How is β-reduction a 2-morphism in Category theory?

According to Categorifying CCCs: Computation as a Process, computation or β-reduction process in untyped-lambda calculus is in fact a 2-morphism in category theory.

Can someone please describe me how is it so and elaborate on it?

P.S. I understand that category theory is a mathematical concept but since this specific question is about β-reduction and lambda calculus I have posted it in computer science section.

Can we think of information theory in terms of “a measure on set of information”?

In information theory, we deal with the quantities $ I(X;Y), H(X),H(Y), H(X|Y), H(Y|X)$ . These are just numbers, but I intuitively think of them as the “measure” of a set of information.

There is at least one special case where this interpretation is exact: suppose there are independent variables $ V_1,…V_n$ , and the variables $ X,Y,Z$ are tuples of $ V_i$ . Then we can literally think of $ I(X;Y)$ as the measure (entropy) of the intersection of $ X$ and $ Y$ .

But it is not obvious to me whether we can define a more general measure of information, such that mutual information can be interpreted as the measure of the intersection, and analogously for the other information quantities. Is this possible?

Using evaluation function in Game theory problems?

Assume you have a game tree and the features(f1, f2, f3…….fn​ ) that describe the state of the game at any node. Also assume that you are using depth-limited minimax and always expand up to a fixed depth d. Say you have the following four evaluation functions:

F1=w1∗f1+w2∗f2+w3∗f3+⋯+fn​

F2=w1∗f1^2+w2∗f2^2+w3∗f3^2+⋯+fn^2

F3=exp⁡{w1∗f1^2+w2∗f2^2+w3∗f3^2+⋯+fn^2}

F4=w1∗f1∗f2+w2∗f2∗f3+w3∗f3∗f4+⋯+wn∗fn−1∗fn

The weights (w1, w2, w3 …… wn​) are same for all evaluation functions. Then select all the statements that are correct:

  1. The optimal sequence of moves for the MAX player would be the different for the function F1 and F2.

  2. The optimal sequence of moves for the MAX player would be the different for the function F2 and F3

  3. The optimal sequence of moves for the MAX player would be the different for the function F1 and F3

  4. The optimal sequence of moves for the MAX player would be necessarily the same for all the functions

  5. None of the above are correct

I need help with this problem that I came across. My understanding on this is that unless weights are changed the evaluation function should be the same. that is, in this case statement 2 and 3 are the only ones correct.

Any ideas ?

There are languages I would define as “fermionic”, see below. What is the usual name for their theory?

What is the usual name of the theory I dubbed “fermionic languages”, for want of a better term?

Provisionally then, say a fermionic alphabet is a set of symbols, each of which is labeled by a spin: either a (signed) integer, making the symbol a boson, or a half integer, making it a fermion. To make things interesting, add the following conditions:

  1. there is at least one fermion in the alphabet,
  2. the alphabet is closed under spin inversion: an involutive mapping from each letter to one with a spin of the same magnitude and sign changed. “Scalar” bosons (those with spin 0) are allowed, not required, to map to themselves.

Of course, we immediately extend this to finite words, the spin of a word being the sum of the spins of its individual letters: this makes the free monoid on the alphabet an U-semigroup. Then, the sub-languages of the free monoid I called fermionic languages are those containing >= 1 fermionic word.

The prototypical spin function would map the generators of a group to their orders, assuming they are all finite: in which case, inversion in the group-theoretic sense is composed of the 2 commuting involutions, to wit, spin inversion of the letters and word reversal. I believe anyone dissecting the involutions of a finitely generated group would give a try to the concept, so there must be a name for it, if only in folklore.

However, I emphatically do NOT require fermion languages to be closed under any of spin inversion, word reversal, nor their composite, which I would call “GT-inversion”. I do not even require the set of spin values of these languages to be closed under change of sign, nor to contain the value 0. All I want is >= 1 fermionic word in the language under study, and immensely valuable properties such as:

  1. the empty word is not among its kin,
  2. spin inversion commutes with word reversal.

information theory, find entropy given Markov chain

There is an information source on the information source alphabet $ A = \{a, b, c\}$ represented by the state transition diagram below:

Markov chain

a) The random variable representing the $ i$ -th output from this information source is represented by $ X_i$ . It is known that the user is now in state $ S_1$ . In this state, let $ H (X_i|s_1)$ denote the entropy when observing the next symbol $ X_i$ , find the value of $ H (X_i|s_1)$ , entropy of this information source, Calculate $ H (X_i|X_{i-1}) $ and $ H (X_i)$ respectively. Assume $ i$ is quite large

How can I find $ H(X_i|s_1)?$ I know that $ $ H(X_i|s_1) = -\sum_{i,s_1} p\left(x_i, s_1\right)\cdot\log_b\!\left(p\left(x_i|s_1\right)\right) = -\sum_{i,j} p\left(x_i, s_1\right)\cdot\log_b\!\left(\frac{p\left(x_i, s_1\right)}{p\left(s_1\right)}\right)$ $ but I don’t know $ p(s_1)$ .

$ $ A=\begin{pmatrix}0.25 & 0.75 & 0\0.5 & 0 & 0.5 \0 & 0.7 & 0.3 \end{pmatrix}.$ $

From matrix I can know that $ p(s_1|s_1)=0.25$ , etc.

But what is the probability of $ s_1$ ? And how can I calculate $ H (X_i|X_{i-1})$ ?