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docs/lectures/compilers/02_arithmetic.md

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+# Arithmetic Grammar
+
+## Syntax of Expressions
+
+An expression can be defined as:
+
+```haskell
+exp ::= int | exp + exp | exp - exp | exp * exp
+		| exp / exp | - exp | ( exp )
+```
+
+$$
+7 + (10/3) \times (-2)
+$$
+
+Applying this to the above expression:
+
+```haskell
+exp -> exp + exp
+-> int + exp
+-> 7 + exp
+-> 7 + exp * exp
+-> 7 + (exp) * exp
+-> 7 + (exp / exp) * exp
+-> 7 + (int / int) * exp
+-> 7 + (10 / 3) * (-exp)
+-> 7 + (10 / 3) * (-int)
+-> 7 + (10 / 3) * (-2)
+```
+
+This grammar is **ambiguous**, this means one input expression could be generated in several different ways.
+
+$$
+5 - 4 \times 7
+$$
+
+```haskell
+exp -> exp - exp 
+-> int - exp 
+-> 5 - exp
+-> 5 - exp * exp
+...
+-> 5 - 4 * 7
+```
+
+However there is another way to derive this expression starting with `*`
+
+```haskell
+exp -> exp * exp
+-> exp - exp * exp
+...
+-> 5 - 4 * 7
+```
+
+These give us two different ASTs, which gives us two different numeric answers. We must use more terminal symbols to follow BIDMAS.
+
+![](img/e.png)
+
+```haskell
+exp  ::= mexp | mexp + exp | mexp - exp
+mexp ::= term | term * mexp | term / mexp --multiplicative expression
+term ::= int | - term | ( exp )
+
+exp -> mexp - exp
+-> term - exp
+-> int - exp
+-> 5 - exp
+-> 5 - mexp
+-> 5 - term * mexp -> 5 - int * mexp -> 5 - 4 * mexp
+-> 5 - 4 * term -> 5 - 4 * int -> 5 - 4 * 7
+```
+
+This grammar is unique (non-ambiguous)
+
+## Semantics of Expressions
+
+On the left hand side the $+$ is just a symbol, however on the right hand side it is an arithmetic sum operation.
+
+$[\![ exp + exp ]\!] = [\![exp ]\!] + [\![exp ]\!]$	| $[\![ exp - exp ]\!] = [\![exp ]\!] - [\![exp ]\!]$ ... same for all binary operations
+
+$[\![ -exp]\!] = - [\![exp ]\!]$
+
+$[\![ x]\!] = x$
+
+$[\![(exp) ]\!] = [\![exp ]\!]$  - This is because parentheses change order of operations, not the operation itself.
+
+Addition can be rewritten:
+
+$$
+[\![exp_1 + exp_2 ]\!] = +([\![exp_1 ]\!], [\![exp_2 ]\!])
+$$
+
+```haskell
+int ::= digit | int digit
+digit ::= 0 | 1 | 2 | 3 ... | 9
+```
+
+$[\![d_0 ]\!] = value(d_0)$
+
+$[\![d_s d_0]\!] = [\![d_s ]\!]\times10 + value(d_0)$
+
+
+
+## Scanners and Parsers
+
+![img](img/f.png)
+
+Scanners take the source language as input and outputs a stream of tokens.
+
+A **token** is a chunk of input; "words" of the language eg. integers, operator symbols, identifiers (function & variable names etc), parenthesis.
+
+The **grammar of tokens is always regular**, this means it can be generated and recognised by a DFA (deterministic finite automata).