Many papers are using SAT, but few mentioned how to convert an addition to CNF.
Since CNF only allows AND OR NOT operation, it is difficult to describe addition operation. For example,
x1 + x2 + x3 + ... +x1599 < 30, xi is binary.
Map these equations into a Boolean circuit.
Apply Tseitin's transformation to the circuit and convert it into DIMACS format.
But is there any way to read the results? I do think it is possible to read the results if all the variables are defined by ourself, so figuring out how to convert a linear constraint to SAT problem is necessary.
If there are 3 or 4 variables, i.e. x1+x2+x3 <3, we can use truth table to solve this conversion. Also, a direct way is that chose 29 (any number smaller than 30) variables from 1600 variables to be 1, the others to be 0. But there are too many possibilities which makes this problem hard to solve.
I have used STP, but it can only give 1 answer. As the increasing number of variables and clauses, it costs a long time for STP to run.
So I tried to use SAT to solve the cnf given by STP, it can give out answers in a minutes. But the results cannot be read.
In the end, I found some paper,
1. Encoding Linear Constraints with Implication Chains to CNF,
2. SAT-Based Techniques for Integer Linear Constraints. This may be helpful.
What you're describing is known as a cardinality constraint. There are many ways of encoding these in CNF. As a starting point, some of these encodings are explained in
http://www.carstensinz.de/papers/CP-2005.pdf and
https://arxiv.org/pdf/1012.3853.pdf.
Many are implemented in the PySAT python toolkit
https://pysathq.github.io/docs/html/api/card.html
Related
[Related to https://codegolf.stackexchange.com/questions/12664/implement-superoptimizer-for-addition from Sep 27, 2013]
I am interested in how to write superoptimizers. In particular to find small logical formulae for sums of bits. This was previously set this as a challenge on codegolf but it seems a lot harder than one might imagine.
I would like to write code that finds the smallest possible propositional logical formula to check if the sum of y binary 0/1 variables equals some value x. Let us call the variables x1, x2, x3, x4 etc. In the simplest approach the logical formula should be equivalent to the sum. That is, the logical formula should be true if and only if the sum equals x.
Here is a naive way to do that. Say y=15 and x = 5. Pick all 3003 different ways of choosing 5 variables and for each make a new clause with the AND of those variables AND the AND of the negation of the remaining variables. You end up with 3003 clauses each of length exactly 15 for a total cost of 45054.
However, if you are allowed to introduce new variables into your solution then you can potentially reduce this a lot by eliminating common subformulae. So in this case your logical formula consists of the y binary variables, x and some new variables. The whole formula would be satisfiable if and only if the sum of the y variables equals x. The only allowed operators are and, or and not.
It turns out there is a clever method for solving this problem when x =1, at least in theory . However, I am looking for a computational intensive method to search for small solutions.
How can you make a superoptimizer for this problem?
Examples. Take as an example two variables where we want a logical formula that is True exactly when they sum to 1. One possible answer is:
(((not y0) and (y1)) or ((y0) and (not y1)))
To introduce a new variable into a formula such as z0 to represent y0 and not y1 then we can introduce a new clause (y0 and not y1) or not z0 and replace y0 and not y1 by z0 throughout the rest of the formula . Of course this is pointless in this example as it makes the expression longer.
Write your desired sum in binary. First look at the least important bit, y0 . Clearly,
x1 xor x2 xor ... xor xn = y0 - that's your first formula. The final formula will be a conjunction of formulae for each bit of the desired sum.
Now, do you know how an adder is implemented? http://en.wikipedia.org/wiki/Adder_(electronics) . Take inspiration from it, group your input into pairs/triples of bits, calculate the carry bits, and use them to make formulae for y1...yk . If you need further hints, let me know.
If I understand what you're asking, you'll want to look into the general topics of logic minimization and/or Boolean function simplification. The references are mostly about general methods for eliminating redundancy in Boolean formulas that are disjunctions ("or"s) of terms that are conjunctions ("and"s).
By hand, the standard method is called a Karnaugh map. The equivalent algorithm expressed in a way that's more amenable to computer implementation is Quine-McKlosky (also called the method of prime implicants). The minimization problem is NP-hard, and QM solves it exactly.
Therefore I think QM is what you want for the "super-optimizer" you're trying to build.
But the combination of NP-hard and exact solution means that QM is impractical for large problems, at least non-trivial ones.
The QM Algorithm lays out the conjunctive terms (called minterms in this context) in a table and conducts searches for 1-bit differences between pairs of terms. These terms can be combined and the factor for the differing bit labeled "don't care" in further combinations. This is repeated with 2-bit, 4-bit, etc. subsets of bits. The exponential behavior results because choices are involved for the combinations of larger bit sets: choosing one rules out another. Therefore it is essentially a search problem.
There is an enormous literature on heuristics to trim the search space, yet find "good" solutions that aren't necessarily optimal. A famous one is Espresso. However, since algorithm improvements translate directly to dollars in semiconductor manufacture, it's entirely possible that the best are proprietary and closely held.
I notice that almost all of new calculators are able to display the roots of quadratic equations in exact form. For example:
x^2-16x+14=0
x1=8+5sqrt2
x2=8-5sqrt2
What algorithm could I use to achieve that? I've been searching around but I found no results related to this problem
Assuming your quadratic equation is in the form
y = ax^2+bx+c
you get the two roots by
x_1,x_2 = ( -b +- sqrt(b^2-4ac)) / 2a
when for one you use the + between b and the square root, and for the other the -.
If you want to take something out of the square root, just compute the factors of the argument and take out the ones with exponent greater than 2.
By the way, the two root you posted are wrong.
The “algorithm” is exactly the same as on paper. Depending on the programming language, it may start with int delta = b*b - 4*a*c;.
You may want to define a datatype of terms and simplifications on them, though, in case the coefficients of the equation are not simply integer but themselves solutions of previous equations. If this is the sort of thing you are after, look up “symbolic computation”. Some languages are better for this purpose than others. I expect that elementary versions of what you are asking is actually used as an example in some ML tutorials, for instance (see chapter 9).
I was trying various methods to implement a program that gives the digits of pi sequentially. I tried the Taylor series method, but it proved to converge extremely slowly (when I compared my result with the online values after some time). Anyway, I am trying better algorithms.
So, while writing the program I got stuck on a problem, as with all algorithms: How do I know that the n digits that I've calculated are accurate?
Since I'm the current world record holder for the most digits of pi, I'll add my two cents:
Unless you're actually setting a new world record, the common practice is just to verify the computed digits against the known values. So that's simple enough.
In fact, I have a webpage that lists snippets of digits for the purpose of verifying computations against them: http://www.numberworld.org/digits/Pi/
But when you get into world-record territory, there's nothing to compare against.
Historically, the standard approach for verifying that computed digits are correct is to recompute the digits using a second algorithm. So if either computation goes bad, the digits at the end won't match.
This does typically more than double the amount of time needed (since the second algorithm is usually slower). But it's the only way to verify the computed digits once you've wandered into the uncharted territory of never-before-computed digits and a new world record.
Back in the days where supercomputers were setting the records, two different AGM algorithms were commonly used:
Gauss–Legendre algorithm
Borwein's algorithm
These are both O(N log(N)^2) algorithms that were fairly easy to implement.
However, nowadays, things are a bit different. In the last three world records, instead of performing two computations, we performed only one computation using the fastest known formula (Chudnovsky Formula):
This algorithm is much harder to implement, but it is a lot faster than the AGM algorithms.
Then we verify the binary digits using the BBP formulas for digit extraction.
This formula allows you to compute arbitrary binary digits without computing all the digits before it. So it is used to verify the last few computed binary digits. Therefore it is much faster than a full computation.
The advantage of this is:
Only one expensive computation is needed.
The disadvantage is:
An implementation of the Bailey–Borwein–Plouffe (BBP) formula is needed.
An additional step is needed to verify the radix conversion from binary to decimal.
I've glossed over some details of why verifying the last few digits implies that all the digits are correct. But it is easy to see this since any computation error will propagate to the last digits.
Now this last step (verifying the conversion) is actually fairly important. One of the previous world record holders actually called us out on this because, initially, I didn't give a sufficient description of how it worked.
So I've pulled this snippet from my blog:
N = # of decimal digits desired
p = 64-bit prime number
Compute A using base 10 arithmetic and B using binary arithmetic.
If A = B, then with "extremely high probability", the conversion is correct.
For further reading, see my blog post Pi - 5 Trillion Digits.
Undoubtedly, for your purposes (which I assume is just a programming exercise), the best thing is to check your results against any of the listings of the digits of pi on the web.
And how do we know that those values are correct? Well, I could say that there are computer-science-y ways to prove that an implementation of an algorithm is correct.
More pragmatically, if different people use different algorithms, and they all agree to (pick a number) a thousand (million, whatever) decimal places, that should give you a warm fuzzy feeling that they got it right.
Historically, William Shanks published pi to 707 decimal places in 1873. Poor guy, he made a mistake starting at the 528th decimal place.
Very interestingly, in 1995 an algorithm was published that had the property that would directly calculate the nth digit (base 16) of pi without having to calculate all the previous digits!
Finally, I hope your initial algorithm wasn't pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... That may be the simplest to program, but it's also one of the slowest ways to do so. Check out the pi article on Wikipedia for faster approaches.
You could use multiple approaches and see if they converge to the same answer. Or grab some from the 'net. The Chudnovsky algorithm is usually used as a very fast method of calculating pi. http://www.craig-wood.com/nick/articles/pi-chudnovsky/
The Taylor series is one way to approximate pi. As noted it converges slowly.
The partial sums of the Taylor series can be shown to be within some multiplier of the next term away from the true value of pi.
Other means of approximating pi have similar ways to calculate the max error.
We know this because we can prove it mathematically.
You could try computing sin(pi/2) (or cos(pi/2) for that matter) using the (fairly) quickly converging power series for sin and cos. (Even better: use various doubling formulas to compute nearer x=0 for faster convergence.)
BTW, better than using series for tan(x) is, with computing say cos(x) as a black box (e.g. you could use taylor series as above) is to do root finding via Newton. There certainly are better algorithms out there, but if you don't want to verify tons of digits this should suffice (and it's not that tricky to implement, and you only need a bit of calculus to understand why it works.)
There is an algorithm for digit-wise evaluation of arctan, just to answer the question, pi = 4 arctan 1 :)
I need to prove an inequality (or find a counter example) given several assumptions (also inequalities). Unfortunately the inequality to prove is a quite long and complicated expression. There are about 15 variables and FullSimplify's output fills several A4 pages. For examples with less variables, FindInstance helps to find a counterexample or gives a result of {} if the inequality is true. I also tried to use Reduce in that way:
Reduce[
Implies[
assumtion1 && assumtion2,
inequality
],
Reals
]
For simple examples this outputs "True", if the inequality holds. But in my case, after several hours of running time Mathematica needed 5-6 GB of RAM (and swap) so I had to abort the process.
Is there anything that I could do with Mathematica to improve the performance?
You will find a very nice paper on Mma CAD algorithms here
The cylindrical algebraic decomposition (CAD), which Mma uses, scales with a double exponential behavior on the number of variables.
Newer methods are double exponential in the number of quantifier alternations.
I think you'll have no luck using only the Mma internal engine, but you may roll your own based in the symmetries of your problem (if any)
Imagine, there are two same-sized sets of numbers.
Is it possible, and how, to create a function an algorithm or a subroutine which exactly maps input items to output items? Like:
Input = 1, 2, 3, 4
Output = 2, 3, 4, 5
and the function would be:
f(x): return x + 1
And by "function" I mean something slightly more comlex than [1]:
f(x):
if x == 1: return 2
if x == 2: return 3
if x == 3: return 4
if x == 4: return 5
This would be be useful for creating special hash functions or function approximations.
Update:
What I try to ask is to find out is whether there is a way to compress that trivial mapping example from above [1].
Finding the shortest program that outputs some string (sequence, function etc.) is equivalent to finding its Kolmogorov complexity, which is undecidable.
If "impossible" is not a satisfying answer, you have to restrict your problem. In all appropriately restricted cases (polynomials, rational functions, linear recurrences) finding an optimal algorithm will be easy as long as you understand what you're doing. Examples:
polynomial - Lagrange interpolation
rational function - Pade approximation
boolean formula - Karnaugh map
approximate solution - regression, linear case: linear regression
general packing of data - data compression; some techniques, like run-length encoding, are lossless, some not.
In case of polynomial sequences, it often helps to consider the sequence bn=an+1-an; this reduces quadratic relation to linear one, and a linear one to a constant sequence etc. But there's no silver bullet. You might build some heuristics (e.g. Mathematica has FindSequenceFunction - check that page to get an impression of how complex this can get) using genetic algorithms, random guesses, checking many built-in sequences and their compositions and so on. No matter what, any such program - in theory - is infinitely distant from perfection due to undecidability of Kolmogorov complexity. In practice, you might get satisfactory results, but this requires a lot of man-years.
See also another SO question. You might also implement some wrapper to OEIS in your application.
Fields:
Mostly, the limits of what can be done are described in
complexity theory - describing what problems can be solved "fast", like finding shortest path in graph, and what cannot, like playing generalized version of checkers (they're EXPTIME-complete).
information theory - describing how much "information" is carried by a random variable. For example, take coin tossing. Normally, it takes 1 bit to encode the result, and n bits to encode n results (using a long 0-1 sequence). Suppose now that you have a biased coin that gives tails 90% of time. Then, it is possible to find another way of describing n results that on average gives much shorter sequence. The number of bits per tossing needed for optimal coding (less than 1 in that case!) is called entropy; the plot in that article shows how much information is carried (1 bit for 1/2-1/2, less than 1 for biased coin, 0 bits if the coin lands always on the same side).
algorithmic information theory - that attempts to join complexity theory and information theory. Kolmogorov complexity belongs here. You may consider a string "random" if it has large Kolmogorov complexity: aaaaaaaaaaaa is not a random string, f8a34olx probably is. So, a random string is incompressible (Volchan's What is a random sequence is a very readable introduction.). Chaitin's algorithmic information theory book is available for download. Quote: "[...] we construct an equation involving only whole numbers and addition, multiplication and exponentiation, with the property that if one varies a parameter and asks whether the number of solutions is finite or infinite, the answer to this question is indistinguishable from the result of independent tosses of a fair coin." (in other words no algorithm can guess that result with probability > 1/2). I haven't read that book however, so can't rate it.
Strongly related to information theory is coding theory, that describes error-correcting codes. Example result: it is possible to encode 4 bits to 7 bits such that it will be possible to detect and correct any single error, or detect two errors (Hamming(7,4)).
The "positive" side are:
symbolic algorithms for Lagrange interpolation and Pade approximation are a part of computer algebra/symbolic computation; von zur Gathen, Gerhard "Modern Computer Algebra" is a good reference.
data compresssion - here you'd better ask someone else for references :)
Ok, I don't understand your question, but I'm going to give it a shot.
If you only have 2 sets of numbers and you want to find f where y = f(x), then you can try curve-fitting to give you an approximate "map".
In this case, it's linear so curve-fitting would work. You could try different models to see which works best and choose based on minimizing an error metric.
Is this what you had in mind?
Here's another link to curve-fitting and an image from that article:
It seems to me that you want a hashtable. These are based in hash functions and there are known hash functions that work better than others depending on the expected input and desired output.
If what you want a algorithmic way of mapping arbitrary input to arbitrary output, this is not feasible in the general case, as it totally depends on the input and output set.
For example, in the trivial sample you have there, the function is immediately obvious, f(x): x+1. In others it may be very hard or even impossible to generate an exact function describing the mapping, you would have to approximate or just use directly a map.
In some cases (such as your example), linear regression or similar statistical models could find the relation between your input and output sets.
Doing this in the general case is arbitrarially difficult. For example, consider a block cipher used in ECB mode: It maps an input integer to an output integer, but - by design - deriving any general mapping from specific examples is infeasible. In fact, for a good cipher, even with the complete set of mappings between input and output blocks, you still couldn't determine how to calculate that mapping on a general basis.
Obviously, a cipher is an extreme example, but it serves to illustrate that there's no (known) general procedure for doing what you ask.
Discerning an underlying map from input and output data is exactly what Neural Nets are about! You have unknowingly stumbled across a great branch of research in computer science.