Jan Erik Bellingrath
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janbellingrath.bsky.social
Jan Erik Bellingrath
@janbellingrath.bsky.social
65 followers 280 following 1 posts
PhD candidate in brain-inspired artificial intelligence @UnivToulouse III. Interested in consciousness, deep learning, and computational neuroscience.
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janbellingrath.bsky.social
Love the “all is fog” framing
October 9, 2025 at 6:27 AM
Reposted by Jan Erik Bellingrath
fintanmallory.com
I spent so many hours working through papers like this only for the field to switch to the 'neural network go brrr' paradigm
Lemma 2 Suppose B is a CIS and h a surjective CIS homomorphism from P(Σ∗) → B. Then let G be a context-free grammar. Let φ be the cfg- morphism given by φ(N) = h(L(G,N)). Then for all nonterminals N in G, L(φ(G),φ(N)) ⊆h∗(φ(N)) Proof We will assume for clarity of exposition that the grammar is in CNF – the proof of the more general case is straightforward but less legible. Suppose we have a rule N →PQin G. Therefore L(G,N) ⊇L(G,P)L(G,Q). So h(L(G,N)) ≥h(L(G,P))◦h(L(G,Q)). Then h∗(h(L(G,N))) ≥h∗(h(L(G,P))◦h(L(G,Q))) by monotonicity of h∗. By Proposition 39 h∗(h(L(G,N))) ≥ h∗(h(L(G,P)))◦h∗(h(L(G,Q))). Similarly suppose we have a rule N →a in G. So h(L(G,N)) ≥h({a}), and h∗(h((L(G,N))) ≥h∗(h({a})). Now suppose that N′ ∗ ⇒φ(G) w; we want to show that w∈h∗(N′). We pro- ceed by induction on the length of the derivation. Base case: the production is of length 1. Suppose N′→ais a production in φ(G). Then for any production N →a in G such that N′ = φ(N), h∗(h((L(G,N))) ≥h∗(h({a})) ≥{a}. Suppose it is true for all derivations of length at most k, and let N′ ∗ ⇒φ(G) w be a derivation of length k+ 1 that starts with N′→P′Q′→uv= w, where P′ ∗ ⇒φ(G) u and Q′ ∗ ⇒φ(G) v. these last two derivations must be of length at most k. Then consider any production N → PQ in G such that φ(N) = N′ , φ(P) = P′ and φ(Q) = Q′ (of which there must be at least one). Now by the inductive hypothesis, u∈h∗(P′) and v∈h∗(Q′). So uv∈h∗(P′)h∗(Q′) = h∗(h(L(G,P)))◦h∗(h(L(G,Q))). Now since h∗(h(L(G,N))) ≥h∗(h(L(G,P)))◦ h∗(h(L(G,Q))) this means that uv ∈h∗(h(L(G,N))). So w ∈h∗(φ(N). By induction, it is therefore true for derivations of any length.
April 20, 2025 at 11:52 AM