we like as globals; for example, numpy.