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Due to the argument order of `_⊙_`, the children got merged in the wrong
order. Hence, the merge order of the children kind of reversed after
each level. Notice that the new argument order agrees with the usual
`List.foldl` argument order.
Example of what changed:
Previously, we had
((0 -< rose-leaf 1 ∷ [] >- ∷ []) ⊕ (0 -< rose-leaf 2 ∷ [] >- ∷ []))
≡ (0 -< rose-leaf 2 ∷ rose-leaf 1 ∷ [] >- ∷ [])
and
((0 -< 1 -< rose-leaf 2 ∷ [] >- ∷ [] >- ∷ []) ⊕ (0 -< 1 -< rose-leaf 3 ∷ [] >- ∷ [] >- ∷ []))
≡ (0 -< 1 -< rose-leaf 2 ∷ rose-leaf 3 ∷ [] >- ∷ [] >- ∷ [])
now we have
((0 -< rose-leaf 1 ∷ [] >- ∷ []) ⊕ (0 -< rose-leaf 2 ∷ [] >- ∷ []))
≡ (0 -< rose-leaf 1 ∷ rose-leaf 2 ∷ [] >- ∷ [])
and
((0 -< 1 -< rose-leaf 2 ∷ [] >- ∷ [] >- ∷ []) ⊕ (0 -< 1 -< rose-leaf 3 ∷ [] >- ∷ [] >- ∷ []))
≡ (0 -< 1 -< rose-leaf 2 ∷ rose-leaf 3 ∷ [] >- ∷ [] >- ∷ [])
In the paper, we do not presume the standard library so we define `_⊕'_` without `foldl`.
The `FeatureAlgebra` concept is based on the paper "An algebraic foundation for automatic feature-based program synthesis" by Apel et. al. which is a little vague and inconsistent with the order of their composition operator. While discussing the buggy children order fixed in f3ccb0f, we discovered that the distant idempotence law is dictated by the order during composition. Using our current definition of `_⊕_`, we have ((0 -< rose-leaf 1 ∷ [] >- ∷ []) ⊕ (0 -< rose-leaf 2 ∷ rose-leaf 3 ∷ [] >- ∷ [])) ≡ (0 -< rose-leaf 1 ∷ rose-leaf 2 ∷ rose-leaf 3 ∷ [] >- ∷ []) which gives rise to the law i₁ ⊕ i₂ ⊕ i₁ ≡ i₁ ⊕ i₂ In contrast, with the alternative definition of `_⊕_`, with swapped composition order, we have ((0 -< rose-leaf 1 ∷ [] >- ∷ []) ⊕ (0 -< rose-leaf 2 ∷ rose-leaf 3 ∷ [] >- ∷ [])) ≡ (0 -< rose-leaf 2 ∷ rose-leaf 3 ∷ rose-leaf 1 ∷ [] >- ∷ []) which gives rise to the law i₂ ⊕ i₁ ⊕ i₂ ≡ i₁ ⊕ i₂ Although Apel et. al. state the distant idempotence law as i₂ ⊕ i₁ ⊕ i₂ ≡ i₁ ⊕ i₂ we decided to use the more intuitive composition order.
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I love how this PR did not have to touch any of the existing proofs on FSTs, which we just merged. 🚀 |
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I am beginning to review in-file this time. |
pmbittner
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May 28, 2024
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I just pushed my review. I also integrated some minor improvements myself, including enhanced documentation and extraction of some utility lemmas.
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Nice! |
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