# Parsing and Patterns

The logic compiler generates Haskell code to implement a calculus. This includes a data-type to represent formulae, a function to resolve those formulae into sub-formulae, functions to format them for output, miscellaneous utility functions and a parser to parse formulae in the language the calculus operates with. As the code stands currently, the logic language allows the user to define operators with arbitrary arity and fixity. One might, if one were of a mind, define a 4-fix **#** operator with an arity of 3. A formula with such an operator might look like:

`α`

_{1} α_{2} α_{3} # α_{4}

As the code stands, the generated parser is built using the buildExpressionParser facility of the Parsec parser library. Unfortunately, `buildExpressionParser`

is only capable of handling expression languages with unary prefix, binary infix and unary postfix operators, meaning that we can’t generate parsers for all calculi that can be defined with our language.

To fix this, I’m going to have to write a more general expression parser generator (hopefully using `buildExpressionParser`

as a base) that can handle operators of arbitrary arity and fixity. Either that, restrict the system to only unary and binary operators which would make implementing some [conceivable] logics very difficult, if not impossible.

On the other hand, `buildExpressionParser`

has greatly simplified the pattern handling code in the compiler. In our language (which is so simple as to not need a name), we define rules such as:

`Resolve "a->b" to "~a" or "b".`

The patterns (the bits in quote marks) are formulae using the operators of the calculus, with arbitrary alphabetic names for variables (which can stand for any sub-formula - we can make no distinction at this point between sub-formulae and propositions as I haven’t got round to implementing that yet). Our compiler generates *at run time* a parser for the formulae of the calculus and uses it to parse the rule patterns and generate appropriate snippets of code which we use for pattern-matching in the Haskell code implementing the calculus. For example, the rule above would be translated to the Haskell code

`resolve (Impl a b) = Or [(Neg a), b]`

Once we’ve parsed the pattern and generated a function, we get the pattern matching at run-time for free from Haskell.