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Languages & Machines/Introduction.md

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+---
+type: theoretical
+ty:
+---
+
+## The foundations of computation
+Looking for answers for basic questions like:
+- Computability?
+- Power $\leftrightarrow$ Programming constructs?
+
+Which leads us to fundamental concepts like:
+- State
+- Transitions
+- Non-determinism
+- Undecideability
+
+
+## Models
+### Finite memory
+Finite automata, regexp
+![](Pasted%20image%2020250414190100.png)
+### Finite memory with stack
+Push down automata
+![](Pasted%20image%2020250414190119.png)
+### Unrestricted
+Turing machines
+
+![](Pasted%20image%2020250414190144.png)
+
+
+
+## Grammars
+
+![](Pasted%20image%2020250414190229.png)
+
+
+On a higher level, it seems like grammars and machines are very different, but parsing a language (a set of strings) is quite similar to computation.
+
+## State-based systems and glossary
+An FSM can be a specification for OOP.
+
+- States ($q_0,\ldots, q_n$)
+- Transitions ($a,b,c,\ldots,z$)
+- We can interpret the transitions as class methods and specify the sequences of allowed invocations - **typestate**
+
+
+
+## Notation
+- $x \in X, X\subseteq Y$
+- $\forall x \in X: P(x), \exists x \in X: P(x)$
+- $R \subseteq X \times Y$ is a relation between $X$ and $Y$
+- $xRy \equiv (x,y) \in R$ 
+- $G = (V, E)$, where $E \subseteq V\times V$ is a directed graph
+
+Part of [Relations and Digraphs](Relations%20and%20Digraphs.md)

Languages & Machines/Regular languages.md

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+---
+type: math
+---
+## Induction
+Similar (if not the same) to:
+- [Mathematical Proofs (Induction)](Mathematical%20Proofs%20(Induction).md)
+- [Structural Proofs](Proofs.md)
+
+
+- Base case $0\in \mathbb{N}$
+- Inductive step - if $n\in \mathbb{N} \implies n+1\in \mathbb{N}$
+- We allow a finite number of steps
+
+
+I.e.
+Given $f (n) = n(n + 1)$ for all $n\in N$, then $f (n)$ is even.
+
+**Base case:** $f(0) = 0\times 1 = 0$, which is even
+**I.S.:**
+$$
+f(n+1) = (n+1)(n+2)= n(n+1)+2(n+1) = f(n) + 2(n+1) \blacksquare
+$$
+
+## Strings and Languages
+Literally the same as [Mathematical Data Structures](Mathematical%20Data%20Structures.md), but on strings
+
+How to define the reversal of a string, inductively?
+
+
+Let $w$ be a finite string. We define $w^R$ by induction on $|w|$:
+
+**B.C.:**
+$|w| = 0$, then, trivially, $w = \epsilon \therefore w^R = \epsilon$
+
+**I.S.:**
+$|w| = n \geq 1$, so $w = u a$ with $|u| = n-1$,