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I have to prove that not all ternary linear codes contain the all zero codeword.
Some term explanations:
A linear code is a code that the sum or difference of any two codewords must also be a codeword. In other words, for all x, y in C, x +/- y is also in C.
A ternary codeword in this exercise is a codeword containing 0, 1 or 2. Eg, 1011, 1012, 012, ...
An all zero codeword is something like 0000, 000, 00000, ...
My attempt:
The addition of codewords must be done MOD 3 which means there are three allowable numbers in this MOD 3 world including 0, 1 and 2. For any ternary linear codes, there are cases when all the additions of any two codewords yeild non-zero codewords.
I am not sure about the proof. I have an idea that it must deal with the MOD 3 but not sure how to explain.
Thank you so much!
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Can anyone help me with the Mathematics of sparse categorical cross entropy loss function? I have searched for the derivation, explanation (Mathematical) but couldn't find any
I know it is not the right place to ask question like this. But I am helpless.
It is just cross entropy loss. The "sparse" refers to the representation it is expecting for efficiency reasons. E.g. in keras it is expected that label provided is an integer i*, an index for which target[i*] = 1.
CE(target, pred) = -1/n SUM_k [ SUM_i target_ki log pred_ki ]
and since we have sparse target, we have
sparse-CE(int_target, pred) = -1/n SUM_k [ log pred_k{int_target_k} ]
So instead of summing over label dimension we just index, since we know all remaining ones are 0s either way.
And overall as long as targets are exactly one class we have:
CE(target, pred) = CE(onehot(int_target), pred) = sparse-CE(int_target, pred)
The only reason for this distinction is efficiency. For regular classification with ~10-100 classes it does not really matter, but imagine word-level language models where we have thousands of classes.
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Converting units and measurements from one system to another can be achieved in most programming languages in one or another way. But, Can we express something like "12.Pounds.ToKilograms()" in any programming language?
Not exactly in that syntax but you may want to take a look at Frink: https://frinklang.org
Frink syntax is similar to Google Calculator or Wolfram Alpha but not exactly the same. Whereas Google and Wolfram Alpha uses the in keyword to trigger unit conversion Frink uses the -> operator. So in frink, the following is valid source code:
// Calculate length of Wifi antenna:
lightspeed / (2.4GHz) / 4 -> inches
As I mentioned, this syntax is similar to Google. For reference, the same calculation in Google syntax is speed of light / 2.4GHz / 4 in inches. Frink predates both Google calculator and Wolfram Alpha. I was first aware of frink sometime in the early 2000s.
Frink is unit aware. A number in frink always has unit even if that unit is simply "scalar" (no units). So to declare a variable that is 12 pounds you'd do:
var x = 12 pounds
To convert you'd do:
x -> kg
Or you can simply write the expression:
12 pounds -> kg
In Smalltalk you could express this as
12 pounds inKilograms
Notice however that it is up to you to program both messages pounds and inKilograms (there are libraries that do that kind of things also). But the key point is that the expression above is perfectly valid in Smalltalk (even if these messages do not exist).
I can't say I've ever seen this as a valid expression. 12 would have to be defined as a type. Let's assume 12 is an integer. An integer type only knows that it is an integer and in most languages there are built in functions/ methods for integers. For this to be a valid expression you would need to define a weight type and within that object you could define methods of conversion or inherit child types with conversion methods.
You could do it in Ruby because of it's syntactic sugar.
Actually there's a ruby gem - alchemist - you should check it out.
For example you could write:
10.miles.to.meters
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Let there be the following definition of gradient descent cost function
with the hypothesis function defined as
what I've come up with for multivariate linear regression is
theta = theta - alpha * 1/m * ([theta', -1]*[X';y']*X)';
h_theta = 1/(2*m)* (X*theta - y)'*(X*theta-y);
(octave notation, ' means matrix transpose, [A, n] means adding a new column to matrix A with scalar value n, [A; B] means appending matrix B to matrix A row-wise)
It's doing its job correctly how far I can tell (the plots look ok), however I have a strong feeling that it's unnecessarily complicated.
How to write it with as little matrix operations as possible (and no element-wise operations, of course)?
I don't think that is unnecessarily complicated, and instead this is what you want. Matrix operations are good because you don't have to loop over elements yourself or do element-wise operations. I remember taking a course online and my solution seems pretty similar.
The way you have it is the most efficient way of doing it as it is fully vectorized. It can be done by having a for loop over the summation and so on, however this is very inefficient in terms of processing power.
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Update:
Surprised that it is being so heavily downvoted...
The question is coding-related and before asking this question I have googled for "42" in combination with:
site:msdn.micrsoft.com
"code example"
"c#"
"magic number"
And I am not an expert/fan of Western culture/literature.
Also found, Why are variables “i” and “j” used for counters? [duplicate] which was not closed but even protected.
I feel that everybody knows it, except me...
What is the origin of ubiquitous magic digit 42 used all over the code samples and samples?
How have you come using 42? because I have not ever come or ever used 42
After some search, I found MSDN doc on it: Magic Numbers: Integers:
"Aside from a book/movie reference, developers often use this as an arbitrary value"
Well, this did not explain me anything.
Which movies and books have I missed for all those years of being involved in development, coding and programming and around-IT related activities like rwquirements analysis, system administration, etc??
Some references to some texts using code snippets with 42 (just C#-related):
Jérôme Laban. C# Async Tips and Tricks, Part 3: Tasks and the Synchronization Context
var t = Task.Delay(TimeSpan.FromSeconds(1))
.ContinueWith
(
_ => Task.Delay(TimeSpan.FromSeconds(42))
);
MSDN Asynchronous Agents Library
send(_target, 42);
Quickstart: Calling asynchronous APIs in C# or Visual Basic
Office.context.document.setSelectedDataAsync(
"<html><body>hello world</body></html>",
{coercionType: "html", asyncContext: 42},
function(asyncResult) {
write(asyncResult.status + " " + asyncResult.asyncContext);
Asynchronous Programming in C++ Using PPL
task<int> myTask = someOtherTask.then([]() { return 42; });
Boxing and Unboxing (C# Programming Guide)
Console.WriteLine(String.Concat("Answer", 42, true));
How To: Override the ToString Method (C# Programming Guide)
int x = 42;
Trace Listeners
// Use this example when debugging.
System.Diagnostics.Debug.WriteLine("Error in Widget 42");
// Use this example when tracing.
System.Diagnostics.Trace.WriteLine("Error in Widget 42");
|| Operator (C# Reference
// The following line displays True, because 42 is evenly
// divisible by 7.
Console.WriteLine("Divisible returns {0}.", Divisible(42, 7));
// The following line displays False, because 42 is not evenly
// divisible by 5.
Console.WriteLine("Divisible returns {0}.", Divisible(42, 5));
// The following line displays False when method Divisible
// uses ||, because you cannot divide by 0.
// If method Divisible uses | instead of ||, this line
// causes an exception.
Console.WriteLine("Divisible returns {0}.", Divisible(42, 0));
WIKIPedia C Sharp (programming language)
int foo = 42; // Value type.
It's from The Hitch Hiker's Guide to the Galaxy.
In The Hitchhiker's Guide to the Galaxy (published in 1979), the
characters visit the legendary planet Magrathea, home to the
now-collapsed planet-building industry, and meet Slartibartfast, a
planetary coastline designer who was responsible for the fjords of
Norway. Through archival recordings, he relates the story of a race of
hyper-intelligent pan-dimensional beings who built a computer named
Deep Thought to calculate the Answer to the Ultimate Question of Life,
the Universe, and Everything. When the answer was revealed to be 42,
Deep Thought explained that the answer was incomprehensible because
the beings didn't know what they were asking. It went on to predict
that another computer, more powerful than itself would be made and
designed by it to calculate the question for the answer. (Later on,
referencing this, Adams would create the 42 Puzzle, a puzzle which
could be approached in multiple ways, all yielding the answer 42.)
The answer is, as people already have stated, The Hitchhiker's Guide to the Galaxy.
I made a little experiment and put a couple of numbers in the search field, and these are the results:
It seems like 42 beats its neighbors clearly, but it can't touch regular numbers like 40, 45 and 50, no matter how magical it is.
It would be interesting to do the same search in source code only.
Dude!
It's the Answer to the Ultimate Question of Life, the Universe, and Everything! As computed by Deep Thought supercomputer, which took 7.5 million years!
http://en.wikipedia.org/wiki/The_answer_to_life_the_universe_and_everything#Answer_to_the_Ultimate_Question_of_Life.2C_the_Universe_and_Everything_.2842.29
Check this out. 42 is the ultimate answer to the ultimate question of life the universe and everything
This is from The Hitch hikers Guide to the Galaxy and is:
The Answer to the Ultimate Question of Life, the Universe, and Everything
WikiLink
Refer The Hitch Hiker's Guide to the Galaxy.
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I am looking for open-source implementations of tautology checkers written in GNU Prolog (implementation for SWI-Prolog is acceptable as well, but GNU Prolog is preferred).
I'd like to feed program input with queries like:
A and (B or C) iff (A or B) and (A or C).
or
3^2 * (X + 2) == (9 * X) + 18.
of course, notation can be different, like this:
(3 power 2) mul (X plus 2) equals (9 mul X) plus 18.
What I expect as result, is boolean answer , like "Yes/No", "Equals/Different", "Prove found/Failed to find prove" or similar.
I've found tautology checker for GNU-Prolog on ftp://ftp.cs.yorku.ca/pub/peter/SVT/GNU/ , but licence is not attached and no clue how to apply Gnu Prolog Arithmetic constraints and Arithmetic capabilities in order to extend just logical model with arithmetic.
Any other simmilar solvers?
Some examples how arithmetic capabilities might be used in order to extend model.
Thanks, Greg.
P.S. According arithmetic, I am looking for partial support - I want to handle only some basic cases, which I can code by hand with simple heuristics (gnu-prolog integer functions examples welcome as well) if proposed solution will handle classical logic properly and open-source code will be nice to extend :).
P.P.S As #larsmans noted, according to Gödel's incompleteness theorems there is no way to prove "all" formulas. That's why I am looking for something that proves, what can be proven from given set of axioms and rules, as I am looking for Gnu Prolog program, I am looking for examples of such sets of axioms and rules ;). Of course checker may fail - I am expecting it will work in "some" cases :). - On the other hand, there is Gödel's completeness theorem ;).
If you want an extensible theorem prover in Prolog, check out the family of lean theorem provers, of which leanCOP is the main representative; it handles classical first-order logic in 555 bytes of Prolog.
Version 1.0 is the following program:
prove(M,I) :- append(Q,[C|R],M), \+member(-_,C),
append(Q,R,S), prove([!],[[-!|C]|S],[],I).
prove([],_,_,_).
prove([L|C],M,P,I) :- (-N=L; -L=N) -> (member(N,P);
append(Q,[D|R],M), copy_term(D,E), append(A,[N|B],E),
append(A,B,F), (D==E -> append(R,Q,S); length(P,K), K<I,
append(R,[D|Q],S)), prove(F,S,[L|P],I)), prove(C,M,P,I).
The leanCOP website has more readable versions, with more features. You'll have to implement equality and arithmetic yourself.
You could use constraint logic programming for your problem. Equality
gives you directly a result (example GNU Prolog):
?- X*X #= 4.
X = 2
For inequality you will typically need to ask the system to generate
instanciations after the constraints have been set up. This is typically
done via a labeling directive (example GNU Prolog):
?- 27 * (X + 2) #\= (9 * X) + 18.
X = _#22(0..14913080)
?- 27 * (X + 2) #\= (9 * X) + 18, fd_labeling(X).
X = 0 ? ;
X = 1 ? ;
X = 2 ? ;
Etc..
A list that shows which prologs do have which kind of CLP can be found here. Just
check the CLP multi column.
Overview of Prolog Systems, Ulrich Neumerkel
Bye
I've found some tautology checkers, in Mathematical Logic for Computer Science by Ben-Ari, Mordechai, unfortunately they cover boolean logic. What advantage, his implementations are in Prolog, and present a variety of approaches related with automated proving, or solving such equations (or, automatically verifying proofs).