I want to write the grammar for a list containing sequentially increasing integers like this
1 X
1 X 2 X
1 X 2 X 3 X
1 X 2 X 3 X 4 X
// and so on
I want to use a recursive definition to avoid defining separate rules for each case manually.
Without the increasing integers, I could write the following recursive grammar (in McKeeman Form)
NumberedList
Int X
Int X NumberedList
How can I specify the increasing integers?
No context-free grammar can represent this language because it is not context-free. Each expansion depends on the preceding expansion, which is an example of what the "context" in "context-free" is referring to.
It would be straightforward but tedious to write a context-sensitive grammar. That's the result of having to write out decimal arithmetic as a set of string substitution rules.
Related
Let's say I have a vector V, and I want to either turn this vector into multiple m x n matrices, or get multiple m x n matrices from this Vector V.
For the most basic example: Turn V = collect(1:75) into 3 5x5 matrices.
As far as I am aware this can be done by first using reshape reshape(V, 5, :) and then looping through it. Is there a better way in Julia without using a loop?
If possible, a solution that can easily change between row-major and column-major results is preferrable.
TL:DR
m, n, n_matrices = 4, 2, 5
V = collect(1:m*n*n_matrices)
V = reshape(V, m, n, :)
V = permutedims(V, [2,1,3])
display(V)
From my limited knowledge about Julia:
When doing V = collect(1:m*n), you initialize a contiguous array in memory. From V you wish to create a container of m by n matrices. You can achieve this by doing reshape(V, m, n, :), then you can access the first matrix with V[:,:,1]. The "container" in this case is just another array (thus you have a three dimensional array), which in this case we interpret as "an array of matrices" (but you could also interpret it as a box). You can then transpose every matrix in your array by swapping the first two dimensions like this: permutedims(V, [2,1,3]).
How this works
From what I understand; n-dimensional arrays in Julia are contiguous arrays in memory when you don't do any "skipping" (e.g. V[1:2:end]). For example the 2 x 4 matrix A:
1 3 5 7
2 4 6 8
is in memory just 1 2 3 4 5 6 7 8. You simply interpret the data in a specific way, where the first two numbers makes up the first column, then the second two numbers makes the next column so on so forth. The reshape function simply specifies how you want to interpret the data in memory. So if we did reshape(A, 4, 2) we basically interpret the numbers in memory as "the first four values makes the first column, the second four values makes the second column", and we would get:
1 5
2 6
3 7
4 8
We are basically doing the same thing here, but with an extra dimension.
From my observations it also seems to be that permutedims in this case reallocates memory. Also, feel free to correct me if I am wrong.
Old answer:
I don't know much about Julia, but in Python using NumPy I would have done something like this:
reshape(V, :, m, n)
EDIT: As #BatWannaBe states, the result is technically one array (but three dimensional). You can always interpret a three dimensional array as a container of 2D arrays, which from my understanding is what you ask for.
I would like to write the vector of monomials [1,x,y,z,x^2,xy,xz,y^2...,z^2]. My main problem is to write the number 1 as a monomial, and not as integer.
Per DICOM specification, a UID is defined by: 9.1 UID Encoding Rules. In other words the following are valid DICOM UIDs:
"1.2.3.4.5"
"1.3.6.1.4.35045.103501438824148998807202626810206788999"
"1.2.826.0.1.3680043.2.1143.5028470438645158236649541857909059554"
while the following are illegal DICOM UIDs:
".1.2.3.4.5"
"1..2.3.4.5"
"1.2.3.4.5."
"1.2.3.4.05"
"12345"
"1.2.826.0.1.3680043.2.1143.50284704386451582366495418579090595540"
Therefore I know that the string is at most 64 bytes, and should match the following regex [0-9\.]+. However this regex is really a superset, since there are a lot less than (10+1)^64 (=4457915684525902395869512133369841539490161434991526715513934826241L) possibilities.
How would one computes precisely the number of possibilities to respect the DICOM UID rules ?
Reading the org root / suffix rule clearly indicates that I need at least one dot ('.'). In which case the combination is at least 3 bytes (char) in the form: [0-9].[0-9]. In which case there are 10x10=100 possibilities for UID of length 3.
Looking at the first answer, there seems to be something unclear about:
The first digit of each component shall not be zero unless the
component is a single digit.
What this means is that:
"0.0" is valid
"00.0" or "1.01" are not valid
Thus I would say a proper expression would be:
(([1-9][0-9]*)|0)(\.([1-9][0-9]*|0))+
Using a simple C code, I could find:
f(0) = 0
f(1) = 0
f(2) = 0
f(3) = 100
f(4) = 1800
f(5) = 27100
f(6) = 369000
f(7) = 4753000
f(8) = 59049000
The validation of the Root UID part is outside the scope of this question. A second validation step could take care of rejecting some OID that cannot possibly be registered (some people mention restriction on first and second arc for example). For simplicity we'll accept all possible (valid) Root UID.
While my other answer takes good care of this specific application, here is a more generic approach. It takes care of situations where you have a different regular expression describing the language in question. It also allows for considerably longer string lengths, since it only requires O(log n) arithmetic operations to compute the number of combinations for strings of length up to n. In this case the number of strings grows so quickly that the cost of these arithmetic operations will grow dramatically, but that may not be the case for other, otherwise similar situations.
Build a finite state automaton
Start with a regular expression description of your language in question. Translate that regular expression into a finite state automaton. In your case the regular expression can be given as
(([1-9][0-9]*)|0)(\.([1-9][0-9]*|0))+
The automaton could look like this:
Eliminate ε-transitions
This automaton usually contains ε-transitions (i.e. state transitions which do not correspond to any input character). Remove those, so that one transition corresponds to one character of input. Then add an ε-transition to the accepting state(s). If the accepting states have other outgoing transitions, don't add ε-loops to them, but instead add an ε-transition to an accepting state with no outgoing edges and then add the loop to that. This can be seen as padding the input with ε at its end, without allowing ε in the middle. Taken together, this transformation ensures that performing exactly n state transitions corresponds to processing an input of n characters or less. The modified automaton might look like this:
Note that both the construction of the first automaton from the regular expression and the elimination of ε-transitions can be performed automatically (and perhaps even in a single step. The resulting automata might be more complicated than what I constructed here manually, but the principle is the same.
Ensuring unique paths
You don't have to make the automaton deterministic in the sense that for every combination of source state and input character there is only one target state. That's not the case in my manually constructed one either. But you have to make sure that every complete input has only one possible path to the accepting state, since you'll essentially be counting paths. Making the automaton deterministic would ensure this weaker property, too, so go for that unless you can ensure unique paths without this. In my example the length of each component clearly dictates which path to use, so I didn't make it deterministic. But I've included an example with a deterministic approach at the end of this post.
Build transition matrix
Next, write down the transition matrix. Associate the rows and columns with your states (in order a, b, c, d, e, f in my example). For each arrow in your automaton, write the number of characters included in the label of that arrow in the column associated with the source state and the row associated with the target state of that arrow.
⎛ 0 0 0 0 0 0⎞
⎜ 9 10 0 0 0 0⎟
⎜10 10 0 10 10 0⎟
⎜ 0 0 1 0 0 0⎟
⎜ 0 0 0 9 10 0⎟
⎝ 0 0 0 10 10 1⎠
Read result off that matrix
Now applying this matrix with a column vector once has the following meaning: if the number of possible ways to arrive in a given state is encoded in the input vector, the output vector gives you the number of ways one transition later. Take the 64th power of that matrix, concentrate on the first column (since ste start situation is encoded as (1,0,0,0,0,0), meaning only one way to end up in the start state) and sum up all the entries that correspond to accepting states (only the last one in this case). The bottom left element of the 64th power of this matrix is
1474472506836676237371358967075549167865631190000000000000000000000
which confirms my other answer.
Compute matrix powers efficiently
In order to actually compute the 64th power of that matrix, the easiest approach would be repeated squaring: after squaring the matrix 6 times you have an exponent of 26 = 64. If in some other scenario your exponent (i.e. maximal string length) is not a power of two, you can still perform exponentiation by squaring by multiplying the relevant squares according to the bit pattern of the exponent. This is what makes this approach take O(log n) arithmetic operations to compute the result for string length n, assuming a fixed number of states and therefore fixed cost for each matrix squaring.
Example with deterministic automaton
If you were to make my automaton deterministic using the usual powerset construction, you'd end up with
and sorting the states as a, bc, c, d, cf, cef, f one would get the transition matrix
⎛ 0 0 0 0 0 0 0⎞
⎜ 9 10 0 0 0 0 0⎟
⎜ 1 0 0 0 0 0 0⎟
⎜ 0 1 1 0 1 1 0⎟
⎜ 0 0 0 1 0 0 0⎟
⎜ 0 0 0 9 0 10 0⎟
⎝ 0 0 0 0 1 1 1⎠
and could sum the last three elements of the first column of its 64th power to obtain the same result as above.
Single component
Start by looking for ways to form a single component. The corresponding regular expression for a single component is
0|[1-9][0-9]*
so it is either zero or a non-zero digit followed by arbitrary many zero digits. (I had missed the possible sole zero case at first, but the comment by malat made me aware of this.) If the total length of such a component is to be n, and you write h(n) to denote the number of ways to form such a component of length exactly n, then you can compute that h(n) as
h(n) = if n = 1 then 10 else 9 * 10^(n - 1)
where the n = 1 case allows for all possible digits, and the other cases ensure a non-zero first digit.
One or more components
Subsection 9.1 only writes that a UID is a bunch of dot-separated number components, as outlined above. So in regular expressions that would be
(0|[1-9][0-9]*)(\.(0|[1-9][0-9]*))*
Suppose f(n) is the number of ways to write a UID of length n. Then you have
f(n) = h(n) + sum h(i) * f(n-i-1) for i from 1 to n-2
The first term describes the case of a single component, while the sum takes care of the case where it consists of more than one component. In that case you have a first component of length i, then a dot which accounts for the -1 in the formula, and then the remaining digits form one or more components which is expressed via the recursive use of f.
Two or more components
As the comment by cneller indicates, the part of section 9 before subsection 9.1 indicates that there has to be at least two components. So the proper regular expression would be more like
(0|[1-9][0-9]*)(\.(0|[1-9][0-9]*))+
with a + at the end indicating that we want at least one repetition of the parenthesized expression. Deriving an expression for this simply means leaving out the one-component-only case in the definition of f:
g(n) = sum h(i) * f(n-i-1) for i from 1 to n-2
If you sum all the g(n) for n from 3 (the minimal possible UID length) through 64 you get the number of possible UIDs as
1474472506836676237371358967075549167865631190000000000000000000000
or approximately 1.5e66. Which is considerably less than the 4.5e66 you get from your computation, in terms of absolute difference, although it's definitely on the same order of magnitude. By the way, your estimate doesn't explicitely mention UIDs shorter than 64, but you can always consider padding them with dots in your setup. I did the computation using a few lines of Python code:
f = [0]
g = [0]
h = [0, 10] + [9 * (10**(n-1)) for n in range(2, 65)]
s = 0
for n in range(1, 65):
x = 0
if n >= 3:
for i in range(1, n - 1):
x += h[i] * f[n-i-1]
g.append(x)
f.append(x + h[n])
s += x
print(h)
print(f)
print(g)
print(s)
I usually have no problem with vectorization in r, but I am having a tough time in the example below where there are both iterative and non-iterative components in the for loop.
In the code below, I have a calculation that I have to perform based on a set of constants (Dini), a vector of values (Xs), where the ith value of the output vector (Ys) is also dependent on i-1 value:
Dini=128 #constant
Xs=c(6.015, 5.996, 5.989, 5.911, 5.851, 5.851, 5.858, 5.851)
Y0=125.73251 #starting Y value
Ys=c(Y0) #starting of output vector, first value is known
for (Vi in Xs[2:length(Xs)]){
ytm1=Ys[length(Ys)]
y=(955.74301-2*((Dini+ytm1-Vi)^2-ytm1^2)^0.5+2*ytm1*acos(ytm1/(Dini+ytm1-Vi)))/pi/2
Ys=c(Ys, y)
}
df=data.frame(Xs, Ys)
df
Xs Ys
1 6.015 125.7325
2 5.996 125.7273
3 5.989 125.7251
4 5.911 125.7036
5 5.851 125.6859
6 5.851 125.6849
7 5.858 125.6868
8 5.851 125.6850
For this case, where there is a mix of both iterative and non iterative components in the for loop, my mind has got twisted in a non-vectorized knot.
Any suggestions?
You might want to look into use Reduce in this case. For example
Ys<-Reduce(function(prev, cur) {
(955.74301-2*((Dini+prev-cur)^2-prev^2)^0.5 + 2*prev*acos(prev/(Dini+prev-cur)))/pi/2
}, Xs, init=Y0, accumulate=T)[-1]
From the ?Reduce help page: "Reduce uses a binary function to successively combine the elements of a given vector and a possibly given initial value." This makes it easier to create vectors where a given value depends on a previous value.
I would like to check whether 2 vectors are the same in APL. Right now I am using the following solution (comparing element by element, summing the elements and comparing with size of vector a):
a←1 2 3
b←1 2 3
(+/a=b)=⍴a ⍝ it needs to return 0 or 1
Is there any quicker or more idiomatic solution?
You can use the match function which compares its entire arguments rather than equals which is a scalar function that compares the elements of each argument:
a←1 2 3
b←1 2 3 4 5
c←1 2 3
a≡b
0
a≡c
1
The match primitive, as mentioned above, returns 1 if the arguments are exactly identical. This means that they have the exact same rank, shape, data type, and content. In a few cases match will return a false negative because of data-type issues (division resulting in a floating point representation, even though it is within the comparison tolerance of an integer), or because a scaler will not match a 1-element vector.
^/a=b
will return a 1 if all elements of a test equal to corresponding elements of b, but it will fail with a LENGTH error if a and b are of different lengths, and it will use scaler extension, so that if a is 1 1 1 and b is a scaler 1, the result will be 1.
Match is usually better for this, and it is also more efficient on large arrays.