I remember solving a lot of indefinite integration problems. There are certain standard methods of solving them, but nevertheless there are problems which take a combination of approaches to arrive at a solution.
But how can we achieve the solution programatically.
For instance look at the online integrator app of Mathematica. So how do we approach to write such a program which accepts a function as an argument and returns the indefinite integral of the function.
PS. The input function can be assumed to be continuous(i.e. is not for instance sin(x)/x).
You have Risch's algorithm which is subtly undecidable (since you must decide whether two expressions are equal, akin to the ubiquitous halting problem), and really long to implement.
If you're into complicated stuff, solving an ordinary differential equation is actually not harder (and computing an indefinite integral is equivalent to solving y' = f(x)). There exists a Galois differential theory which mimics Galois theory for polynomial equations (but with Lie groups of symmetries of solutions instead of finite groups of permutations of roots). Risch's algorithm is based on it.
The algorithm you are looking for is Risch' Algorithm:
http://en.wikipedia.org/wiki/Risch_algorithm
I believe it is a bit tricky to use. This book:
http://www.amazon.com/Algorithms-Computer-Algebra-Keith-Geddes/dp/0792392590
has description of it. A 100 page description.
You keep a set of basic forms you know the integrals of (polynomials, elementary trigonometric functions, etc.) and you use them on the form of the input. This is doable if you don't need much generality: it's very easy to write a program that integrates polynomials, for example.
If you want to do it in the most general case possible, you'll have to do much of the work that computer algebra systems do. It is a lifetime's work for some people, e.g. if you look at Risch's "algorithm" posted in other answers, or symbolic integration, you can see that there are entire multi-volume books ("Manuel Bronstein, Symbolic Integration Volume I: Springer") that have been written on the topic, and very few existing computer algebra systems implement it in maximum generality.
If you really want to code it yourself, you can look at the source code of Sage or the several projects listed among its components. Of course, it's easier to use one of these programs, or, if you're writing something bigger, use one of these as libraries.
These expert systems usually have a huge collection of techniques and simply try one after another.
I'm not sure about WolframMath, but in Maple there's a command that enables displaying all intermediate steps. If you do so, you get as output all the tried techniques.
Edit:
Transforming the input should not be the really tricky part - you need to write a parser and a lexer, that transforms the textual input into an internal representation.
Good luck. Mathematica is very complex piece of software, and symbolic manipulation is something that it does the best. If you are interested in the topic take a look at these books:
http://www.amazon.com/Computer-Algebra-Symbolic-Computation-Elementary/dp/1568811586/ref=sr_1_3?ie=UTF8&s=books&qid=1279039619&sr=8-3-spell
Also, going to the source wouldn't hurt either. These book actually explains the inner workings of mathematica
http://www.amazon.com/Mathematica-Book-Fourth-Stephen-Wolfram/dp/0521643147/ref=sr_1_7?ie=UTF8&s=books&qid=1279039687&sr=1-7
Related
As the question describe itself "What is the core difference between algorithm and pseudocode?".
algorithm
An algorithm is a procedure for solving a problem in terms of the actions to be executed and the order in which those actions are to be executed. An algorithm is merely the sequence of steps taken to solve a problem. The steps are normally "sequence," "selection, " "iteration," and a case-type statement.
Pseudocode
Pseudocode is an artificial and informal language that helps programmers develop algorithms. Pseudocode is a "text-based" detail (algorithmic) design tool.
The rules of Pseudocode are reasonably straightforward. All statements showing "dependency" are to be indented. These include while, do, for, if, switch. Examples below will illustrate this notion.
I think all the other answers give useful explanations and definitions, but I'm going to give mine.
An algorithm is the idea of how to obtain some result from some input. It is an abstract concept; an algorithm is not something material by itself, but more something like an imagination or a deduction, a thing that only exists in the mind. In the broad sense, any sequence of steps that give you some thing(s) from other thing(s) could be called an algorithm. For example, if the screen of your computer is dirty, "spraying some glass cleaner on it and wipe it with a cloth" could be said to be an algorithm to solve the problem of how to obtain a clean screen form a dirty screen. It is important to note the difference between the problem itself (getting a clean screen) and the algorithm (wiping it with a cloth and cleaner); generally, several different algorithms are possible to solve the same problem. The idea of complexity is inherent to the algorithms itself, not the problem or the particular implementation or execution of the algorithm.
Pseudocode is a language to express algorithms. Since, as said before, algorithms are only concepts, we need to use something to express them and explain them to other people. Pseudocode is a convenient way for many computer science algorithms, because it is usually unambiguous, easy to read and somewhat similar to many programming languages. However, a specific programming language like C or Java can also be used to express and algorithm (it's just less convenient to those not familiarized with that language). In other cases, pseudocode may not be the best way to express an algorithm; for example, many graph and tree algorithms can be explained more easily with drawings or diagrams. In the previous example, the algorithm to get your screen cleaned is probably better expressed in a natural language like English, because it is simple and specific enough for that case.
Obviously, terms are frequently used loosely and exchanged depending on the context, and there's no need to be nitpicky about it, but I think it is important to have the difference clear. An algorithm doesn't stop being an algorithm just because it is written in Python instead of pseudocode. Pseudocode is just a convenient and widespread communication tool to express them.
An algorithm is something (a sequence of steps) you can do. Pseudocode is a notation to describe an algorithm.
Algorithm is something which is represented in mathematical terms. It includes, analysis, complexity considerations(best, average and worstcase analysis etc.).Pseudo code is a human readable representation of a program.
From Wikipedia :
Starting from an initial state and initial input (perhaps empty), the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states, eventually producing "output" and terminating at a final ending state.
With a pseudo language one can implement an algorithm without using a programming language such as C.
An example of pseudo language is Flow Charts.
I'm building a web-based programming language partially inspired by Prolog and Haskell (don't laugh).
It already has quite a bit of functionality, you can check out the prototype at http://www.lastcalc.com/. You can see the source here and read about the architecture here. Remember it's a prototype.
Currently LastCalc cannot simplify expressions or solve equations. Rather than hard-coding this in Java, I would like to enhance the fundamental language such that it can be extended to do these things using nothing but the language itself (as with Prolog). Unlike Prolog, LastCalc has a more powerful search algorithm, Prolog is "depth-first search with backtracking", LastCalc currently uses a heuristic best-first search.
Before delving into this I want to understand more about how other systems solve this problem, particularly Mathematica / Wolfram Alpha.
I assume the idea, at least in the general case, is that you give the system a bunch of rules for manipulation of equations (like a*(b+c) = a*b + a+c) specify the goal (eg. isolate variable x) and then let it loose.
So, my questions are:
Is my assumption correct?
What is the search strategy for applying rules? eg. depth first, breadth first, depth first with iterative deepening, some kind of best first?
If it is "best first", what heuristics are used to determine whether it is likely that a particular rule application has got us closer to our goal?
I'd also appreciate any other advice (except for "give up" - I regularly ignore that piece of advice and doing so has served me well ;).
I dealt with such questions myself some time ago. I then found this document about simplification of expressions. It is titled Rule-based Simplification of Expressions and shows some details about simplification in Mupad, which later became a part of Matlab.
According to this document, your assumption is correct. There is a set of rules for manipulation of expressions. A heuristic quality metric is is used as a target function for simplification.
Wolfram alpha is developed by Mathematica
mathematica is stephen wolphram's brainchild. Mathematica 1.0 was released in 1988. mathematica is much like maple and they both rely heavily on older software libraries like LaPack.
The libraries that these programs are, based on, and often simply, legacy software. They've been around, and modified, for a very long time.
If you would like to know about the background programs running, sagemath is a free open source alternative; you could possible reverse engineer the solutions to your questions:
SageMath.org
I understand that doing minimization in integer programming is a very complex problem. But what makes this problem so difficult?
If I were to (attempt) to write an algorithm to solve it, what would I need to take into account? I'm only familiar with the branch-and-bound technique for solving it and I'm wondering what sort of roadblocks I will face when attempting to apply this technique programatically.
I'm wondering what sort of roadblocks I will face when attempting to apply this technique programatically.
None in particular (assuming a fairly straightforward implementation without a lot of tricks). The algorithms aren’t complicated – they are complex, that’s a fundamental difference.
Techniques such as branch and bound or branch and cut try to prune the search tree and thus speed up the running time. But the whole problem tree is nevertheless exponentially large, hence the problem.
Like the other said, those problem are very hard and there are no simple solution nor simple algorithm that apply to all classes of problems.
The "classic" way of solving those problem is to do a branch-and-bound and apply the simplex algorithm at each node, as you say in your question. However, I would not recommand implementing this yourself if you are not an expert.
As for a lot of numerical methods, it is very hard to get it right (good parameter values, good optimisations), and a lot have been done (see CPLEX, COIN_OR, etc).
It's not that you can't do it: the branch-and-bound part is pretty straigtforward, but without all the tricks your program will be really slow.
Also, you will need a simplex implementation and this is not something you want to do yourself: you will have to use a third-part lib anyway.
Most likely, wether
if your data set is not that big (try it !), and you are not interested in solving it really fast: use something like COIN-OR or lp_solve with the default method, it will work;
if your data set is really big (and/or you need to find a solution quickly each time), you need to work with an expert in this field.
My main point is that only experienced people will know which algorithm will perform better on your problem, wich form of the model will be the easiest to solve, which method to apply and what kind of optimisations you can try.
If you are interested in those problems, I would recommend this book for an introduction to the math behind all this (with a lot of examples). It is incredibly expansive, so you may want to go to a library instead of buying it: Nemhauser and Wolsey.
Integer programming is NP-hard. That's why it is so difficult.
There is a tutorial that you might be interested.
The first thing you do before you solve any mathematical optimization problem is you categorize it. Except special cases, most of the time, integer programming problems will be np-hard. So instead of using an "algorithm", you will use a "heuristic". The final solution you will find will not be a guaranteed optimum, but it will be a pretty good solution for real life problems.
Your main roadblock will your programming skills. Heuristic programming requires a good level of programming understanding. So instead of programming your own heuristic you are better of using well known package (eg, COIN-OR, free). This way you can focus on your problem instead of the heuristic.
I am brainstorming an idea of developing a high level software to manipulate matrix algebra equations, tensor manipulations to be exact, to produce optimized C++ code using several criteria such as sizes of dimensions, available memory on the system, etc.
Something which is similar in spirit to tensor contraction engine, TCE, but specifically oriented towards producing optimized rather than general code.
The end result desired is software which is expert in producing parallel program in my domain.
Does this sort of development fall on the category of expert systems?
What other projects out there work in the same area of producing code given the constraints?
What you are describing is more like a Domain-Specific Language.
http://en.wikipedia.org/wiki/Domain-specific_language
It wouldn't be called an expert system, at least not in the traditional sense of this concept.
Expert systems are rule-based inference engines, whereby the expertise in question is clearly encapsulated in the rules. The system you suggest, while possibly encapsulating insight about the nature of the problem domain inside a linear algebra model of sorts, would act more as a black box than an expert system. One of the characteristics of expert systems is that they can produce an "explanation" of their reasoning, and such a feature is possible in part because the knowledge representation, while formalized, remains close to simple statements in a natural language; matrices and operations on them, while possibly being derived upon similar observation of reality, are a lot less transparent...
It is unclear from the description in the question if the system you propose would optimize existing code (possibly in a limited domain), or if it would produced optimized code, in that case driven bay some external goal/function...
Well production systems (rule systems) are one of four general approaches to computation (Turing machines, Church recursive functions, Post production systems and Markov algorithms [and several more have been added to that list]) which more or less have these respective realizations: imperative programming, functional programming, rule based programming - as far as I know Markov algorithms don't have an independent implementation. These are all Turing equivalent.
So rule based programming can be used to write anything at all. Also early mathematical/symbolic manipulation programs did generally use rule based programming until the problem was sufficiently well understood (whereupon the approach was changed to imperative or constraint programming - see MACSYMA - hmmm MACSYMA was written in Lisp so perhaps I have a different program in mind or perhaps they originally implemented a rule system in Lisp for this).
You could easily write a rule system to perform the matrix manipulations. You could keep a trace depending on logical support to record the actual rules fired that contributed to a solution (some rules that fire might not contribute directly to a solution afterall). Then for every rule you have a mapping to a set of C++ instructions (these don't have to be "complete" - they sort of act more like a semi-executable requirement) which are output as an intermediate language. Then that is read by a parser to link it to the required input data and any kind of fix up needed. You might find it easier to generate functional code - for one thing after the fix up you could more easily optimize the output code in functional source.
Having said that, other contributors have outlined a domain specific language approach and that is what the TED people did too (my suggestion is that too just using rules).
Both Wolfram Alpha and Bing are now providing the ability to solve complex, algebraic logic problems (ie "solve for x, given this equation"), and not just evaluate simple arithmetic expressions (eg "what's 5+5?"). How is this done?
I can read most types of code that might get thrown at me, so it doesn't really make a difference what you use to explain and represent the algorithm. I find that bash makes a really good pseudo-code, not to mention its actually functional, so that'd be ideal. Also, I'm fairly familiar with its in's and out's. Sorry to go ranting on a tangent, but it really irritates me to see people spend effort on crunching out "pseudocode" when they could be getting something 100% functional for just slightly more effort. Anyways, thanks so much for advance.
There are 2 main methods to solve:
Numeric methods. Numerical methods mean, basically, that the solver tries to change the value of x until the equation is satisfied. More info on numerical methods.
Symbolic math. The solver manipulates the equation as a string of symbols, by a number of formal rules. It's not that different from algebra we learn in school, the solver just knows a lot of different rules. More info on computer algebra.
Wolfram|Alpha (W|A) is based on the Mathematica kernel, combined with a natural language parser (which is also built primarily with Mathematica). They have a whole heap of curated data and associated formula that can be used once the question has been interpreted.
There's a blog post describing some of this which came out at the same time as W|A.
Finally, Bing simply uses the (non-free) API to answer questions via W|A.