Course Notes on Formal Languages and Compilers

22
Backward Deterministic Regular Grammars
5
A regular grammar is called backward deterministic if all rules have different
right hand sides. These grammars have the property that the derivation can
be easily constructed by working backwards - see previous examples.
Recognition Algorithm for Backward Deterministic Regular Grammars
10
Say we wish to recognise whether a string
a 1 a 2 ..........a n ∈ L(G)
where each a j ∈ T
We proceed as follows:
15
20
25
1) Initially set u to Aa 1 a 2 ..........a n where A is the nonterminal
such that A→ε.
If A exists it is unique.
If no such A exists then L(G)= ∅,
and therefore a 1 a 2 ..........a n ∉L(G).
2) While u can be reduced by a rule, do the reduction (and repeat process)
that is if u has the form Aa j a j+1 ........... a n and B→Aa j is a rule, then reduce
u to Ba j+1 .........a n . Note, if such a rule exists, it is unique as all rules have
different right hand sides.
3) Once step 2 can't be repeated we can say
a 1 a 2 ..........a n ∈L(G) if and only if u∈S
(the start set of G).
30
Theorem: Conversion to backward deterministic form
Any regular grammar G can be transformed into a backward deterministic
regular grammar G' such that L(G') = L(G).
35
40
(What this means is that any regular language is easily recongnised by a
sequential syntax analysis process).
Proof: Let G = (N, T, P, S) be a regular grammar. A backward deterministic
regular grammar. G' = (N', T, P', S') is constructed from G using the
following method which is called the set construction.
A non terminal of G' is a non empty set of non terminals of G,
45
that is, N' = {U: U ⊆ N and U ≠ ∅}.
Equivalently, U ∈ N' if and only if U ⊆ N and U ≠ ∅.