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Does the verbal component for spellcasting have to be words?

To make a long story short: for my first D&D campaign, I want to create a warlock that had to give up her voice as a part of her deal with her Patron.

As such, she is incapable of speaking, but she can still produce sounds with her mouth. I was wondering if that would incapacitate her from casting spells with a Verbal component.

Do spells needs a specific phrase to be cast, or does gibberish work?

How to remove any words containing two adjacent characters with different cases?

I have a list of permutations of ABCabc and I want to remove any permutations with two adjacent characters with different cases (uppercase and lowercase).

For example,

  • ABCcab is kept.
  • ABCacb must be removed because Ca contains two adjacent characters with different cases.
  • AbBcaC must be removed as well.

Attempt

Here is my attempt but without filtering.

Select[StringJoin /@ Permutations[Characters@"ABCabc"],....] 

Would a two word domain name with a shared letter between the words be treated the same for SEO as the fuller name?

I found a domain name (2 words) that is not available. Then I found almost the same domain name with a slight difference, the last letter of the first word is the same of the first letter of the second word.

Ex : jamessecret.com (not available)      jamesecret.com (available) 

In terms of SEO, is it seen as the same to Google? Would search engines interpret them the same?

If we want to map abbreviations of full-English words (e.g. map “Jan” to “January”), how can we identify abbreviations which map to multiple words?


Short Version:

How can we construct such a trie which maps abbreviations of names-of-the-month to full-month (we map the abbreviation "mar" to "march")?

  • The set of all abbreviations is formed by:
    • keeping the first letter of the month name. (all abbreviations of "january" begin with "j")
    • deleting 1 or more characters ("jan" deletes "uary" from "january")

The Looonnnnggg Version:

How can we construct such a trie?
What algorithm will build the appropriate trie from the container of verbose strings.

Consider the English names for months of the year:

  • January
  • February
  • March
  • April
  • [… truncated …]
  • October
  • November
  • December

We find it useful say that the English names for months of the year are "verbose" strings.
For any "verbose" string $ v$ , and any string $ a$ we say that $ a$ is an "abbreviation" of $ v$ if and only if all of the following conditions are met:

  • $ a$ non-empty. $ |a| \geq 1$
  • $ a$ can formed by deleting 1 or more characters from "verbose" string $ v$
  • $ a(1) = v(1)$ . Assume that string indexing begins at $ 1$ , and not $ 0$ .

For example, "jan" is an abbreviation of "january."

Suppose you want to write an algorithm which:

  • accepts a list of verbose strings as inputs.
  • the algorithm outputs a "trie" data-structure (information retrieval tree) $ T$ such that:
    • The trie $ T$ accepts any ASCII string as input.
    • An output (leaf node) of the trie should be set of strings $ S$ such that:
      • every string in $ S$ is a verbose string
      • the string fed as input into trie $ T$ is an abbreviation of every verbose string in container $ S$

Some examples of input to the trie and output of the trie are shown below:

  • Example 1

    • Input: "Ma"
    • Output: $ \{$ "March", "May"$ \}$
  • Example 2

    • Input: "Mar"
    • Output: $ \{$ "March"$ \}$
  • Example 3

    • Input: "Decuary"
    • Output: $ \{$ " "$ \}$ ….. the empty-set

The output from the trie should be one of:

  • the empty set
  • a set of one item
  • a set of two or more items

For months of the year, we might write javascript so that an end-user can type in any half-way reasonable date-format, instead getting an error message when they put slashes instead of dashes, etc….

If you do not like the months of the year application, a different use-case would be to write write your own Linux Shell (similar to BASH). Maybe any half-way reasonable abbreviation of "make directory" will map to "mkdir" In that case, we could have many-to-one mapping from high-level shell-commands to low-level Linux commands.

The question is:

How can we construct such a trie?
What algorithm will build the appropriate trie from the container of verbose strings.

Also, can we avoid brute-force generating a list of all aberrations before-hand? The set of all strings form-able by deleting 1 or more characters from the verbose strings is quite large. We would like to avoid combinatorial explosion, if we can.

The programming language (Java, python, C+ + ) does not matter for answering this question.

Number of words of length n for special language

Let $ \Sigma$ be an alphabet and let $ L$ be a language over it with the following properties:

  1. if $ w\in L$ then there exists $ v\in \Sigma^*$ such that $ wv \in L$ and for every $ s\in \Sigma$ the word $ wvs$ does not lie in $ L$
  2. $ wv\in L$ then $ vw \in L$
  3. It is prefix-closed, i.e. prefix of any word is still in the language.

Note that by the definition, it is not cyclic language. I’m trying to compute its growth function, by that I mean $ \gamma_n:= |\{w\in L \mid |w| = n\}|$ . I know about my specific case that it is not regular and my hypothesis is that function $ \Gamma(x) = \sum_{n=1}^\infty \gamma_nx^n$ is not rational. However, I couldn’t find any information about these functions for non-regular languages. Maybe, there’s a formula that connects entropy of language, i.e. $ e(L):= \limsup\limits_{n\to\infty} \frac{\log\gamma_n}{n}$ and the $ \Gamma$ function. Or for such a language there’s a way to describe its growth throughout the growth of the language $ \operatorname{End}(L) = \{ w\in L \mid \forall s\in \Sigma \,ws \text{ is not in } L \}$ .

Is my recursive algorithm for Equivalent Words correct?

Here is my problem.

Problem Given two words and a dictionary, find out whether the words are equivalent.

Input: The dictionary, D (a set of words), and two words v and w from the dictionary.

Output: A transformation of v into w by substitutions such that all intermediate words belong to D. If no transformation is possible, output “v and w are not equivalent.”

I need to write both recursive and dynamic programming algorithm. As for recursion, I came up with this algorithm. Is it correct?

 EquivalentWordsProblem(v, w, D)  1.m <- len (v)  2.n <-  len (w)  3.substitutions = [] #array to save substitutions   4.if m != n:  5.  return "v and w are not equivalent"  6.else  7.for i <- m to 1 <-1 do  8.  for j <- n to j <- 1 do  9.    if v[i] != w[j]:  10.           substituted_word <- v[1…i-1]+v[j] #we substitute v[i] for w[j]  11.              if substituted_word in D:  12.                substitutions.append(substituted_word)  13.                   return EquivalentWordsProblem(v[1…m-i], w, D) #recur on the string of length m - i  14.   else:          return EquivalentWordsProblem(v[1…m-1], w, D) #recur on the string decreasing its length by 1  15.if len(substitutions) != 0:  16.  return substitutions   17.else  18.   return (“v and w are not equivalent”)