2006 Scheme and Functional Programming Papers, University of

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A([spec]): ✓✏ (where L([spec]) = > p push c 0 ❅ ✓✏ ✎☞ ❅❘ q {c 0, . . . , c ✒✑❅ . ✒ n−1}) c n−1 ✒✑ ✍✌ ❅ push c n−1 A(”v”): (where v = c 0 . . . c n−1) ✓✏ ✓✏ ✓✏ > p c0 ✲ p 1 ✲ ci c n−1 ✓✏ɛ ✓✏ ✎☞ . . . ✲ p n−1 ✲ p n ✲ q ✒✑ ✒✑ ✒✑ ✒✑push v ✒✑ ✍✌ A((r)) = A(r) A(r ∗ ) = A(r{0, }) A(r + ) = A(r{1, }) A(r ? ) = A(r{0, 1}) A(r{i}) = A(r{i, i}) A(r{0, }): c 0 ɛ ✓✏ > p ❅ ✓✏ push [ ] ✎☞ ❅❘ q ✒✑❅ ɛ ✎☞ ✎☞ɛ ✲ p 1 A(r) q 1 ✒ ✒✑ ✍✌ cons ✍✌ ✍✌push [ ] ✻ ɛ cons A(r{i, }): (where i ≥ 1) ✓✏ɛ ✎☞ ✎☞ ɛ ✎☞ ✎☞ > p ✲ p 1 A(r) q 1 ✲ · · · ✲ p i−1 A(r) q i−1 ✒✑cons ✍✌ ✍✌ cons ✍✌ ✍✌ ɛ cons ✎☞ ✎☞ ɛ ✓✏ ✎☞ ✲ p i A(r) q i ✲ q ✍✌ ✍✌push [ ] ✒✑ ✍✌ ✻ ɛ cons Figure 4. Construction rules for the automata with commands (Part I). commands causes t to be pushed on σ. These properties can be proved straightforwardly by structural induction on R. 5.5 Using deterministic automata For efficiency reasons, it is preferable to use a DFA instead of a NFA. As explained above, the NFA obtained using function A may be converted into a DFA to allow fast recognition of the lexemes but three tables have to be built in order to be able to translate paths through the DFA back into paths through the original NFA. We assume the conversion of the NFA into a DFA to be a straightforward one. We adopt the point of view that deterministic states are sets of non-deterministic states. Then, our assumption says that the deterministic state that is reached after consuming some word w is exactly the set of non-deterministic states that can be reached by consuming w. 1 1 Note that this assumption precludes full minimization of the DFA. SILex currently does not try to minimize the DFA it builds. The assumption is sufficiently strong to ensure that paths through the NFA can be recovered We may now introduce the three tables Acc, f, and g. Table g indicates how to reach a state q from the non-deterministic start state using only ɛ-transitions. It is defined only for the non-deterministic states that are contained in the deterministic start state. Table f indicates how to reach a non-deterministic state q from some state in a deterministic state s using a path that consumes a single character c. It is usually not defined everywhere. Table Acc indicates, for a deterministic state s, which non-deterministic state in s accepts on behalf of the same rule as s. It is defined only for accepting deterministic states. Let us have a word w = c 0 . . . c n−1 that is accepted by the DFA and let P D = s 0 . . . s n be the path that is taken when w is consumed. Each s i is a deterministic state, s 0 is the start state, and s n is an accepting state. Note that an accepting state does not simply accept, but it accepts on behalf of a certain rule. In fact, an accepting deterministic state may contain more than one accepting but it may happen to be unnecessarily strong. More investigation should be made to find a sufficient and necessary condition on the conversion. 56 Scheme and Functional Programming, 2006
A(r{0, 0}): ✓✏ɛ ✓✏ ✎☞ > p ✲ q ✒✑push [ ] ✒✑ ✍✌ ɛ ✓✏ > p ❅ ✓✏ push [ ] ✎☞ ❅❘ q ✒✑❅ ɛ ✎☞ ✎☞ ɛ ✲ p 1 A(r) q 1 ✒ ✒✑ ✍✌ cons ✍✌ ✍✌ ✻ push [ ] ɛ · · · ✛ A(r{0, j}): cons (where j ≥ 1) ✎☞ ✎☞ ɛ ✲ p k A(r) q k ✲ ✍✌ ✍✌push [ ] ɛ · · · ✛ cons ✎☞ ✎☞ ɛ ✲ p j A(r) q j ✍✌ ✍✌push [ ] A(r{i, j}): (where i ≥ 1) ✓✏ɛ ✎☞ ✎☞ ɛ ✎☞ ✎☞ ɛ ✓✏ ✎☞ > p ✲ p 1 A(r) q 1 ✲ · · · ✲ p i A(r) q i ✲ q ✒✑cons ✍✌ ✍✌ cons ✍✌ ✍✌push [ ] ✒✑ ✍✌ ɛ · · · ✛ ✻ cons ✎☞ ✎☞ ɛ ✲ p k A(r) q k ✲ ✍✌ ✍✌push [ ] ɛ · · · ✛ cons ✎☞ ✎☞ ɛ ✲ p j A(r) q j ✍✌ ✍✌push [ ] A(r 0 . . . r n−1): (where n ≥ 2): ✓✏ɛ ✎☞ ✎☞ ɛ ✎☞ ✎☞ ɛ ✓✏ ✎☞ > p ✲ p 1 A(r ✒✑cons ✍✌ 0) q 1 ✲ · · · ✲ p n A(r ✍✌ cons ✍✌ n−1) q n ✲ q ✍✌push [ ] ✒✑ ✍✌ A(r 0 | . . . | r n−1): (where n ≥ 2) ɛ ✎☞ ✎☞ ✲ ✓✏ sel 0 ✍✌ p 1 A(r 0) ✍✌ q ɛ 1 > p ❅ ✓✏ ✎☞ . ❅❘ q ✒✑❅ ɛ ✎☞ ✎☞ɛ ❅ ✲ p n A(r sel n − 1 ✍✌ n−1) q n ✒ ✒✑ ✍✌ ✍✌ Figure 5. Construction rules for the automata with commands (Part II). non-deterministic states, each on behalf of its corresponding rule. In such a case, the deterministic state accepts on behalf of the rule that has highest priority. The non-deterministic path P N that corresponds to P D is recovered backwards portion by portion. The idea consists in determining non-deterministic states {q i} 0≤i≤n and portions of path {P i} 0≤i≤n such that: each q i is in s i; each P i starts at q i−1, ends at q i, and consumes c i−1, except for P 0, which starts at the non-deterministic start state, ends at q 0, and consumes ɛ; q n is a state that accepts on behalf of the same rule as s n. The recovery is initialized by determining q n directly from s n using the query Acc(s n). Next, the main part of the recovery consists in an iteration, with i going from n down to 1. At step i, given q i, one can determine portion of path P i and intermediate non-deterministic state q i−1. P i is obtained from the query f(s i−1, c i−1, q i). By doing so, q i−1 is also obtained as it is the source state of P i. As the final part of the recovery, P 0 is obtained using the query g(q 0). Then path P N is simply the linkage of all the portions together; i.e. P N = P 0 · . . . · P n. Note that the preceding explanation contains some minor inaccuracies. First, tables f and g do not exactly contain portions of path but reversed ones. Indeed, recall that the NFA presented in this paper are such that commands must be executed in the order in which they are met when following paths backwards. Second, there is no need to recover path P N (or its reverse) explicitly. It is sufficient to keep references to the portions that form P N and to later execute the commands by following the portions one after the other. Better yet, one may eagerly execute the commands contained in each portion as the latter gets determined. This way, it is unnecessary to remember P N nor its portions. Only the current state of the construction stack needs to be preserved. Last, one may observe that the sole purpose of the portions of path stored in tables f and g is to be followed in order to recover the parse tree Scheme and Functional Programming, 2006 57
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Scheme and Functional Programming 2
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Replacing run 10 with run ∗ cause
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