Lambda term can be:
variable
lambda abstraction (for example \x.t)
application. If t and s are lambda terms, then ts is an application.
So, application with abstraction in the left part (for example (\x.t)a) looks good. It looks like function calling. But what does application mean when left part is a variable or other application? What does mean ab, ((\x.x)a)b or a(\x.x) if a and b are variables?
((\x.x)b)c is a function application. Here It applies b to c.
((\x.x)b)c
= bc
a(\x.y) is a function application, applying the function a to its sole argument, which happens to be a function, namely the function that returns y (a free variable).
One of the featues of the lamdba calculus is the ease in which functions can be applied to functions and functions can take other functions as arguments. Your two examples show both cases nicely.
EDIT There are (at least) two versions of the lambda calculus: the untyped and the typed. In the untyped calculus, which you are using here, anything can be applied to anything. In the typed calculus there exist base types which are not functions, such as the type of propositions and the type of "individuals." So you would only be able to write ab if the type of a were a function type mapping the type of b to something.
Related
I am trying to lean deep learning using Julia. In one of the tutorials, which is about MLP, use the below structure for modeling multiple layers in ANN. What does this code mean?
struct Chain
layers
Chain(layers...) = new(layers)
end
This definition in isolation doesn't really "mean" anything; it is just a user defined struct with one field (called layers) and one inner constructor. Usually custom structs like this is used for collecting some data and/or used to define operations on, e.g. you could define a function f operating on this struct like this:
function f(c::Chain)
# do something with the layers in the chain
end
but in order to understand what it is used for in this specific case you probably need to consult the documentation and/or the rest of the code.
One peculiarity in this example:
The inner constructor takes multiple arguments (layers...) and creates a tuple out of them, which is assigned to the property layers.
julia> c = Chain(1, 2, "foo")
Chain((1, 2, "foo"))
As I'm learning Julia, I am wondering how to properly do things I might have done in Python, Java or C++ before. For example, previously I might have used an abstract base class (or interface) to define a family of models through classes. Each class might then have a method like calculate. So to call it I might have model.calculate(), where the model is an object from one of the inheriting classes.
I get that Julia uses multiple dispatch to overload functions with different signatures such as calculate(model). The question I have is how to create different models. Do I use the type system for that and create different types like:
abstract type Model end
type BlackScholes <: Model end
type Heston <: Model end
where BlackScholes and Heston are different types of model? If so, then I can overload different calculate methods:
function calculate(model::BlackScholes)
# code
end
function calculate(model::Heston)
# code
end
But I'm not sure if this is a proper and idiomatic use of types in Julia. I will greatly appreciate your guidance!
This is a hard question to answer. Julia offers a wide range of tools to solve any given problem, and it would be hard for even a core developer of the language to assert that one particular approach is "right" or even "idiomatic".
For example, in the realm of simulating and solving stochastic differential equations, you could look at the approach taken by Chris Rackauckas (and many others) in the suite of packages under the JuliaDiffEq umbrella. However, many of these people are extremely experienced Julia coders, and what they do may be somewhat out of reach for less experienced Julia coders who just want to model something in a manner that is reasonably sensible and attainable for a mere mortal.
It is is possible that the only "right" answer to this question is to direct users to the Performance Tips section of the docs, and then assert that as long as you aren't violating any of the recommendations there, then what you are doing is probably okay.
I think the best way I can answer this question from my own personal experience is to provide an example of how I (a mere mortal) would approach the problem of simulating different Ito processes. It is actually not too far off what you have put in the question, although with one additional layer. To be clear, I make no claim that this is the "right" way to do things, merely that it is one approach that utilizes multiple dispatch and Julia's type system in a reasonably sensible fashion.
I start off with an abstract type, for nesting specific subtypes that represent specific models.
abstract type ItoProcess ; end
Now I define some specific model subtypes, e.g.
struct GeometricBrownianMotion <: ItoProcess
mu::Float64
sigma::Float64
end
struct Heston <: ItoProcess
mu::Float64
kappa::Float64
theta::Float64
xi::Float64
end
Note, in this case I don't need to add constructors that convert arguments to Float64, since Julia does this automatically, e.g. GeometricBrownianMotion(1, 2.0) will work out-of-the-box, as Julia will automatically convert 1 to 1.0 when constructing the type.
However, I might want to add some constructors for common parameterizations, e.g.
GeometricBrownianMotion() = GeometricBrownianMotion(0.0, 1.0)
I might also want some functions that return useful information about my models, e.g.
number_parameter(model::GeometricBrownianMotion) = 2
number_parameter(model::Heston) = 4
In fact, given how I've defined the models above, I could actually be a bit sneaky and define a method that works for all subtypes:
number_parameter(model::T) where {T<:ItoProcess} = length(fieldnames(typeof(model)))
Now I want to add some code that allows me to simulate my models:
function simulate(model::T, numobs::Int, stval) where {T<:ItoProcess}
# code here that is common to all subtypes of ItoProcess
simulate_inner(model, somethingelse)
# maybe more code that is common to all subtypes of ItoProcess
end
function simulate_inner(model::GeometricBrownianMotion, somethingelse)
# code here that is specific to GeometricBrownianMotion
end
function simulate_inner(model::Heston, somethingelse)
# code here that is specific to Heston
end
Note that I have used the abstract type to allow me to group all code that is common to all subtypes of ItoProcess in the simulate function. I then use multiple dispatch and simulate_inner to run any code that needs to be specific to a particular subtype of ItoProcess. For the aforementioned reasons, I hesitate to use the phrase "idiomatic", but let me instead say that the above is quite a common pattern in typical Julia code.
The one thing to be careful of in the above code is to ensure that the output type of the simulate function is type-stable, that is, the output type can be uniquely determined by the input types. Type stability is usually an important factor in ensuring performant Julia code. An easy way in this case to ensure type-stability is to always return Matrix{Float64} (if the output type is fixed for all subtypes of ItoProcess then obviously it is uniquely determined). I examine a case where the output type depends on input types below for my estimate example. Anyway, for simulate I might always return Matrix{Float64} since for GeometricBrownianMotion I only need one column, but for Heston I will need two (the first for price of the asset, the second for the volatility process).
In fact, depending on how the code is used, type-stability is not always necessary for performant code (see eg using function barriers to prevent type-instability from flowing through to other parts of your program), but it is a good habit to be in (for Julia code).
I might also want routines to estimate these models. Again, I can follow the same approach (but with a small twist):
function estimate(modeltype::Type{T}, data)::T where {T<:ItoProcess}
# again, code common to all subtypes of ItoProcess
estimate_inner(modeltype, data)
# more common code
return T(some stuff generated from function that can be used to construct T)
end
function estimate_inner(modeltype::Type{GeometricBrownianMotion}, data)
# code specific to GeometricBrownianMotion
end
function estimate_inner(modeltype::Type{Heston}, data)
# code specific to Heston
end
There are a few differences from the simulate case. Instead of inputting an instance of GeometricBrownianMotion or Heston, I instead input the type itself. This is because I don't actually need an instance of the type with defined values for the fields. In fact, the values of those fields is the very thing I am attempting to estimate! But I still want to use multiple dispatch, hence the ::Type{T} construct. Note also I have specified an output type for estimate. This output type is dependent on the ::Type{T} input, and so the function is type-stable (output type can be uniquely determined by input types). But common with the simulate case, I have structured the code so that code that is common to all subtypes of ItoProcess only needs to be written once, and code that is specific to the subtypes is separted out.
This answer is turning into an essay, so I should tie it off here. Hopefully this is useful to the OP, as well as anyone else getting into Julia. I just want to finish by emphasizing that what I have done above is only one approach, there are others that will be just as performant, but I have personally found the above to be useful from a structural perspective, as well as reasonably common across the Julia ecosystem.
I've got a symbol that represents the name of a function to be called:
julia> func_sym = :tanh
I can use that symbol to get the tanh function and call it using:
julia> eval(func_sym)(2)
0.9640275800758169
But I'd rather avoid the 'eval' there as it will be called many times and it's expensive (and func_sym can have several different values depending on context).
IIRC in Ruby you can say something like:
obj.send(func_sym, args)
Is there something similar in Julia?
EDIT: some more details on why I have functions represented by symbols:
I have a type (from a neural network) that includes the activation function, originally I included it as a funcion:
type NeuralLayer
weights::Matrix{Float32}
biases::Vector{Float32}
a_func::Function
end
However, I needed to serialize these things to files using JLD, but it's not possible to serialize a Function, so I went with a symbol:
type NeuralLayer
weights::Matrix{Float32}
biases::Vector{Float32}
a_func::Symbol
end
And currently I use the eval approach above to call the activation function. There are collections of NeuralLayers and each can have it's own activation function.
#Isaiah's answer is spot-on; perhaps even more-so after the edit to the original question. To elaborate and make this more specific to your case: I'd change your NeuralLayer type to be parametric:
type NeuralLayer{func_type}
weights::Matrix{Float32}
biases::Vector{Float32}
end
Since func_type doesn't appear in the types of the fields, the constructor will require you to explicitly specify it: layer = NeuralLayer{:excitatory}(w, b). One restriction here is that you cannot modify a type parameter.
Now, func_type could be a symbol (like you're doing now) or it could be a more functionally relevant parameter (or parameters) that tunes your activation function. Then you define your activation functions like this:
# If you define your NeuralLayer with just one parameter:
activation(layer::NeuralLayer{:inhibitory}) = …
activation(layer::NeuralLayer{:excitatory}) = …
# Or if you want to use several physiological parameters instead:
activation{g_K,g_Na,g_l}(layer::NeuralLayer{g_K,g_Na,g_l} = f(g_K, g_Na, g_l)
The key point is that functions and behavior are external to the data. Use type definitions and abstract type hierarchies to define behavior, as is coded in the external functions… but only store data itself in the types. This is dramatically different from Python or other strongly object-oriented paradigms, and it takes some getting used to.
But I'd rather avoid the 'eval' there as it will be called many times and it's expensive (and func_sym can have several different values depending on context).
This sort of dynamic dispatch is possible in Julia, but not recommended. Changing the value of 'func_sym' based on context defeats type inference as well as method specialization and inlining. Instead, the recommended approach is to use multiple dispatch, as detailed in the Methods section of the manual.
I am trying to create a function
import Language.Reflection
foo : Type -> TT
I tried it by using the reflect tactic:
foo = proof
{
intro t
reflect t
}
but this reflects on the variable t itself:
*SOQuestion> foo
\t => P Bound (UN "t") (TType (UVar 41)) : Type -> TT
Reflection in Idris is a purely syntactic, compile-time only feature. To predict how it will work, you need to know about how Idris converts your program to its core language. Importantly, you won't be able to get ahold of reflected terms at runtime and reconstruct them like you would with Lisp. Here's how your program is compiled:
Internally, Idris creates a hole that will expect something of type Type -> TT.
It runs the proof script for foo in this state. We start with no assumptions and a goal of type Type -> TT. That is, there's a term being constructed which looks like ?rhs : Type => TT . rhs. The ?foo : ty => body syntax shows that there's a hole called foo whose eventual value will be available inside of body.
The step intro t creates a function whose argument is t : Type - this means that we now have a term like ?foo_body : TT . \t : Type => foo_body.
The reflect t step then fills the current hole by taking the term on its right-hand side and converting it to a TT. That term is in fact just a reference to the argument of the function, so you get the variable t. reflect, like all other proof script steps, only has access to the information that is available directly at compile time. Thus, the result of filling in foo_body with the reflection of the term t is P Bound (UN "t") (TType (UVar (-1))).
If you could do what you are wanting here, it would have major consequences both for understanding Idris code and for running it efficiently.
The loss in understanding would come from the inability to use parametricity to reason about the behavior of functions based on their types. All functions would effectively become potentially ad-hoc polymorphic, because they could (say) run differently on lists of strings than on lists of ints.
The loss in performance would come from representing enough type information to do the reflection. After Idris code is compiled, there is no type information left in it (unlike in a system such as the JVM or .NET or a dynamically typed system such as Python, where types have a runtime representation that code can access). In Idris, types can be very large, because they can contain arbitrary programs - this means that far more information would have to be maintained, and computation occurring at the type level would also have to be preserved and repeated at runtime.
If you're wanting to reflect on the structure of a type for further proof automation at compile time, take a look at the applyTactic tactic. Its argument should be a function that takes a reflected context and goal and gives back a new reflected tactic script. An example can be seen in the Data.Vect source.
So I suppose the summary is that Idris can't do what you want, and it probably never will be able to, but you might be able to make progress another way.
I understand what the concept of currying is, and know how to use it. These are not my questions, rather I am curious as to how this is actually implemented at some lower level than, say, Haskell code.
For example, when (+) 2 4 is curried, is a pointer to the 2 maintained until the 4 is passed in? Does Gandalf bend space-time? What is this magic?
Short answer: yes a pointer is maintained to the 2 until the 4 is passed in.
Longer than necessary answer:
Conceptually, you're supposed to think about Haskell being defined in terms of the lambda calculus and term rewriting. Lets say you have the following definition:
f x y = x + y
This definition for f comes out in lambda calculus as something like the following, where I've explicitly put parentheses around the lambda bodies:
\x -> (\y -> (x + y))
If you're not familiar with the lambda calculus, this basically says "a function of an argument x that returns (a function of an argument y that returns (x + y))". In the lambda calculus, when we apply a function like this to some value, we can replace the application of the function by a copy of the body of the function with the value substituted for the function's parameter.
So then the expression f 1 2 is evaluated by the following sequence of rewrites:
(\x -> (\y -> (x + y))) 1 2
(\y -> (1 + y)) 2 # substituted 1 for x
(1 + 2) # substituted 2 for y
3
So you can see here that if we'd only supplied a single argument to f, we would have stopped at \y -> (1 + y). So we've got a whole term that is just a function for adding 1 to something, entirely separate from our original term, which may still be in use somewhere (for other references to f).
The key point is that if we implement functions like this, every function has only one argument but some return functions (and some return functions which return functions which return ...). Every time we apply a function we create a new term that "hard-codes" the first argument into the body of the function (including the bodies of any functions this one returns). This is how you get currying and closures.
Now, that's not how Haskell is directly implemented, obviously. Once upon a time, Haskell (or possibly one of its predecessors; I'm not exactly sure on the history) was implemented by Graph reduction. This is a technique for doing something equivalent to the term reduction I described above, that automatically brings along lazy evaluation and a fair amount of data sharing.
In graph reduction, everything is references to nodes in a graph. I won't go into too much detail, but when the evaluation engine reduces the application of a function to a value, it copies the sub-graph corresponding to the body of the function, with the necessary substitution of the argument value for the function's parameter (but shares references to graph nodes where they are unaffected by the substitution). So essentially, yes partially applying a function creates a new structure in memory that has a reference to the supplied argument (i.e. "a pointer to the 2), and your program can pass around references to that structure (and even share it and apply it multiple times), until more arguments are supplied and it can actually be reduced. However it's not like it's just remembering the function and accumulating arguments until it gets all of them; the evaluation engine actually does some of the work each time it's applied to a new argument. In fact the graph reduction engine can't even tell the difference between an application that returns a function and still needs more arguments, and one that has just got its last argument.
I can't tell you much more about the current implementation of Haskell. I believe it's a distant mutant descendant of graph reduction, with loads of clever short-cuts and go-faster stripes. But I might be wrong about that; maybe they've found a completely different execution strategy that isn't anything at all like graph reduction anymore. But I'm 90% sure it'll still end up passing around data structures that hold on to references to the partial arguments, and it probably still does something equivalent to factoring in the arguments partially, as it seems pretty essential to how lazy evaluation works. I'm also fairly sure it'll do lots of optimisations and short cuts, so if you straightforwardly call a function of 5 arguments like f 1 2 3 4 5 it won't go through all the hassle of copying the body of f 5 times with successively more "hard-coding".
Try it out with GHC:
ghc -C Test.hs
This will generate C code in Test.hc
I wrote the following function:
f = (+) 16777217
And GHC generated this:
R1.p[1] = (W_)Hp-4;
*R1.p = (W_)&stg_IND_STATIC_info;
Sp[-2] = (W_)&stg_upd_frame_info;
Sp[-1] = (W_)Hp-4;
R1.w = (W_)&integerzmgmp_GHCziInteger_smallInteger_closure;
Sp[-3] = 0x1000001U;
Sp=Sp-3;
JMP_((W_)&stg_ap_n_fast);
The thing to remember is that in Haskell, partially applying is not an unusual case. There's technically no "last argument" to any function. As you can see here, Haskell is jumping to stg_ap_n_fast which will expect an argument to be available in Sp.
The stg here stands for "Spineless Tagless G-Machine". There is a really good paper on it, by Simon Peyton-Jones. If you're curious about how the Haskell runtime is implemented, go read that first.