Recursive structure of MOS scales: Difference between revisions
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If w₂ and w₃ (which have the same length) contain the same number of complete chunks, one case is (X denotes a chunk boundary, < > are chunk boundaries that are also the boundary of the word, [] are non-chunk-boundary word boundaries.) | If w₂ and w₃ (which have the same length) contain the same number of complete chunks, one case is (X denotes a chunk boundary, < > are chunk boundaries that are also the boundary of the word, [] are non-chunk-boundary word boundaries.) | ||
Case 1.1: | |||
w2: <L ... sXL ... sXL ... sXL ... sXL ...] | w2: <L ... sXL ... sXL ... sXL ... sXL ...] | ||
w3: <L ... sXL ... sXL ... sXL ... s> | w3: <L ... sXL ... sXL ... sXL ... s> | ||
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This implies that the last chunk is bigger than the first one, a contradiction because w₂ begins in λ. | This implies that the last chunk is bigger than the first one, a contradiction because w₂ begins in λ. | ||
Case 1.2: | |||
w1: s[something with k s's] | w1: s[something with k s's] | ||
w2: <L ... sXL ... sXL ... sXL ... s> | w2: <L ... sXL ... sXL ... sXL ... s> | ||
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So we must have | So we must have | ||
Case 1.3: | |||
w2: <L ... sXL ... sXL ... sXL ... sXL ...] | w2: <L ... sXL ... sXL ... sXL ... sXL ...] | ||
w3: [... sXL ... sXL ... sXL ... sXL ... s> | w3: [... sXL ... sXL ... sXL ... sXL ... s> | ||
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or | or | ||
Case 1.4: | |||
w2: <L ... sXL ... sXL ... sXL ... sX] | w2: <L ... sXL ... sXL ... sXL ... sX] | ||
w3: [... sXL ... sXL ... sXL ... sXL ... | w3: [... sXL ... sXL ... sXL ... sXL ... s> | ||
(⇒ w₃ has one more s). | (⇒ w₃ has one more s). | ||
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If w₂ has more complete chunks than w₃: the only case to consider is when w₂ has one more chunk. | If w₂ has more complete chunks than w₃: the only case to consider is when w₂ has one more chunk. | ||
w2: <L ... sXL ... sXL ... sXL ... sXL ... | Case 2.1: | ||
w2: <L ... sXL ... sXL ... sXL ... sXL ... s> | |||
w3: <L ... sXL ... sXL ... sXL ... s> | w3: <L ... sXL ... sXL ... sXL ... s> | ||
⇒ w₂ has more s's. | ⇒ w₂ has more s's. | ||
Case 2.2: | |||
w2: <L ... sXL ... sXL ... sXL ... sXL ... s> | |||
w3: [sXL ... sXL ... sXL ... sXL ... s> | |||
⇒ do the same trick as in | |||
Case 2.3: | |||
w2: <L ... sXL ... sXL ... sXL ... sXL ... sXL ...] | w2: <L ... sXL ... sXL ... sXL ... sXL ... sXL ...] | ||
w3: [ ... sXL ... sXL ... sXL ... sXL ... s> | w3: [ ... sXL ... sXL ... sXL ... sXL ... s> | ||
Case 2.4: | |||
w2: <L ... sXL ... sXL ... sXL ... sXL ... sXL ...] | w2: <L ... sXL ... sXL ... sXL ... sXL ... sXL ...] | ||
w3: <L ... sXL ... sXL ... sXL ... s> | w3: <L ... sXL ... sXL ... sXL ... s> | ||
Both are contradictions since the first chunk is λ so has to at least as big as the last one. | Both 2.3 and 2.4 are contradictions since the first chunk is λ so has to at least as big as the last one. | ||
=== Preservation of generators === | === Preservation of generators === | ||