I have OWL ontology TBOX only (no instances), I need to formally express it using algebra to define some structure.
I have looked for that, I found some representations
(C, P, A) the reference is http://www.dblab.ece.ntua.gr/pubs/uploads/TR-2007-20.pdf
where C is the set of classes, P is the set of properties and A is the set of axioms.
A is used to express subsumption, restrictions etc...
C includes primitive types used in data properties definition
(C, P, Sub, Func) the reference is http://dl.acm.org/citation.cfm?id=1871946
where C is the set of classes, P is the set of properties, Sub is subsumption relationships and Func relates each class to its applicable properties
Actually, I'm not sure what is the right representation. could you refer my to some reference please if any?
Related
To me, the dcterms:identifier property seems like a legitimate inverse functional property. When two things share the same identifier, I think it is safe to conclude that it is the same thing.
Is there any compelling reason not to define it as such (owl:InverseFunctionalProperty) in my ontology?
If you need to stay in OWL 2 DL, then it's not a good idea to declare data properties to be inverse functional - only object properties can be declared as such without violating the constraints and end up in OWL 2 FULL.
dcterms:identifier has a range of rdfs:Literal defined here
You could use a HasKey axiom to achieve similar results: keys were introduced in OWL 2 for the purpose of identifying one or more properties whose values are identifiers for the referring individuals, and both object and data properties can be used.
I'm reading through Paulson's ML For the Working Programmer and am a bit confused about the distinction between datatypes and structures.
On p. 142, he defines a type for binary trees as follows:
datatype 'a tree = Lf
| Br of 'a * 'a tree * 'a tree;
This seems to be a recursive definition where 'a denotes some fixed type. So any time I see 'a, it must refer to the same type throughout.
On p. 148, he discusses a structure for binary trees:
"...we have been following an imaginary ML session in which we typed in the tree functions one at a time. Now we ought to collect the most important of those functions into a structure, called Tree. We really must do so, because one of our functions (size) clashes with a built-in function. One reason for using structures is to prevent such name clashes.
We shall, however, leave the datatype declaration of tree outside of the structure. If it were inside, we should be forced to refer to the constructors by Tree.Lf and Tree.Br, which would make our patters unreadable. Thus, in the sequel, imagine that we have made the following declarations:
datatype 'a tree = Lf
| Br of 'a * 'a tree * 'a tree;
structure Tree =
struct
fun size Lf = 0
| size (Br( v, t1, t2)) = 1 + size t1 + size t2;
fun depth...
etc...
end;
I'm a little confused.
1) What is the relationship between a datatype and a structure?
2) What is the role of "struct" within the structure definition?
3) Later on, Paulson discusses a structure for dictionaries as binary search trees. He does the following:
structure Dict : DICTIONARY =
struct
type key = string;
type 'a t = (key * 'a) tree;
val empty = Lf;
<a bunch of functions for dictionaries>
This makes me think struct specifies the different primitive or compound types involved int he definition of a Dict.
That's a really fuzzy definition though. Anyone like to clarify?
Thanks for the help,
bclayman
A structure is a module. Everything between the struct and end keywords forms the body of this module. Similarly, you can view a signature as the description of an abstract module interface. Ascribing a signature to a structure (like the : DICTIONARY syntax does in your example) limits the exports of the module to what is specified in that signature (by default, everything would be accessible). That allows you to hide implementation details of a module.
However, ML modules are much richer than that. They can be arbitrarily nested. There are also functors, which are effectively functions from modules to modules ("parameterised modules", if you want). Altogether, the module language in ML forms a full functional language on its own, with structures as the basic entities, functors over them, and signatures describing the "types" of such modules. This little language is a layer on top of the so-called core language, where ordinary values and types live.
So, to answer your individual questions:
1) There is no specific relationship between the datatype and the structure. The latter simply uses the former.
2) struct-end is simply a keyword pair to delimit the structure body (languages in C tradition would probably use curly braces there).
3) As explained above, a structure is a basic module. It can contain (and export) arbitrary other language entities, including other modules. By grouping definitions together, and potentially hiding some of them through a signature ascription, you can express namespacing and encapsulation (in particular, abstract data types).
I should also note that Paulson's book is outdated regarding its description of modules, as it predates the current language version. In particular, it does not describe how to express abstract data types through modules, but instead introduces the obsolete abstype declaration which nobody has been using in almost 20 years. A more extensive and up-to-date introduction to modular programming in ML can be found in Harper's Programming in Standard ML.
In this example, the datatype 'a tree is describing a binary tree (https://en.wikipedia.org/wiki/Binary_tree) that is capable of storing any value of a single type. The 'a in the definition is a variant type which will later be constrained down to a concrete type wherever tree is used with a different type. This allows you to define the structure of a tree once and then use it with any type later on.
The Tree structure is separate from the datatype definition. It is being used to group functions together that operate on the 'a tree datatype. It is being used right now as a way to modularize the code and, as it points out, to prevent namespace clashes.
struct is just an identifier keyword to let the compiler know where your structure definition starts while the end keyword is used to let the compiler know where the definition ends.
The dictionary structure is defining a dictionary (a key -> value data structure) that uses a tree as the internal data structure. Once again, the structure is a collection of functions that will be used to create and operate on dictionaries. The types within the dictionary structure compose the type of the internal data structure that makes up the dictionary. The following functions define the public interface that you're exposing to allow clients to work with dictionaries.
I am studying OWL and I am trying to build an Ontology using Protege.
I found an option called Equivalent To in Protege.
What is that option for please? Is it for dividing the space of instances? or is it to set the Object properties that a class can have?
Equivalent to applies to class expressions, object properties and data properties.
Equivalence in class expressions
In class expressions, equivalence means that two classes have the same individuals in any interpretation (i.e., the two classes are alternate names, or equivalent definitions, for the same set of individuals).
Equivalence in data and object properties
For object and data properties, asserting that two properties are equivalent means that their domains and ranges apply to both properties, and that every assertion using one property can be rewritten as using the other.
Example
For example, suppose I declare a hasOwner object property and an ownedBy as equivalent, then: MyCar hasOwner Me implies MyCar ownedBy Me.
It's folk knowledge that OCaml modules are "just" existential types. That there's some kind of parity between
module X = struct type t val x : t end
and
data 'a spec = { x : 'a }
data x = X : 'a spec
and this isn't untrue exactly.
But as I just evidenced, OCaml has both modules and existential types. My question is:
How do they differ?
Is there anything which can be implemented in one but not the other?
When would you use one over the other (in particular comparing first-class modules with existential types)?
Completing gsg's answer on your third point.
There are two kinds of way to use modules:
As a structuring construct, when you declare toplevel modules. In that case you are not really manipulating existential variables. When encoding the module system in system-F, you would effectively represent the abstract types by existential variables, but morally, it is closer to a fresh singleton type.
As a value, when using first class modules. In that case you are clearly manipulating existential types.
The other representations of existential types are through GADT's and with objects. (It is also possible to encode existential as the negation of universal with records, but its usage are completely replaced by first class modules).
Choosing between those 3 cases depend a bit in the context.
If you want to provide a lot of functions for your type, you will prefer modules or objects. If only a few, you may find the syntax for modules or objects too heavywheight and prefer GADT. GADT's can also reveal a the structure of your type, for instance:
type _ ty =
| List : ty -> ty list
| Int : int list
type exist = E : 'a ty * 'a -> exist
If you are in that kind of case, you do not need to propagate the function working on that type, so you will end up with something a lot lighter with GADT's existentials. With modules this would look like
module type Exist = sig
type t
val t : t ty
end
module Int_list : Exist = struct
type t = int list
let t = List Int
end
let int_list = (module Int_list:Exist)
And if you need sub-typing or late binding, go for the objects. This can often be encoded with modules but this tend to be tedious.
It's specifically abstract types that have existential type. Modules without abstract types can be explained without existentials, I think.
Modules have features other than abstract types: they act as namespaces, they are structurally typed, they support operations like include and module type of, they allow private types, etc.
A notable difference is that functors allow ranging over types of any (fixed) arity, which is not possible with type variables because OCaml lacks higher kinded types:
module type M = sig
type 'a t
val x : 'a t
end
I'm not quite sure how to answer your last question. Modules and existentials are different enough in practice that the question of when to substitute one for the other hasn't come up.
I am a C# developer. Coming from OO side of the world, I start with thinking in terms of interfaces, classes and type hierarchies. Because of lack of OO in Haskell, sometimes I find myself stuck and I cannot think of a way to model certain problems with Haskell.
How to model, in Haskell, real world situations involving class hierarchies such as the one shown here: http://www.braindelay.com/danielbray/endangered-object-oriented-programming/isHierarchy-4.gif
First of all: Standard OO design is not going to work nicely in Haskell. You can fight the language and try to make something similar, but it will be an exercise in frustration. So step one is look for Haskell-style solutions to your problem instead of looking for ways to write an OOP-style solution in Haskell.
But that's easier said than done! Where to even start?
So, let's disassemble the gritty details of what OOP does for us, and think about how those might look in Haskell.
Objects: Roughly speaking, an object is the combination of some data with methods operating on that data. In Haskell, data is normally structured using algebraic data types; methods can be thought of as functions taking the object's data as an initial, implicit argument.
Encapsulation: However, the ability to inspect an object's data is usually limited to its own methods. In Haskell, there are various ways to hide a piece of data, two examples are:
Define the data type in a separate module that doesn't export the type's constructors. Only functions in that module can inspect or create values of that type. This is somewhat comparable to protected or internal members.
Use partial application. Consider the function map with its arguments flipped. If you apply it to a list of Ints, you'll get a function of type (Int -> b) -> [b]. The list you gave it is still "there", in a sense, but nothing else can use it except through the function. This is comparable to private members, and the original function that's being partially applied is comparable to an OOP-style constructor.
"Ad-hoc" polymorphism: Often, in OO programming we only care that something implements a method; when we call it, the specific method called is determined based on the actual type. Haskell provides type classes for compile-time function overloading, which are in many ways more flexible than what's found in OOP languages.
Code reuse: Honestly, my opinion is that code reuse via inheritance was and is a mistake. Mix-ins as found in something like Ruby strike me as a better OO solution. At any rate, in any functional language, the standard approach is to factor out common behavior using higher-order functions, then specialize the general-purpose form. A classic example here are fold functions, which generalize almost all iterative loops, list transformations, and linearly recursive functions.
Interfaces: Depending on how you're using an interface, there are different options:
To decouple implementation: Polymorphic functions with type class constraints are what you want here. For example, the function sort has type (Ord a) => [a] -> [a]; it's completely decoupled from the details of the type you give it other than it must be a list of some type implementing Ord.
Working with multiple types with a shared interface: For this you need either a language extension for existential types, or to keep it simple, use some variation on partial application as above--instead of values and functions you can apply to them, apply the functions ahead of time and work with the results.
Subtyping, a.k.a. the "is-a" relationship: This is where you're mostly out of luck. But--speaking from experience, having been a professional C# developer for years--cases where you really need subtyping aren't terribly common. Instead, think about the above, and what behavior you're trying to capture with the subtyping relationship.
You might also find this blog post helpful; it gives a quick summary of what you'd use in Haskell to solve the same problems that some standard Design Patterns are often used for in OOP.
As a final addendum, as a C# programmer, you might find it interesting to research the connections between it and Haskell. Quite a few people responsible for C# are also Haskell programmers, and some recent additions to C# were heavily influenced by Haskell. Most notable is probably the monadic structure underlying LINQ, with IEnumerable being essentially the list monad.
Let's assume the following operations: Humans can speak, Dogs can bark, and all members of a species can mate with members of the same species if they have opposite gender. I would define this in haskell like this:
data Gender = Male | Female deriving Eq
class Species s where
gender :: s -> Gender
-- Returns true if s1 and s2 can conceive offspring
matable :: Species a => a -> a -> Bool
matable s1 s2 = gender s1 /= gender s2
data Human = Man | Woman
data Canine = Dog | Bitch
instance Species Human where
gender Man = Male
gender Woman = Female
instance Species Canine where
gender Dog = Male
gender Bitch = Female
bark Dog = "woof"
bark Bitch = "wow"
speak Man s = "The man says " ++ s
speak Woman s = "The woman says " ++ s
Now the operation matable has type Species s => s -> s -> Bool, bark has type Canine -> String and speak has type Human -> String -> String.
I don't know whether this helps, but given the rather abstract nature of the question, that's the best I could come up with.
Edit: In response to Daniel's comment:
A simple hierarchy for collections could look like this (ignoring already existing classes like Foldable and Functor):
class Foldable f where
fold :: (a -> b -> a) -> a -> f b -> a
class Foldable m => Collection m where
cmap :: (a -> b) -> m a -> m b
cfilter :: (a -> Bool) -> m a -> m a
class Indexable i where
atIndex :: i a -> Int -> a
instance Foldable [] where
fold = foldl
instance Collection [] where
cmap = map
cfilter = filter
instance Indexable [] where
atIndex = (!!)
sumOfEvenElements :: (Integral a, Collection c) => c a -> a
sumOfEvenElements c = fold (+) 0 (cfilter even c)
Now sumOfEvenElements takes any kind of collection of integrals and returns the sum of all even elements of that collection.
Instead of classes and objects, Haskell uses abstract data types. These are really two compatible views on the problem of organizing ways of constructing and observing information. The best help I know of on this subject is William Cook's essay Object-Oriented Programming Versus Abstract Data Types. He has some very clear explanations to the effect that
In a class-based system, code is organized around different ways of constructing abstractions. Generally each different way of constructing an abstraction is assigned its own class. The methods know how to observe properties of that construction only.
In an ADT-based system (like Haskell), code is organized around different ways of observing abstractions. Generally each different way of observing an abstraction is assigned its own function. The function knows all the ways the abstraction could be constructed, and it knows how to observe a single property, but of any construction.
Cook's paper will show you a nice matrix layout of abstractions and teach you how to organize any class as an ADY or vice versa.
Class hierarchies involve one more element: the reuse of implementations through inheritance. In Haskell, such reuse is achieved through first-class functions instead: a function in a Primate abstraction is a value and an implementation of the Human abstraction can reuse any functions of the Primate abstraction, can wrap them to modify their results, and so on.
There is not an exact fit between design with class hierarchies and design with abstract data types. If you try to transliterate from one to the other, you will wind up with something awkward and not idiomatic—kind of like a FORTRAN program written in Java.
But if you understand the principles of class hierarchies and the principles of abstract data types, you can take a solution to a problem in one style and craft a reasonably idiomatic solution to the same problem in the other style. It does take practice.
Addendum: It's also possible to use Haskell's type-class system to try to emulate class hierarchies, but that's a different kettle of fish. Type classes are similar enough to ordinary classes that a number of standard examples work, but they are different enough that there can also be some very big surprises and misfits. While type classes are an invaluable tool for a Haskell programmer, I would recommend that anyone learning Haskell learn to design programs using abstract data types.
Haskell is my favorite language, is a pure functional language.
It does not have side effects, there is no assignment.
If you find to hard the transition to this language, maybe F# is a better place to start with functional programming. F# is not pure.
Objects encapsulate states, there is a way to achieve this in Haskell, but this is one of the issues that takes more time to learn because you must learn some category theory concepts to deeply understand monads. There is syntactic sugar that lets you see monads like non destructive assignment, but in my opinion it is better to spend more time understanding the basis of category theory (the notion of category) to get a better understanding.
Before trying to program in OO style in Haskell, you should ask yourself if you really use the object oriented style in C#, many programmers use OO languages, but their programs are written in the structured style.
The data declaration allows you to define data structures combining products (equivalent to structure in C language) and unions (equivalent to union in C), the deriving part o the declaration allows to inherit default methods.
A data type (data structure) belongs to a class if has an implementation of the set of methods in the class.
For example, if you can define a show :: a -> String method for your data type, then it belong to the class Show, you can define your data type as an instance of the Show class.
This is different of the use of class in some OO languages where it is used as a way to define structures + methods.
A data type is abstract if it is independent of it's implementation. You create, mutate, and destroy the object by an abstract interface, you do not need to know how it is implemented.
Abstraction is supported in Haskell, it is very easy to declare.
For example this code from the Haskell site:
data Tree a = Nil
| Node { left :: Tree a,
value :: a,
right :: Tree a }
declares the selectors left, value, right.
the constructors may be defined as follows if you want to add them to the export list in the module declaration:
node = Node
nil = Nil
Modules are build in a similar way as in Modula. Here is another example from the same site:
module Stack (Stack, empty, isEmpty, push, top, pop) where
empty :: Stack a
isEmpty :: Stack a -> Bool
push :: a -> Stack a -> Stack a
top :: Stack a -> a
pop :: Stack a -> (a,Stack a)
newtype Stack a = StackImpl [a] -- opaque!
empty = StackImpl []
isEmpty (StackImpl s) = null s
push x (StackImpl s) = StackImpl (x:s)
top (StackImpl s) = head s
pop (StackImpl (s:ss)) = (s,StackImpl ss)
There is more to say about this subject, I hope this comment helps!