Given two (4-1) multiplexers
How can I get the truth table of f(x1,x2,x3,x4) function??
A 4-1 multiplexer has the following general truth-table:
A1 A0 Y
0 0 I0
0 1 I1
1 0 I2
1 1 I3
The two control inputs A0 and A1 select which of the four inputs is switched through to the output.
To get your question solved, start with the left-hand multiplexer and write a truth-table for it.
In a second step write the overall truth-table by plugging in the intermediate signal values in the general table shown above.
The resulting truth-table has four input columns X1, X2, X3, X4.
There is one output column Y. Rather than using intermediate truth-tables you could figure out the output value for each of the 16 input combinations.
Related
Suppose, we want to design an arithmetic and logic unit (ALU) so that it performs the following operations:
Say,
Two inputs are A and B. cin represents carry in. s2,s1 and s0 are selection variables such that:
s2/cin s1 s0 Function
0 0 0 Transfer A
0 0 1 Addition
1 0 0 A+1
1 0 1 Addition with carry
x 1 0 A OR B
x 1 1 A XOR B
will the following circuit perform according to the above logic or any modification needed? The circuit has been designed for two stages.
Dude!, write a verilog code and see RTL view, It gives clear idea whether it works or not, because you can see functional simulation too.
In a logic circuit, I have an 8-bit data vector that is fed into an ECC IC which I am supposed to develop the logic for and that contains a vector of 5 Parity Bits. My first step to develop the logic (with logic gates, XOR), is to figure out which parity bit is going to check for which Data bits (since they are interlaced). I am using even parity, and following general hamming code rules (a parity bit in every 2^n ), I get the following sequence of output:
P1 P2 D1 P3 D2 D3 D4 P4 D5 D6 D7 D8 P5
Following the General Hamming Algorithm:
For each parity bit, Position 1,2,4,8,16 and so on... (Powers of 2), we skip for the first position n (n-1) and we check 1 bit, then we skip another one, the check another one, etc... we repeat the same process for the other bits, but this time checking/skipping every 2^n, where n is the position they occupy in the output array (P1 P2 D1 P3 D2 D3 D4 P4 D5 D6 D7 D8 P5)
Following that convention, I get:
P1 Checks data bits -> XOR(3 5 7 9 10 12)
P2 Checks data bits -> XOR(3 6 7 10 11)
P3 Checks data bits -> XOR(5 6 10 11 12)
P4 Checks data bits -> XOR(9 10 11)
Am I right? The thing that confuses me is that if I should start checking counting the parity bit as one of the 2^n bits that are supposed to be checked, or 1 bit after that specific parity bit. Pretty much sums up to if it is inclusive or not.
Thank you for your help in advance!
Cheers!
You can follow this sheme. The bits marked in each row must sum up to 0 (mod 2) in other words for the marked positions in each row the number of set bits must be even.
P1 P2 D1 P3 D2 D3 D4 P4 D5 D6 D7 D8
x x x x x x
x x x x x x
x x x x x
x x x x x
I don't understand why you have P5 in the scheme.
My task is to write a 16 bit ALU in verilog. I found difficulties when I do the part that needs to rotate the operand and doing the 2's complement addition and subtraction. I know how to work that out by paper and pencil but i cant figure out ways to do it in Verilog.
for example:
A is denoted as a15 a14 a13 a12 a11 a10 a9 a8 a7 a6 a5 a4 a3 a2 a1 a0
if i am going to rotate 4 bits,
the answer would be
a11 a10 a9 a8 a7 a6 a5 a4 a3 a2 a1 a0 a15 a14 a13 a12
i tried concatenation but it turns out to be incorrect.
need you all help...
The following will work using one shifter:
assign A_out = {A_in,A_in} >> (16-shift[3:0]);
When shift is 0 the left A_in is selected. As shift increase the left A_in shifts to the left and the MSBs of the right A_in fills in.
If synthesizing, then you may want to use muxes, as dynamic shift logic tends require more gates. A 16-bit barrel shifter will require 4 levels of 2-to-1 muxes.
wire [15:0] tmp [3:1];
assign tmp[3] = shift[3] ? { A_in[ 7:0], A_in[15: 8]} : A_in;
assign tmp[2] = shift[2] ? {tmp[3][11:0],tmp[3][15:12]} : tmp[3];
assign tmp[1] = shift[1] ? {tmp[2][13:0],tmp[2][15:14]} : tmp[2];
assign A_out = shift[0] ? {tmp[1][14:0],tmp[1][15 ]} : tmp[1];
assign A_out = A_in << bits_to_rotate;
Where bits_to_rotate can be a variable value (either a signal or a reg).
This will infer a generic shifter using multiplexers, or a barrel shifter, whatever suits better the target hardware. The synthetizer will take care about that.
Oh, well. If you want to rotate instead of shift, the thing is just a bit trickier:
assign A_out = (A_in << bits_to_rotate) | (A_in >> ~bits_to_rotate);
Why is concatenation incorrect? This should do what you ask.
assign A_out[15:0] = {A_in[11:0], A_in[15:12]};
The best way I found to do this is finding a pattern. When you want to rotate left an 8 bit signal 1 position (8'b00001111 << 1) the result is 8'b00011110) also when you want to rotate left 9 positions (8'b00001111 << 9) the result is the same, 8'b00011110, and also rotating 17 positions, this reduces your possibilities to next table:
So if you look, the three first bits of all numbers on tale equivalent to rotate 1 position (1,9,17,25...249) are equal to 001 (1).
The three first bits of all numbers on table equivalent to rotate 6 positions (6,14,22,30...254) are equal to 110 (6).
So you can apply a mask (8'b00000111) to determine the correct shifting by making zero all other bits:
reg_out_temp <= reg_in_1 << (reg_in_2 & 8'h07);
reg_out_temp shall be the double of reg_in_1, in this case reg_out_temp shall be 16 bit and reg_in_1 8 bit, so you can get the carried bits to the other byte when you shift the data so you can combine them using an OR expression:
reg_out <= reg_out_temp[15:8] | reg_out_temp[7:0];
So by two clock cycles you have the result. For a 16 bit rotation, your mask shall be 8'b00011111 (8'h1F) because your shifts goes from 0 to 16, and your temporary register shall be of 32 bits.
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I have asked this question in cstheory too
I have a form distribution problem. There is n forms in c categories (each form in 1 category). And there is u users, which each user can receive forms from at least one category (but maybe more than one category).
The goal is to distribute forms between users, so each user receive the same amount of forms. I prefer to equally use categories.
For example:
If categories are:
C1 : 20 forms
C2 : 3 forms
C3 : 8 forms
C4 : 2 forms
And users are:
U1 with access to C1 and C2
U2 with access to C2
U3 with access to C3
U4 with access to C1 and C3
U5 with access to C2 and C4
The answer should be:
U1: 1 x C1 + 1 x C2 | 2 x C1 (preferred)
U2: 2 x C2
U3: 2 x C3
U4: 1 x C1 + 1 x C3 | 2 x C1 (preferred) | 2 x C3
U5: 2 x C4
And 23 forms remains.
Do you have any suggestion on how can I write such algorithm?
There could be a second question, which in that some Categories have a SHOULD CONTRIBUTE option. Which if set, all remaining forms in that category will distribute between users who have access to that. for example if C1 have this option enabled, the answer should be:
U1: 1 x C1 + 1 x C2 + 9 C1
U2: 2 x C2
U3: 2 x C3
U4: 2 x C3 (to minimize remaining forms in C3 category) + 10 C1
U5: 2 x C4
and remaining forms would be 0 in C1, 0 in C2, 4 in C3 and 0 in C4.
I think its kinda Bin Packing algorithm, but I am not sure and I don't know how to solve it! :(
Note: The above answers are not best answers, these are just what I think!
It seems to me that if you fix a number N of forms per user and ask the question: can we give N forms to each user? then you can turn this into a http://en.wikipedia.org/wiki/Maximum_flow_problem problem, where each user can receive flow/forms from their subset of categories, and there is an outflow of capacity N from each user. Also, if you can solve this problem for N you can solve it for all lesser values of N.
So you could solve the first problem by running max-flow lg (maximum N) times, using a binary chop to find out what the best possible value of N is. Since you can solve it by max flow, you can also solve it by linear programming. Doing it this way, perhaps just for the critical value of N, might allow you to favour some assignments over others, or perhaps to see where there are neighbouring feasible solutions, and then see if you can mix them to use categories equally.
Example - Create a source, and link it to each of the categories Ci, with the capacity of the link being the number of forms available in that category, so C1 gets a link from the source of capacity 20. Create links with their source's capacity between users and categories, where the user has access to the category, so U1 gets links to C1 and C2, but U2 only gets a link to C2. Now create links of capacity N from each user to a single sink. If there is an assignment of forms to users that gives every user N forms, then this will produce a maximum flow that fills every link from user to sink, and you can look at the flows between users and categories to see how to assign forms. You could start off with N = 3, because user 2 only has access to a total of 3 forms, so the answer can't be greater than that. That won't work because you have said the right answer has N = 2, so the max flow won't fill all the N=3 capacity links. So your program tries again at 3/2 = 1, and finds a solution - you have provided a solution for N = 2, so there must be one for N = 1. Now the program knows there is a solution for N = 1 but not one for N = 3 so it tries one halfway between at N = (1 + 3) / 2 = 2, and finds your solution. There is one for N = 2 but not for N = 3 so the N = 2 solution is the best you can do.
Question
Do you think genetic algorithms worth trying out for the problem below, or will I hit local-minima issues?
I think maybe aspects of the problem is great for a generator / fitness-function style setup. (If you've botched a similar project I would love hear from you, and not do something similar)
Thank you for any tips on how to structure things and nail this right.
The problem
I'm searching a good scheduling algorithm to use for the following real-world problem.
I have a sequence with 15 slots like this (The digits may vary from 0 to 20) :
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
(And there are in total 10 different sequences of this type)
Each sequence needs to expand into an array, where each slot can take 1 position.
1 1 0 0 1 1 1 0 0 0 1 1 1 0 0
1 1 0 0 1 1 1 0 0 0 1 1 1 0 0
0 0 1 1 0 0 0 1 1 1 0 0 0 1 1
0 0 1 1 0 0 0 1 1 1 0 0 0 1 1
The constraints on the matrix is that:
[row-wise, i.e. horizontally] The number of ones placed, must either be 11 or 111
[row-wise] The distance between two sequences of 1 needs to be a minimum of 00
The sum of each column should match the original array.
The number of rows in the matrix should be optimized.
The array then needs to allocate one of 4 different matrixes, which may have different number of rows:
A, B, C, D
A, B, C and D are real-world departments. The load needs to be placed reasonably fair during the course of a 10-day period, not to interfere with other department goals.
Each of the matrix is compared with expansion of 10 different original sequences so you have:
A1, A2, A3, A4, A5, A6, A7, A8, A9, A10
B1, B2, B3, B4, B5, B6, B7, B8, B9, B10
C1, C2, C3, C4, C5, C6, C7, C8, C9, C10
D1, D2, D3, D4, D5, D6, D7, D8, D9, D10
Certain spots on these may be reserved (Not sure if I should make it just reserved/not reserved or function-based). The reserved spots might be meetings and other events
The sum of each row (for instance all the A's) should be approximately the same within 2%. i.e. sum(A1 through A10) should be approximately the same as (B1 through B10) etc.
The number of rows can vary, so you have for instance:
A1: 5 rows
A2: 5 rows
A3: 1 row, where that single row could for instance be:
0 0 1 1 1 0 0 0 0 0 0 0 0 0 0
etc..
Sub problem*
I'de be very happy to solve only part of the problem. For instance being able to input:
1 1 2 3 4 2 2 3 4 2 2 3 3 2 3
And get an appropriate array of sequences with 1's and 0's minimized on the number of rows following th constraints above.
Sub-problem solution attempt
Well, here's an idea. This solution is not based on using a genetic algorithm, but some ideas could be used in going in that direction.
Basis vectors
First of all, you should generate what I think of as the basis vectors. For instance, if your sequence were 3 numbers long rather than 15, the basis vectors would be:
v1 = [1 1 0]
v2 = [0 1 1]
v3 = [1 1 1]
Any solution for sequence length 3 would be a linear combination of these three vectors using only positive integers. In other words, the general solution would be
a*v1 + b*v2 + c*v3
where a, b and c are positive integers. For the sequence [1 2 1], the solution is v1 = 1, v2 = 1, v3 = 0. What you first want to do is find all of the possible basis vectors of length 15. From my rough calculations I think that there are somewhere between 300-400 basis vectors of length 15. I can give you some tips towards generating them if you want.
Finding solutions
Now, what you want to do is sort these basis vectors by their sums/magnitudes. Then in searching for your solution, you start with the basis vectors which have the largest sums. We start with the vectors that have the largest sums because they lead to having less total rows. We also have an array, veccoefs, which contains an entry for the linear coefficient for each basis vector. At the beginning of searching for the solution, all the veccoefs are 0.
So we take the first basis vector (the one with the largest sum/magnitude) and subtract this vector from the sequence until we either create an unsolvable result ( having a 0 1 0 in it for instance) or any of the numbers in the result is negative. We store the number of times we subtract the vector in veccoefs. We use the result after subtracting the basis vector from the sequence as the sequence for the next basis vector. If there are only zeros left in the result, then we stop the loop.
I'm not sure of the efficiency/accuracy of this method, but it might at least give you some ideas.
Other possible solutions
Another idea for solving this is to use the basis vectors and form the problem as an optimization/least squares problem. You form a matrix of the basis vectors such that the basic problem will be minimizing Sum[(Ax - b)^2] where A is the matrix of basis vectors, b is the input sequence, and x are the basis vector coefficients. However, you also want to minimize the number of rows, so you can add a term like x^T*x to the minimization function where x^T is the transpose of x. The hard part in my opinion is finding differentiable terms to add that will encourage integer vector coefficients. If you can think of a way to do that, then optimization could very well be a good way to do this.
Also, you might consider a Metropolis-type Monte Carlo solution. You would choose randomly whether to add a vector, remove a vector, or substitute a vector at each step. The vector to be added/removed/substituted would be chosen randomly. The probability of this change to be accepted would be a ratio of the suitabilities of the solutions before the change and after the change. The suitability could be equal to the difference between the current solution and the sequence, squared and summed, minus the number of rows/basis vectors involved in the solution. You would need to put in appropriate constants to for various terms to try to get the acceptance rate around 50%. I kind of doubt that this will work very well, but I thought that you should still consider it when looking for possible solutions.
GA can be applied to this problem, but it won't be 5 minute task. You need to put several things together, without knowing which implementation of each of them is best.
So:
Solution representation - how you will represent possible solution? Using matrix seems to be most straight forward. Using collection of one dimensional arrays is possible also.
But you have some constrains, so maybe SuperGene concept is worth considering?
You must use proper mutation/crossover operators for given gene representation.
How will you enforce constrains on solutions? Destroying those that are not proper? What if they contain valuable information? Maybe let them stay in population but add some penalty to fitness, so they will contribute to offspring, but won't go into next generations?
Anyway I think that GA can be applied to this problem. Is it worth? Usually GA are not best algorithm, but they are decent algorithm if others fail. I would go with GA, just because it would be most fun but I would look for alternative solution (just in case).
P.S. Personal insight: I was solving N Queens Problem, for 70 < N < 100 (board NxN, N queens). Algorithm was working fine for lower N (maybe it was trying all combination?), but with N in this range, I couldn't find proper solution. Fitness quickly jumped to about 90% of max, but in the end there were always two queens conflicting. But it was very naive implementation.