I tried logcontract(log(a)/2) but it left the expression unchanged. How would I get Maxima to do that?
Unfortunately there is no satisfactory way to accomplish that.
Maxima's simplification rules only work in one direction. Given an identity, e.g. f(a) = g(a), Maxima implements either f(a) --> g(a) or g(a) --> f(a). There isn't a way to switch from one to the other.
Given a rule f(a) --> g(a), it is possible to disable it, but it is clumsy. Let me know if you want to go down that road.
Related
I have created a grammar to read a file of equations then created AST nodes for each rule.My question is how can I do simplification or substitute vales on the equations that the parser is able to read correctly. in which stage? before creating AST nodes or after?
Please provide me with ideas or tutorials to follow.
Thank you.
I'm assuming you equations are something like simple polynomials over real-value variables, like X^2+3*Y^2
You ask for two different solutions to two different problems that start with having an AST for at least one equation:
How to "substitute values" into the equation and compute the resulting value, e.g, for X==3 and Y=2, substitute into the AST for the formula above and compute 3^2+3*2^2 --> 21
How to do simplification: I assume you mean algebraic simplification.
The first problem of substituting values is fairly easy if yuo already have the AST. (If not, parse the equation to produce the AST first!) Then all you have to do is walk the AST, replacing every leaf node containing a variable name with the corresponding value, and then doing arithmetic on any parent nodes whose children now happen to be numbers; you repeat this until no more nodes can be arithmetically evaluated. Basically you wire simple arithmetic into a tree evaluation scheme.
Sometimes your evaluation will reduce the tree to a single value as in the example, and you can print the numeric result My SO answer shows how do that in detail. You can easily implement this yourself in a small project, even using JavaCC/JJTree appropriately adapted.
Sometimes the formula will end up in a state where no further arithmetic on it is possible, e.g., 1+x+y with x==0 and nothing known about y; then the result of such a subsitution/arithmetic evaluation process will be 1+y. Unfortunately, you will only have this as an AST... now you need to print out the resulting AST in order for the user to see the result. This is harder; see my SO answer on how to prettyprint a tree. This is considerably more work; if you restrict your tree to just polynomials over expressions, you can still do this in small project. JavaCC will help you with parsing, but provides zero help with prettyprinting.
The second problem is much harder, because you must not only accomplish variable substitution and arithmetic evaluation as above, but you have to somehow encode knowledge of algebraic laws, and how to match those laws to complex trees. You might hardwire one or two algebraic laws (e.g., x+0 -> x; y-y -> 0) but hardwiring many laws this way will produce an impossible mess because of how they interact.
JavaCC might form part of such an answer, but only a small part; the rest of the solution is hard enough so you are better off looking for an alternative rather than trying to build it all on top of JavaCC.
You need a more organized approach for this: a Program Transformation System (PTS). A typical PTS will allow you specify
a grammar for an arbitrary language (in your case, simply polynomials),
automatically parses instance to ASTs and can regenerate valid text from the AST. A good PTS will let you write source-to-source transformation rules that the PTS will apply automatically the instance AST; in your case you'd write down the algebraic laws as source-to-source rules and then the PTS does all the work.
An example is too long to provide here. But here I describe how to define formulas suitable for early calculus classes, and how to define algebraic rules that simply such formulas including applying some class calculus derivative laws.
With sufficient/significant effort, you can build your own PTS on top of JavaCC/JJTree. This is likely to take a few man-years. Easier to get a PTS rather than repeat all that work.
Let's say in the example lower case is constant and upper case is variable.
I'd like to have programs that can "intelligently" do specified tasks like algebra, but teaching the program new methods should be easy using symbols understood by humans. For example if the program told these facts:
aX+bX=(a+b)X
if a=bX then X=a/b
Then it should be able to perform these operations:
2a+3a=5a
3x+3x=6x
3x=1 therefore x=1/3
4x+2x=1 -> 6x=1 therefore x= 1/6
I was trying to do similar things with Prolog as it can easily "understand" variables, but then I had too many complications, mainly because two describing a relationship both ways results in a crash. (not easy to sort out)
To summarise: I want to know if a program which can be taught algebra by using mathematic symbols only. I'd like to know if other people have tried this and how complicated it is expected to be. The purpose of this is to make programming easier (runtime is not so important)
It depends on what do you want machine to do and how intelligent it should be.
Your question is mostly about AI but not ML. AI deals with formalization of "human" tasks while ML (though being a subset of AI) is about building models from data.
Described program may be implemented like this:
Each fact form a pattern. Program given with an expression and some patterns can try to apply some of them to expression and see what happens. If you want your program to be able to, for example, solve quadratic equations given rule like ax² + bx + c = 0 → x = (-b ± sqrt(b²-4ac))/(2a) then it'd be designed as follows:
Somebody gives a set of rules. Rule consists of a pattern and an outcome (solution or equivalent form). Think about the pattern as kind of a regular expression.
Then the program is asked to show some intelligence and prove its knowledge via doing something with a given expression. Here comes the major part:
you build a graph of expressions by applying possible rules (if a pattern is applicable to an expression you add new vertex with the corresponding outcome).
Then you run some path-search algorithm (A*, for example) to find sequence of transformations leading to the form like x = ...
I think this is an interesting question, although it off topic in SO (tool recommendation)
But nevertheless, because it captured my imagination, I wrote couple of function using R that can solve stuff like that quite easily
First, you'll have to install R, after words you'll need to download package called stringr
So in R console run
install.packages("stringr")
library(stringr)
And then you can define the following functions that I wrote
FirstFunc <- function(temp){
paste0(eval(parse(text = gsub("[A-Z]", "", temp))), unique(str_extract_all(temp, "[A-Z]")[[1]]))
}
SecondFunc <- function(temp){
eval(parse(text = strsplit(temp, "=")[[1]][2])) / eval(parse(text = gsub("[[:alpha:]]", "", strsplit(temp, "=")[[1]][1])))
}
Now, the first function will solve equations like
aX+bX=(a+b)X
While the second will solve equations like
4x+2x=1
For example
FirstFunc("3X+6X-2X-3X")
will return
"4X"
Now this functions is pretty primitive (mostly for the propose of illustration) and will solve equation that contain only one variable type, something like FirstFunc("3X-2X-2Y") won't give the correct result (but the function could be easily modified)
The second function will solve stuff like
SecondFunc("4x-2x=1")
will return
0.5
or
SecondFunc("4x+2x*3x=1")
will return
0.1
Note that this function also works only for one unknown variable (x) but could be easily modified too
as far as I understand the FOLLOW-Set is there to tell me at the first possible moment if there is an error in the input stream. Is that right?
Because otherwise I'm wondering what you actually need it for. Consider you're parser has a non-terminal on top of the stack (in our class we used a stack as abstraction for LL-Parsers)
i.e.
[TOP] X...[BOTTOM]
The X - let it be a non-terminal - is to be replaced in the next step since it is at the top of the stack. Therefore the parser asks the parsing table what derivation to use for X. Consider the input is
+ b
Where + and b are both terminals.
Suppose X has "" i.e. empty string in its FIRST set. And it has NO + in his FIRST set.
As far as I see it in this situation, the parser could simply check that there is no + in the FIRST set of X and then use the derivation which lets X dissolve into an "" i.e. empty string since it is the only way how the parser possibly can continue parsing the input without throwing an error. If the input stream is invalid the parser will recognize it anyway at some moment later in time. I understand that the FOLLOW set can help here to right away identify whether parsing can continue without an error or not.
My question is - is that really the only role that the FOLLOW set plays?
I hope my question belongs here - I'm sorry if not. Also feel free to request clarification in case something is unclear.
Thank you in advance
You are right about that. The parser could eventually just continue parsing and would eventually find a conflict in another way.
Besides that, the FOLLOW set can be very convenient in reasoning about a grammar. Not by the parser, but by the people that constructs the grammar. For instance, if you discover, that any FIRST/FIRST or FIRST/FOLLOW conflict exists, you have made an ambiguous grammar, and may want to revise it.
E -> A | B
A -> a | c
B -> b | c
My answer is no because it has a reduce/reduce conflict, can anyone else verify this?
Also I gained my answer through constructing the transition diagram, is there a simpler way of finding this out?
Thanks for the help!
P.S Would a Recursive Descent be able to parse this?
You're right -- starting from a 'c' in the input there's no way to decide whether to treat that as an 'A' or a 'B'. I doubt there's anything that can really parse this properly -- it's simply ambiguous. Using a different type of parser won't help; you really need to change the language.
There are some formal methods for detecting such ambiguities, but I can hardly imagine bother with them for a grammar this small. One easy way to spot this particular problem is to mentally arrange it into a tree:
The two lines coming up out of the 'c' box represent the reduce/reduce conflict. There's no reason to prefer one route from 'c' to 'E' over the other, so the grammar is ambiguous.
First of all sorry for my English.
I would like to use a backtracking algorithm in Erlang. It would serve as a guessing to solve partially filled sudokus. A 9x9 sudoku is stored as a list of 81 elements, where every element stores the possible number which can go into that cell.
For a 4x4 sudoku my initial solution looks like this:
[[1],[3],[2],[4],[4],[2],[3],[1],[2,3],[4],[1],[2,3],[2,3],[1],[4],[2,3]]
This sudoku has 2 solutions. I have to write out both of them. After that initial solution reached, I need to implement a backtracking algorithm, but I don't know how to make it.
My thought is to write out the fixed elements into a new list called fixedlist which will change the multiple-solution cells to [].
For the above mentioned example the fixedlist looks like this:
[[1],[3],[2],[4],[4],[2],[3],[1],[],[4],[1],[],[],[1],[4],[]]
From here I have a "sample", I look for the lowest length in the solutionlist which is not equal to 1, and I try the first possible number of this cell and I put it to that fixedlist. Here I have an algorithm to update the cells and checks if it is still a solvable sudoku or not. If not, I don't know how to step back one and try a new one.
I know the pseudo code of it and I can use it for imperative languages but not for erlang. (prolog actually implemented backtrack algorithm, but erlang didn't)
Any idea?
Re: My bactracking functions.
These are the general functions which provide a framework for handling back-tracking and logical variables similar to a prolog engine. You must provide the function (predicates) which describe the program logic. If you write them as you would in prolog I can show you how to translate them into erlang. Very briefly you translate something like:
p :- q, r, s.
in prolog into something like
p(Next0) ->
Next1 = fun () -> s(Next0) end,
Next2 = fun () -> r(Next1) end,
q(Next2).
Here I am ignoring all other arguments except the continuations.
I hope this gives some help. As I said if you describe your algorithms I can help you translate them, I have been looking for a good example. You can, of course, just as well do it by yourself but this provides some help.