The Grammar Hammer of 2012

Normal Form In order to fit any grammar into the conditions required by the previously described matching techniques, we demand the following normalisation: 1. lack of labels for production rules 2. lack of named subexpressions 3. lack of terminal symbols 4. maximal outward factoring of inner choices 5. lack of horizontal production rules 6. lack of separator lists 7. lack of trivially defined nonterminals (with α, ε or φ) 8. no mixing of chain and non-chain production rules 9. the nonterminal call graph is connected, and its top nonterminals are the starting symbols of the grammar It can be shown that transforming any grammar into its Abstract Normal Form is in fact a grammar mutation (see §3.8.1). In the prototype, I have implemented it to effectively generate bidirectional grammar transformation steps, so the normalisation preserves any information that it needs to abstract from. Grammar design mutation Some grammar design smells (terminology per [Sto12a]) like yaccification (per [SV99; BSV98]) or layered expressions (per [LZ09a]) have shown to be persistent enough to survive all normalisations and cause problems for establishing nominal and structural mappings. They can be identified and dealt with by automated analyses and mutations, but so far I have to proof that they are the only possible obstacles, and no guarantees about any other smells problematic for guided grammar convergence. 3.1.1 Generalisation of production signatures The method of establishing nonterminal mappings of different grammars of the same intended language, can be generalised as follows. Suppose that we have a metalanguage. Without loss of generality, let us assume that each grammar definition construct that is present in it, can be referred to by a single symbol: “,”, “?”, “*”, etc and uses prefix notation. This metasyntactic alphabet Λ will form the foundation of our footprints and signatures. Let us also assume that all metasymbols are unary or are encoded as unary, except for two composition constructs: a sequential “,” and an alternative “|”, which take a list of symbols. Then, a footprint of any nonterminal n in an expression x is a multiset of metasymbols that are used for occurrences of n within x: πn(x) =  {1} if x = n {μ} if x = μ(n), μ ∈ Λ ⋃ e∈L πn(e) if x = ,(L) ∅ otherwise, also if x = |(L) Our previously given definition of a production signature can still be used with this generally redefined footprints. It is well known that language equivalence is undecidable. Any formulation of the grammar equivalence problem, that is based on language equivalence, is thus also undecidable. Grammar convergence [LZ09a; LZ11] is a practically reformulated grammar equivalence problem that uses automated grammar transformation steps programmed by a human expert. By using these generalised metasyntactic signatures, we can infer converging transformation steps automatically, thus eliminating the weakest link of the present methodology. However, this is not the only application of the generalisation. The most trivial use of metasyntactic footprints and signatures would lie in grammarware metrics. Research on software metrics applied to context-free grammars has never been an extremely popular topic, but it did receive some attention in the 1970s [Gru71], 1980s [Kel81] and even recently [PM04; Čre+10]. Using quantitative aspects of metasyntactic footprints and signatures (numbers of different footprints within