Given the different types of axioms available in OWL, is it possible to assert that a class is not "empty"? Or in other words, can we assert that there exists at least one individual that is part of the specified class?
So, basically I am looking for an equivalence of:
ObjectAssertNotEmpty(a:SomeClass)
In an RDF serialization, all it takes is to identify such an individual:
[] a a:SomeClass .
I presume the OWL syntax should be able to specify it this way:
ClassAssertion( a:SomeClass a:SomeIndividual )
It doesn't matter that you call the individual a:SomeIndividual here, it could very well be owl:sameAs any other individual. Once you state that something is in the class, it follows that the class is not empty.
That is exactly what the model theoretic underpinning of Description Logics aim to do is to assume that all concepts (or OWL classes) are not empty. When a reasoner is run over an ontology, it will give an error if a class is found to be empty. Such a class is deemed to be unsatisfiable. That is the basis of satisfiability checking (and consistency checking, since satisfiability checking can be translated to consistency checking).
See An introduction to Description Logics text for details.
Here is a simple example ontology to illustrate this:
Class: A
Class: NotA
EquivalentTo: not (A)
Class: B
EquivalentTo: A and NotA
Individual: b
Types: B
Note that class B will be equivalent to owl:Nothing which is the empty set. Now setting individual b to be an instance of B will lead to an inconsistency with explanation as shown below.
The answer as given by #IS4 will only work if SomeClass is found to be satisfiable (that is not empty), otherwise the reasoner will give an inconsistency.
Related
I am trying to use fuzzy logic to weight and extract the best sentences for the query. I have extracted the following features which they can be used in fuzzy logic:
Each sentence has cosine value.
How many proper-noun is in the sentence.
the position of the sentence in the document.
sentence length.
I want to use the above features to apply the fuzzy logic. for instance, i want to create the rule base something like the following
if cosineValue >= 0.9 && numberOfPropernoun >=1
THEN the sentence is important
I am not quite sure how to start implementing the rule base, the facts and inference engine. It would like someone to guide me to implement this in python. Please note that I am not familiar with logic programming languages. I would like to implement it in python
This is just a sketch; I'm not even going to try this code because I'm not sure what you want.
Make a class for your features:
Features = namedtuple('Features', ['cosine', 'nouns', 'position', ...])
etc.
Now imagine you are building your AST. What grammar does your language have? Well, you have conditions, and your conditions have consequences, and your conditions can be combined by boolean operators, so let's make some basic ones:
class CosineValue(object):
def evaluate(self, features):
return features.cosine
class Nouns(object):
def evaluate(self, features):
return features.nouns
... etc.
Now you need to combine these AST nodes with some operations
class GreaterThan(object):
def __init__(self, property, value):
self.property, self.value = property, value
def evaluate(self, sentence):
return property.evaluate(sentence) > self.value
Now GreaterThan(CosineValue(), 0.9) is an object (an abstract syntax tree, actually) that represents cosineValue > 0.9. You can evaluate it like so:
expr = GreaterThan(CosineValue(), 0.9)
expr.evaluate(Features(cosine=0.95, ...)) # returns True
expr.evaluate(Features(cosine=0.40, ...)) # returns False
These objects don't look like much, but what they are doing is reifying your process. Their structure encodes what formerly would have been code. Think about this, because this is the only hard part about what you are trying to do: comprehending how you can delay computation by turning it into structure, and how you can play with when values become part of your computation. You were probably stuck thinking about how to write those "if" statements and keeping them separate from the code and the runtime values you need to run them against. Now you should be able to see how, but it's a more advanced way of thinking about programming.
Now you need to build your if/then structure. I'm not sure what you want here either but I would say your if/then is going to be a class that takes an expression like we've just created as one argument and a "then" case, and does the test and either performs or does not perform the "then" case. Probably you will need if/then/else, or else a way to track if it fired, or a way to evaluate your if into a value. You will have to think about this part; nobody can tell you based on what you wrote above what you should be doing.
To make your conditions more powerful, you will need to add some more classes for boolean operators that take conditions as arguments, but it should be straightforward; you'll have And and Or, they'll both take two Condition arguments and their evaluation will do the sensible thing. You could make a Condition superclass, and then add some methods like And and Or to simplify generating these structures.
Finally, if you want to parse something like what you have above, you should try out pyparsing, but make sure you have the AST figured out first or it will be an uphill battle. Or look at what they have; maybe they have some primitives for this, I haven't dealt with pyparsing in a long time.
Best of luck, and please ask a better question next time!
I know this topic has been discussed several times, but there is something still unclear to me.
I've read this question applicative-order/call-by-value and normal-order/call-by-name differences and there is something I would to clarify once and for all:
Call-by-name
As normal order, but no reductions are performed inside abstractions. For example λx.(λx.x)x is in normal form according to this strategy, although it contains the redex (λx.x)x.
In call by name, the expression λx.(λx.x)x is said to be in normal form; is this because "(λx.x)x" is considered to be the body (since the scope of λ extends as far as possible to the right)? And so on the other side, if I apply the normal order, what would be the result?
In call by name, the expression λx.(λx.x)x is said to be in normal form; is this because "(λx.x)x" is considered to be the body (since the scope of λ extends as far as possible to the right)?
Yes, you are right.
And so on the other side, if I apply the normal order, what would be the result?
You do reduction inside the body: (λx.x)x -> x, so the whole thing reduces to the identity function:
λx.(λx.x)x -> λx.x
To clarify it a bit further, let me do this one more time, renaming the variables to conform with the Barendregt variable convention: λx.(λx.x)x =α λx.(λy.y)x:
λx.(λy.y)x -> λx.[y := x](y) = λx.x
I don't know if you call this statement or not, but I have this question
what is the difference between these two statments :
A ⊑ B ⊓ C
and
A ASSERTA_SYMBOL = B ⊓ C
sorry I don't know how to write the ASSERTA_SYMBOL, but it is in this image
a real example is this:
Both expressions describe or define a concept (or class or set).
The difference is subclass vs equivalent class.
An elephant is one kind of grey, large animals. There might be other kinds.
A happy father is a man who has at least one daughter and any man with at least one daughter is a happy father. There are no others.
⊑ represents a subconcept relationship, ≐ means agreement (sometimes called the same as constructor).
Good morning,
I want to understand how can I describe something using the first order logic.
For example I want to describe what is a film (an entity) and what is an attribute (for example actor: Clooney) for the film. How can I describe that using the first order logic?
******* UPDATE ********
What I need to explain in first logic order is:
ENTITY: an element, an abstraction or an object that can be described with a set of properties or attributes. So I think that I must says that the entity has got a set of attributes with their respective values. An Entity describes an element, an abstraction or an object.
ATTRIBUTE: an attribute has always got a value and it always associated to an entity. It describes a specific feature/property of the entity.
DOCUMENT: a pure text description (pure text it not contains any html tags). Every document describes only ONE entity through its attribute.
To state that an object has a certain property you would use a single place predicate. For example, to state that x is a film you could write Film(x). If you want to attribute some value to an object you can use two (or more) place predicate. Using your example you could say that Clooney starred in a film as Starred(clooney, x).
There are certain conventions that people use. For example, predicates start with capital letters (Actor, Film, FatherOf) and constants start with a lower case letter (x, clooney, batman). Constants denote objects and predicates say something about the objects. In case of predicates with more than one argument the first argument is usually the subject about which you are making the statement. That way you can naturally read the logical formula as a sentence in normal language. For example, FatherOf(x, y) would read as "x is the father of y".
Answer for the update:
I am not sure whether you can do that in first order logic. You could describe an Entity as something that has certain properties by formula such as
\forall x (Entity(x) ==> Object(x) | Element(x) | Abstraction(x))
This is a bit more difficult for the Attribute. In first order logic an attribute ascribes some quality to an object or relates it to another object. You could probably use a three place predicate as in:
\forall attribute (\exists object (\exists value (Has(object, attribute, value))))
As to the document, that would be just a conjunction of such statements. For example, the description of George Clooney could be the following:
Entity(clooney) & Has(clooney, starred, gravity) & Has(clooney, bornIn, lexington) & ...
The typical way to do this is to explain that a specific object exists and this object has certain attributes. For example:
(∃x)(property1(x) & property2(x) & ~property3(x))
aka: There exists a thing that satisfies properties 1 and 2 but does not satisfy property 3.
Your current question formulation makes it unclear as to what you mean by attributes and documents. Perhaps towards your idea of attributes: it's possible to describe as the domain of property1 all the entities that satisfy it; so, for example, the domain of blue is all blue objects.
First-order logic has nothing to do with HTML -- are you trying to use HTML to represent an entity in first-order logic somehow? It remains incredibly unclear what your question is.
So as I understand it, the === operator tests to see if the RHS object is a member of the LHS object. That makes sense. But how does this work in Ruby? I'm looking at the Ruby docs and I only see === defined in Object, I don't see it in Integer itself. Is it just not documented?
Integer is a class, which (at least in Ruby) means that it is just a boring old normal object like any other object, which just happens to be an instance of the Class class (instead of, say, Object or String or MyWhateverFoo).
Class in turn is a subclass of Module (although arguably it shouldn't be, because it violates the Liskov Substition Principle, but that is a discussion for another forum, and is also a dead horse that has already been beaten many many times). And in Module#=== you will find the definition you are looking for, which Class inherits from Module and instances of Class (like Integer) understand.
Module#=== is basically defined symmetric to Object#kind_of?, it returns true if its argument is an instance of itself. So, 3 is an instance of Integer, therefore Integer === 3 returns true, just as 3.kind_of?(Integer) would.
So as I understand it, the === operator tests to see if the RHS object is a member of the LHS object.
Not necessarily. === is a method, just like any other method. It does whatever I want it to do. And in some cases the "is member of" analogy breaks down. In this case it is already pretty hard to swallow. If you are a hardcore type theory freak, then viewing a type as a set and instances of that type as members of a set is totally natural. And of course for Array and Hash the definition of "member" is also obvious.
But what about Regexp? Again, if you are formal languages buff and know your Chomsky backwards, then interpreting a Regexp as an infinite set of words and Strings as members of that set feels completely natural, but if not, then it sounds kind of weird.
So far, I have failed to come up with a concise description of precisely what === means. In fact, I haven't even come up with a good name for it. It is usually called the triple equals operator, threequals operator or case equality operator, but I strongly dislike those names, because it has absolutely nothing to do with equality.
So, what does it do? The best I have come up with is: imagine you are making a table, and one of the column headers is Integer. Would it make sense to write 3 in that column? If one of the column headers is /ab*a/, would it make sense to write 'abbbba' in that column?
Based on that definition, it could be called the subsumption operator, but that's even worse than the other examples ...
It's defined on Module, which Class is a subclass of, which Integer is an instance of.
In other words, when you run Integer === 3, you're calling '===' (with the parameter 3) on the object referred to to by the constant Integer, which is an instance of the class named Class. Since Class is a subclass of Module and doesn't define its own ===, you get the implementation of === defined on Module.
See the API docs for Module for more information.
Umm, Integer is a subclass of Object.