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Updated Sep 22, commits 2 contributors Kotlin. What follows is the exact definition used for the Algebra domain.
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It defines tokens or lexemes which make up the elements of the input language for our Algebra domain. Tokens are defined by a token name, a regular expression over Unicode that determines the allowable sequence of characters for the token, an internal representation for the token value if any , and a conversion procedure to generate that internal representation if any.
The conversion code must also capture lexical formatting data leading zero count on decimal numbers, number radix that will be needed to regenerate text for that lexeme by the prettyprinter. Note that a library of predefined conversions is defined by DMS, that covers most interesting cases. What follows is the exact definition of the grammar provided to DMS for the Algebra domain. Fundamentally it is a classic context-free grammar. We've been living without them for 15 years and it just isn't a loss. The grammar terms used are either lexemes as defined and exactly as spelled in the lexical definition, or nonterminals appropriate for the domain.
No special machinery is needed to build ASTs. There is no YACC cruft here, no way to get the tree shape wrong, and if you decide to change the grammar, no conceptual or implementation cost except for changing the grammar rules themselves. Grammar changes are truly easy to do. Interleaved in the grammar definition are prettyprinting rules.
The final result shown on this page comes from DMS's prettyprinter. Because we are defining simple mathmatics, we have no need for more sophisticated indentation policies, but DMS has a quite sophisticated language for expressing aligned "box layouts" for prettyprinted source text. What follows below this discussion is the exact input used to drive DMS to transform Algebra examples. What is being defined here are source-to-source transformation rules.
These say, in essence, if you see X, replace it by Y. This implements the foundational idea from mathematics of equality: if A is equal to B, one can substitute B for A. Individual rules are named so we can talk about them e. The patterns are written using the surface syntax of the defined language Algebra augumented with metavariables.
Surface syntax patterns in rules are written inside meta-domain quotes " The condition places extra, often semantic, constraints on whether a rule can apply, above and beyond pure syntax matching. It only applied to ASTs which represent product nonterminals, and it produces a product nonterminal when translating from one language to another, the source and target nonterminals may be very different.
The pattern looks for two subtrees which are connected by a multiplication tree ; further, because we used the same metavariable in both places, both subtrees must be identical. The replacement pattern builds a tree involving the exponentiation operator. You can see metaprogram control in the forms of rulesets.
These group rules together, enabling a ruleset to be used as a monolith. Rulesets can be combined to make bigger rules; see the Simplication ruleset. DMS reads these rules using the parser generated for Algebra, and stores them away for later use by a DMS application. It is fundamentally important to understand that these rules are not text-string matching and substitution.
Rather, they specify, the external string form of the corresponding ASTs that they represent.
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The transformation process matches the tree patterns that the rules specify, against the parsed tree of an instance text processed by DMS; when a match occurs, the matched tree is replaced by the tree corresponding to the right hand side of the rule. Consequence: the pattern matching is perfect; it can only match the proper AST nodes of interest.
These rules should be very unsurprising; they are what you learned in 9th grade algebra, written down in precise form. What's nice is that you write them down so nicely! The above rules implement basic algebraic simplification. We don't claim completenees; this is a demo small enough for an interested reader to absorb. The following ruleset implement differentiation over algebraic equations. Similarly, we don't make any claim of completeness here. We hope it is pretty obvious how differentiation is implemented. The above information is all definitional; it is provided to DMS to give it the ability to parse, simplify, differentiate, and print out results.
In this section, we run DMS with these definitions. For our example, we build a formal document a single file containing the following "Algebra" domain equations specifying that we want to differentiate a series of formulas:.