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Search found 221 results on 9 pages for 'algebra'.

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• How to solve generic algebra using solver/library programmatically? Matlab, Mathematica, Wolfram etc?

  - by DevDevDev

  I'm trying to build an algebra trainer for students. I want to construct a representative problem, define constraints and relationships on the parameters, and then generate a bunch of Latex formatted problems from the representation. As an example: A specific question might be: If y < 0 and (x+3)(y-5) = 0, what is x? Answer (x = -3) I would like to encode this as a Latex formatted problem like. If $y<0$ and $(x+constant_1)(y+constant_2)=0$ what is the value of x? Answer = -constant_1 And plug into my problem solver constant_1 > 0, constant_1 < 60, constant_1 = INTEGER constant_2 < 0, constant_2 > -60, constant_2 = INTEGER Then it will randomly construct me pairs of (constant_1, constant_2) that I can feed into my Latex generator. Obviously this is an extremely simple example with no real "solving" but hopefully it gets the point across. Things I'm looking for ideally in priority order * Solve algebra problems * Definition of relationships relatively straight forward * Rich support for latex formatting (not just writing encoded strings) Thanks!

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• Mathematically Find Max Value without Conditional Comparison

  - by Cnich

  ----------Updated ------------ codymanix and moonshadow have been a big help thus far. I was able to solve my problem using the equations and instead of using right shift I divided by 29. Because with 32bits signed 2^31 = overflows to 29. Which works! Prototype in PHP $r = $x - (($x - $y) & (($x - $y) / (29))); Actual code for LEADS (you can only do one math function PER LINE!!! AHHHH!!!) DERIVDE1 = IMAGE1 - IMAGE2; DERIVED2 = DERIVED1 / 29; DERIVED3 = DERIVED1 AND DERIVED2; MAX = IMAGE1 - DERIVED3; ----------Original Question----------- I don't think this is quite possible with my application's limitations but I figured it's worth a shot to ask. I'll try to make this simple. I need to find the max values between two numbers without being able to use a IF or any conditional statement. In order to find the the MAX values I can only perform the following functions Divide, Multiply, Subtract, Add, NOT, AND ,OR Let's say I have two numbers A = 60; B = 50; Now if A is always greater than B it would be simple to find the max value MAX = (A - B) + B; ex. 10 = (60 - 50) 10 + 50 = 60 = MAX Problem is A is not always greater than B. I cannot perform ABS, MAX, MIN or conditional checks with the scripting applicaiton I am using. Is there any way possible using the limited operation above to find a value VERY close to the max?

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• Metric 3d reconstruction

  - by srand

  I'm trying to reconstruct 3D points from 2D image correspondences. My camera is calibrated. The test images are of a checkered cube and correspondences are hand picked. Radial distortion is removed. After triangulation the construction seems to be wrong however. The X and Y values seem to be correct, but the Z values are about the same and do not differentiate along the cube. The 3D points look like as if the points were flattened along the Z-axis. What is going wrong in the Z values? Do the points need to be normalized or changed from image coordinates at any point, say before the fundamental matrix is computed? (If this is too vague I can explain my general process or elaborate on parts)

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• Rotation Matrix calculates by column not by row

  - by pinnacler

  I have a class called forest and a property called fixedPositions that stores 100 points (x,y) and they are stored 250x2 (rows x columns) in MatLab. When I select 'fixedPositions', I can click scatter and it will plot the points. Now, I want to rotate the plotted points and I have a rotation matrix that will allow me to do that. The below code should work: theta = obj.heading * pi/180; apparent = [cos(theta) -sin(theta) ; sin(theta) cos(theta)] * obj.fixedPositions; But it wont. I get this error. ??? Error using == mtimes Inner matrix dimensions must agree. Error in == landmarkslandmarks.get.apparentPositions at 22 apparent = [cos(theta) -sin(theta) ; sin(theta) cos(theta)] * obj.fixedPositions; When I alter forest.fixedPositions to store the variables 2x250 instead of 250x2, the above code will work, but it wont plot. I'm going to be plotting fixedPositions constantly in a simulation, so I'd prefer to leave it as it, and make the rotation work instead. Any ideas? Also, fixed positions, is the position of the xy points as if you were looking straight ahead. i.e. heading = 0. heading is set to 45, meaning I want to rotate points clockwise 45 degrees. Here is my code: classdef landmarks properties fixedPositions %# positions in a fixed coordinate system. [x, y] heading = 45; %# direction in which the robot is facing end properties (Dependent) apparentPositions end methods function obj = landmarks(numberOfTrees) %# randomly generates numberOfTrees amount of x,y coordinates and set %the array or matrix (not sure which) to fixedPositions obj.fixedPositions = 100 * rand([numberOfTrees,2]) .* sign(rand([numberOfTrees,2]) - 0.5); end function obj = set.apparentPositions(obj,~) theta = obj.heading * pi/180; [cos(theta) -sin(theta) ; sin(theta) cos(theta)] * obj.fixedPositions; end function apparent = get.apparentPositions(obj) %# rotate obj.positions using obj.facing to generate the output theta = obj.heading * pi/180; apparent = [cos(theta) -sin(theta) ; sin(theta) cos(theta)] * obj.fixedPositions; end end end P.S. If you change one line to this: obj.fixedPositions = 100 * rand([2,numberOfTrees]) .* sign(rand([2,numberOfTrees]) - 0.5); Everything will work fine... it just wont plot.

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• Sparse constrained linear least-squares solver

  - by Jacob

  This great SO answer points to a good sparse solver, but I've got constraints on x (for Ax = b) such that each element in x is >=0 an <=N. The first thing which comes to mind is an QP solver for large sparse matrices. Also, A is huge (around 2e6x2e6) but very sparse with <=4 elements per row. Any ideas/recommendations? I'm looking for something like MATLAB's lsqlin but with huge sparse matrices.

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• Static Typing and Writing a Simple Matrix Library

  - by duckworthd

  Aye it's been done a million times before, but damnit I want to do it again. I'm writing a simple Matrix Library for C++ with the intention of doing it right. I've come across something that's fairly obvious in mathematics, but not so obvious to a strongly typed system -- the fact that a 1x1 matrix is just a number. To avoid this, I started walking down the hairy path of matrices as a composition of vectors, but also stumbled upon the fact that two vectors multiplied together could either be a number or a dyad, depending on the orientation of the two. My question is, what is the right way to deal with this situation in a strongly typed language like C++ or Java?

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• How do you calculate the reflex angle given to vectors in 3D space?

  - by Reimund

  I want to calculate the angle between two vectors a and b. Lets assume these are at the origin. This can be done with theta = arccos(a . b / |a| * |b|) However arccos gives you the angle in [0, pi], i.e. it will never give you an angle greater than 180 degrees, which is what I want. So how do you find out when the vectors have gone past the 180 degree mark? In 2D I would simply let the sign of the y-component on one of the vectors determine what quadrant the vector is in. But what is the easiest way to do it in 3D?

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• Finding coordinates of a point between two points?

  - by Nicros

  Doing some 3D stuff in wpf- want to use a simpler test to see if everything is working (before moving to curves). The basic question is given two points x1,y1,z1 and x2,y2,z2 I have calculated the distance between the points. But how to find the coordinates of another point (x3,y3,z3) that lies on that line at some distance? I.e. if my line is 100 long between -50,0,0 and 50,0,0 what are the coordinates of the point at 100 * 0.1 along the line? I think this is a simple formula but I haven't found it yet....

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• Examples of monoids/semigroups in programming

  - by jkff

  It is well-known that monoids are stunningly ubiquitous in programing. They are so ubiquitous and so useful that I, as a 'hobby project', am working on a system that is completely based on their properties (distributed data aggregation). To make the system useful I need useful monoids :) I already know of these: Numeric or matrix sum Numeric or matrix product Minimum or maximum under a total order with a top or bottom element (more generally, join or meet in a bounded lattice, or even more generally, product or coproduct in a category) Set union Map union where conflicting values are joined using a monoid Intersection of subsets of a finite set (or just set intersection if we speak about semigroups) Intersection of maps with a bounded key domain (same here) Merge of sorted sequences, perhaps with joining key-equal values in a different monoid/semigroup Bounded merge of sorted lists (same as above, but we take the top N of the result) Cartesian product of two monoids or semigroups List concatenation Endomorphism composition. Now, let us define a quasi-property of an operation as a property that holds up to an equivalence relation. For example, list concatenation is quasi-commutative if we consider lists of equal length or with identical contents up to permutation to be equivalent. Here are some quasi-monoids and quasi-commutative monoids and semigroups: Any (a+b = a or b, if we consider all elements of the carrier set to be equivalent) Any satisfying predicate (a+b = the one of a and b that is non-null and satisfies some predicate P, if none does then null; if we consider all elements satisfying P equivalent) Bounded mixture of random samples (xs+ys = a random sample of size N from the concatenation of xs and ys; if we consider any two samples with the same distribution as the whole dataset to be equivalent) Bounded mixture of weighted random samples Which others do exist?

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• Alternative to as3isolib?

  - by tedw4rd

  Hi everyone, I've been working on a Flash game that involves an isometric space. I've been using as3isolib for a while now, and I'm less than impressed with how easy it is to use. Whether I'm approaching it the wrong way or it's just not that great to use is a question for another post. Anyways, I've been thinking of a different way to approach the problem of isometric positions, and I think I've got an idea that might work. Essentially, each object that is to be rendered to the iso-space maintains a 3-coordinate position. Those items are then registered with a camera that projects that 3-coordinate position to a 2-coordinate point on the screen according to the math on this Wikipedia article. Then, the MovieClip is added to the stage (or to the camera's MovieClip, perhaps) at that point, and at a child index of the point's y-value. That way, I figure objects that are closer to the camera will be "above" the objects further away, and will get rendered over them. So my question, then, is two-fold: Do you think this idea will work the way I think it will? Are there any existing 3D matrix/vector packages that I should look at? I know there's a Matrix3 class in Flex 3, but we're not using Flex for this game. Thanks!

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• Algorithm(s) for rearranging simple symbolic algebraic expressions

  - by Gabe Johnson

  Hi, I would like to know if there is a straightforward algorithm for rearranging simple symbolic algebraic expressions. Ideally I would like to be able to rewrite any such expression with one variable alone on the left hand side. For example, given the input: m = (x + y) / 2 ... I would like to be able to ask about x in terms of m and y, or y in terms of x and m, and get these: x = 2*m - y y = 2*m - x Of course we've all done this algorithm on paper for years. But I was wondering if there was a name for it. It seems simple enough but if somebody has already cataloged the various "gotchas" it would make life easier. For my purposes I won't need it to handle quadratics. (And yes, CAS systems do this, and yes I know I could just use them as a library. I would like to avoid such a dependency in my application. I really would just like to know if there are named algorithms for approaching this problem.)

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• Sparse quadratic program solver

  - by Jacob

  This great SO answer points to a good sparse solver, but I've got constraints on x (for Ax = b) such that each element in x is >=0 an <=N. The first thing which comes to mind is an QP solver for large sparse matrices. Also, A is huge (around 2e6x2e6) but very sparse with <=4 elements per row. Any ideas/recommendations?

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• rotate a plane around a diagonal

  - by compie

  I would like to rotate a plane, not around a single (X or Y) axis, but around the diagonal (45 degrees between X and Y). How do I calculate the Rx and Ry given the Rdiagonal? (Rdiagonal is the amount of rotation I would like to achieve around the diagonal axis).

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• Solve equation from string to result in C

  - by Alexandre Cassagne

  Hi, I would like to know if anyone has info or experience on how to do something which sounds simple but doesn't look like it when trying to program it. The idea is : give a string containing an equation, such as : "2*x = 10" for example (this is simple, but it could get very complex, such as sqrt(54)*35=x^2; and so on....) and the program would return x = 5 and possibly give a log of how he got there. Is this doable ? If so, does anyone have a lead ? For info there is this site (http://www.numberempire.com/equationsolver.php) which does the same thing in PHP, but isn't open source. Thanks for any help !

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• What is SVD(singular value decomposition)

  - by shiva

  How does it actually reduce noise..can you suggest some nice tutorials?

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• Simple encryption - Sum of Hashes in C

  - by Dogbert

  I am attempting to demonstrate a simple proof of concept with respect to a vulnerability in a piece of code in a game written in C. Let's say that we want to validate a character login. The login is handled by the user choosing n items, (let's just assume n=5 for now) from a graphical menu. The items are all medieval themed: eg: _______________________________ | | | | | Bow | Sword | Staff | |-----------|-----------|-------| | Shield | Potion | Gold | |___________|___________|_______| The user must click on each item, then choose a number for each item. The validation algorithm then does the following: Determines which items were selected Drops each string to lowercase (ie: Bow becomes bow, etc) Calculates a simple string hash for each string (ie: `bow = b=2, o=15, w=23, sum = (2+15+23=40) Multiplies the hash by the value the user selected for the corresponding item; This new value is called the key Sums together the keys for each of the selected items; this is the final validation hash IMPORTANT: The validator will accept this hash, along with non-zero multiples of it (ie: if the final hash equals 1111, then 2222, 3333, 8888, etc are also valid). So, for example, let's say I select: Bow (1) Sword (2) Staff (10) Shield (1) Potion (6) The algorithm drops each of these strings to lowercase, calculates their string hashes, multiplies that hash by the number selected for each string, then sums these keys together. eg: Final_Validation_Hash = 1*HASH(Bow) + 2*HASH(Sword) + 10*HASH(Staff) + 1*HASH(Shield) + 6*HASH(Potion) By application of Euler's Method, I plan to demonstrate that these hashes are not unique, and want to devise a simple application to prove it. in my case, for 5 items, I would essentially be trying to calculate: (B)(y) = (A_1)(x_1) + (A_2)(x_2) + (A_3)(x_3) + (A_4)(x_4) + (A_5)(x_5) Where: B is arbitrary A_j are the selected coefficients/values for each string/category x_j are the hash values for each string/category y is the final validation hash (eg: 1111 above) B,y,A_j,x_j are all discrete-valued, positive, and non-zero (ie: natural numbers) Can someone either assist me in solving this problem or point me to a similar example (ie: code, worked out equations, etc)? I just need to solve the final step (ie: (B)(Y) = ...). Thank you all in advance.

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• mystified by qr.Q(): what is an orthonormal matrix in "compact" form?

  - by gappy

  R has a qr() function, which performs QR decomposition using either LINPACK or LAPACK (in my experience, the latter is 5% faster). The main object returned is a matrix "qr" that contains in the upper triangular matrix R (i.e. R=qr[upper.tri(qr)]). So far so good. The lower triangular part of qr contains Q "in compact form". One can extract Q from the qr decomposition by using qr.Q(). I would like to find the inverse of qr.Q(). In other word, I do have Q and R, and would like to put them in a "qr" object. R is trivial but Q is not. The goal is to apply to it qr.solve(), which is much faster than solve() on large systems.

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• linear combinations in python/numpy

  - by nmaxwell

  greetings, I'm not sure if this is a dumb question or not. Lets say I have 3 numpy arrays, A1,A2,A3, and 3 floats, c1,c2,c3 and I'd like to evaluate B = A1*c1+ A2*c2+ A3*c3 will numpy compute this as for example, E1 = A1*c1 E2 = A2*c2 E3 = A3*c3 D1 = E1+E2 B = D1+E3 or is it more clever than that? In c++ I had a neat way to abstract this kind of operation. I defined series of general 'LC' template functions, LC for linear combination like: template<class T,class D> void LC( T & R, T & L0,D C0, T & L1,D C1, T & L2,D C2) { R = L0*C0 +L1*C1 +L2*C2; } and then specialized this for various types, so for instance, for an array the code looked like for (int i=0; i<L0.length; i++) R.array[i] = L0.array[i]*C0 + L1.array[i]*C1 + L2.array[i]*C2; thus avoiding having to create new intermediate arrays. This may look messy but it worked really well. I could do something similar in python, but I'm not sure if its nescesary. Thanks in advance for any insight. -nick

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• Rejigging a floating point equation ...

  - by Jamie

  I'd like to know if there is a way to improve the accuracy of calculating a slope. (This came up a few months back here). It seems by changing: float get_slope(float dXa, float dXb, float dYa, float dYb) { return (dXa - dXb)/(dYa - dYb); } to float get_slope(float dXa, float dXb, float dYa, float dYb) { return dXa/(dYa - dYb) - dXb/(dYa - dYb); } might be an improvement. Suggestions? Edit: It's precision I'm after, not efficiency.

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• O'Reilly book clarification on 2d linear system

  - by Eric

  The Oreilly book "Learning openCV" states at page 356 : Quote Before we get totally lost, let’s consider a particular realistic situation of taking measurements on a car driving in a parking lot. We might imagine that the state of the car could be summarized by two position variables, x and y, and two velocities, vx and vy. These four variables would be the elements of the state vector xk. Th is suggests that the correct form for F is: x = [ x; y; vx; vy; ]k F = [ 1, 0, dt, 0; 0, 1, 0, dt; 0, 0, 1, 0; 0, 0, 0, 1; ] It seems natural to put 'dt' just there in the F matrix but I just don't get why. What if I have a n states system, how would I spray some "dt" in the F matrix?

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• Where can I find BLAS example code (in Fortran)?

  - by Feynman

  I have been searching for decent documentation on blas, and I have found some 315 pages of dense material that ctrl-f does not work on. It provides all the information regarding what input arguments the routines take, but there are a LOT of input arguments and I could really use some example code. I am unable to locate any. I know there has to be some or no one would be able to use these libraries! Specifically, I use ATLAS installed via macports on a mac osx 10.5.8 and I use gfortran from gcc 4.4 (also installed via macports). I am coding in Fortran 90. I am still quite new to Fortran, but I have a fair amount of experience with mathematica, matlab, perl, and shell scripting. I would like to be able to initialize and multiply a dense complex vector by a dense symmetric (but not hermitian) complex matrix. The elements of the matrix are defined through a mathematical function of the indices--call it f(i,j). Could anyone provide some code or a link to some code?

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• Permutations distinct under given symmetry (Mathematica 8 group theory)

  - by Yaroslav Bulatov

  Given a list of integers like {2,1,1,0} I'd like to list all permutations of that list that are not equivalent under given group. For instance, using symmetry of the square, the result would be {{2, 1, 1, 0}, {2, 1, 0, 1}}. Approach below (Mathematica 8) generates all permutations, then weeds out the equivalent ones. I can't use it because I can't afford to generate all permutations, is there a more efficient way? Update: actually, the bottleneck is in DeleteCases. The following list {2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0} has about a million permutations and takes 0.1 seconds to compute. Apparently there are supposed to be 1292 orderings after removing symmetries, but my approach doesn't finish in 10 minutes removeEquivalent[{}] := {}; removeEquivalent[list_] := ( Sow[First[list]]; equivalents = Permute[First[list], #] & /@ GroupElements[group]; DeleteCases[list, Alternatives @@ equivalents] ); nonequivalentPermutations[list_] := ( reaped = Reap@FixedPoint[removeEquivalent, Permutations@list]; reaped[[2, 1]] ); group = DihedralGroup[4]; nonequivalentPermutations[{2, 1, 1, 0}]

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• Solving linear system over integers with numpy

  - by A. R. S.

  I'm trying to solve an overdetermined linear system of equations with numpy. Currently, I'm doing something like this (as a simple example): a = np.array([[1,0], [0,1], [-1,1]]) b = np.array([1,1,0]) print np.linalg.lstsq(a,b)[0] [ 1. 1.] This works, but uses floats. Is there any way to solve the system over integers only? I've tried something along the lines of print map(int, np.linalg.lstsq(a,b)[0]) [0, 1] in order to convert the solution to an array of ints, expecting [1, 1], but clearly I'm missing something. Could anyone point me in the right direction?

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• Triangulation & Direct linear transform

  - by srand

  Following Hartley/Zisserman's Multiview Geometery, Algorithm 12: The optimal triangulation method (p318), I got the corresponding image points xhat1 and xhat2 (step 10). In step 11, one needs to compute the 3D point Xhat. One such method is Direct Linear Transform (DLT), mentioned in 12.2 (p312) and 4.1 (p88). The homogenous method (DLT), p312-313, states that it finds a solution as the unit singular vector corresponding to the smallest singular value of A, thus, A = [xhat1(1) * P1(3,:)' - P1(1,:)' ; xhat1(2) * P1(3,:)' - P1(2,:)' ; xhat2(1) * P2(3,:)' - P2(1,:)' ; xhat2(2) * P2(3,:)' - P2(2,:)' ]; [Ua Ea Va] = svd(A); Xhat = Va(:,end); plot3(Xhat(1),Xhat(2),Xhat(3), 'r.'); However, A is a 16x1 matrix, resulting in a Va that is 1x1. What am I doing wrong (and a fix) in getting the 3D point? For what its worth sample data: xhat1 = 1.0e+009 * 4.9973 -0.2024 0.0027 xhat2 = 1.0e+011 * 2.0729 2.6624 0.0098 P1 = 699.6674 0 392.1170 0 0 701.6136 304.0275 0 0 0 1.0000 0 P2 = 1.0e+003 * -0.7845 0.0508 -0.1592 1.8619 -0.1379 0.7338 0.1649 0.6825 -0.0006 0.0001 0.0008 0.0010 A = <- my computation 1.0e+011 * -0.0000 0 0.0500 0 0 -0.0000 -0.0020 0 -1.3369 0.2563 1.5634 2.0729 -1.7170 0.3292 2.0079 2.6624

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• prolog sets problem, stack overflow

  - by garm0nboz1a

  Hi. I'm gonna show some code and ask, what could be optimized and where am I sucked? sublist([], []). sublist([H | Tail1], [H | Tail2]) :- sublist(Tail1, Tail2). sublist(H, [_ | Tail]) :- sublist(H, Tail). less(X, X, _). less(X, Z, RelationList) :- member([X,Z], RelationList). less(X, Z, RelationList) :- member([X,Y], RelationList), less(Y, Z, RelationList), \+less(Z, X, RelationList). lessList(X, LessList, RelationList) :- findall(Y, less(X, Y, RelationList), List), list_to_set(List, L), sort(L, LessList), !. list_mltpl(List1, List2, List) :- findall(X, ( member(X, List1), member(X, List2)), List). chain([_], _). chain([H,T | Tail], RelationList) :- less(H, T, RelationList), chain([T|Tail], RelationList), !. have_inf(X1, X2, RelationList) :- lessList(X1, X1_cone, RelationList), lessList(X2, X2_cone, RelationList), list_mltpl(X1_cone, X2_cone, Cone), chain(Cone, RelationList), !. relations(List, E) :- findall([X1,X2], (member(X1, E), member(X2, E), X1 =\= X2), Relations), sublist(List, Relations). semilattice(List, E) :- forall( (member(X1, E), member(X2, E), X1 < X2), have_inf(X1, X2, List) ). main(E) :- relations(X, E), semilattice(X, E). I'm trying to model all possible graph sets of N elements. Predicate relations(List, E) connects list of possible graphs(List) and input set E. Then I'm describing semilattice predicate to check relations' List for some properties. So, what I have. 1) semilattice/2 is working fast and clear ?- semilattice([[1,3],[2,4],[3,5],[4,5]],[1,2,3,4,5]). true. ?- semilattice([[1,3],[1,4],[2,3],[2,4],[3,5],[4,5]],[1,2,3,4,5]). false. 2) relations/2 is working not well ?- findall(X, relations(X,[1,2,3,4]), List), length(List, Len), writeln(Len),fail. 4096 false. ?- findall(X, relations(X,[1,2,3,4,5]), List), length(List, Len), writeln(Len),fail. ERROR: Out of global stack ^ Exception: (11) setup_call_catcher_cleanup('$bags':'$new_findall_bag'(17852886), '$bags':fa_loop(_G263, user:relations(_G263, [1, 2, 3, 4|...]), 17852886, _G268, []), _G835, '$bags':'$destroy_findall_bag'(17852886)) ? abort % Execution Aborted 3) Mix of them to finding all possible semilattice does not work at all. ?- main([1,2]). ERROR: Out of local stack ^ Exception: (15) setup_call_catcher_cleanup('$bags':'$new_findall_bag'(17852886), '$bags':fa_loop(_G41, user:less(1, _G41, [[1, 2], [2, 1]]), 17852886, _G52, []), _G4767764, '$bags':'$destroy_findall_bag'(17852886)) ?

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