====== THEOCS Lecture A/1: Introduction to Theoretical Computer Science and Languages ======
== 03/10/2024 ==

{{ :theocs:lecture01_a_slides_1_.pdf | Lecture 1 Slides}}

  * **Alan Turing ❤️ 🏳️‍🌈**
    * Proven that computational power does not determine the problems it can run. All classical computers regardless of powers can compute the same problems.
  * In late 1950s. N. Chomsky begain the study of formal "grammars" - they have a close relationship to abstract automata.
  * S. Cook seperated problems that **can be solved efficiently** from (CONT)

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  * Set of symbols - alphabet - which are combined into acceptable strings
    * A finite, nonempty set of symbols is called **an alphabet**
      * $\Sigma = \{a,b,c\}$  is an example of a alphabet
  * Rules which tell us how to combine strings in sensible ways are called the grammar
    * A **sting** is a **finite sequence** of symbols from the alphabet
      * $abc, aaa, bb$ are examples of strings on $\Sigma$
      * A string with no symbols is called an **empty string** and is denoted $\Lambda$
  * Using alphabet, strings, grammar makes a language
    * If $\Sigma$ is an alphabet, then a **language over $\Sigma$** is a set of strings (including empty string $\Lambda$) whose symbols come from $\Sigma$.
      * If $\Sigma = \{a,b\}$, then $L = \{ab,aaaab,abbb,a\}$ is an example of a language over $\Sigma$
      * If $\Sigma$ is an alphabet, then $\Sigma^{\ast}$ denoted the infinite set of all strings made up from $\Sigma$ (including an empty string $\Lambda$
      * Looking at $\Sigma^{\ast}$, a lanuage over $\Sigma$ is any subset of $\Sigma^{\ast}$

  * Four simple examples of languages over an alphabet $\Sigma$ are the sets:
    * $\emptyset, \{\Lambda\}, \Sigma, \Sigma^{\ast}$
  * If $L = \{aa,bb,ab\} and M = \{ab,aabb\} then$ 

  * $L \cdot \{\Lambda\} = \{\Lambda\} \cdot L = L$
  * $L \cdot \emptyset = \emptyset \cdot L = \emptyset$
  * $L \cdot M \neq M \cdot L$
    * **TODO: Insert examples**
  * $L \cdot (M \cdot N) = (L \cdot M ) \cdot N$
  * $L^0 = \{\Lambda\},\ L^n = L \cdot L^{n-1},\ if\ n > 0$ 
    * **TODO: Insert examples**
  * If L is a language of $\Sigma$ (i.e. $L \subset \Sigma^\ast$) then the **closure of L** is the language denoted by $L^\ast$ 
  * **TODO: Positive closure**

====== Lecture A/2: Grammars ======

{{ :theocs:lecture02_a_1_.pdf |Lecture 2 Slides}}