By William J. Dally
APRIL 2, 1990, CAMBRIDGE, MASSACHUSETTS Theory or perform in VLSI structures: hugely parallel architectures, special-purpose VLSI chips and structures, sensory structures, checking out, and fault tolerance. Design and Automation: synthesis and silicon compilation, structure and routing, research and simulation, novel layout equipment. Circuits and units: cutting edge electric circuits, technique and gadget modeling, computerized semiconductor production, wafer-scale platforms.
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Extra info for Advanced Research in VLSI: Proceedings of the 6th MIT Conference
After 4 years of refereeing by a team of 12 referees, an abridged version was published only recently . The referees declared that they were 99% certain of the correctness of the proof. The programs were merely given a “diagonal look”. edu/∼thales/ﬂyspeck) to formalize the whole proof with a theorem prover. This paper is the ﬁrst deﬁnite contribution to ﬂyspeck. Hales’ proof goes roughly like this: any potential counter example (denser packing) gives rise to a tame plane graph, where tameness is a very speciﬁc notion; enumerate all (ﬁnitely many) tame graphs (by computer); for each of them check (again by computer) that it cannot constitute a counter example.
A graph is tame if it is plane and satisﬁes 8 conditions: 1. The size of each face is at least 3 and at most 8: tame 1 g ≡ ∀ f ∈F g. 3 ≤ |vertices f | ∧ |vertices f | ≤ 8 2. Every 3-cycle is a face or the opposite of a face: tame 2 g ≡ ∀ a b c. is-cycle g [a, b, c] ∧ distinct [a, b, c] −→ (∃ f ∈F g. vertices f ∼ = [a, b, c] ∨ vertices f ∼ = [c, b, a]) where is-cycle g vs ≡ hd vs ∈ set (neighbors g (last vs)) ∧ is-path g vs, function neighbors does the obvious and is-path g  = True is-path g (u · vs) = (case vs of  ⇒ True | v · ws ⇒ v ∈ set (neighbors g u) ∧ is-path g vs) 3.
3) that same (tameEnum 0 800000 ) Tri, same (tameEnum 1 8000000 ) Quad, same (tameEnum 2 20000000 ) Pent, same (tameEnum 3 4000000 ) Hex, same (tameEnum 4 1000000 ) Hept, and same (tameEnum 5 2000000 ) Oct, where same is an executable check of equivalence (modulo )3 of two lists of fgraphs. Corollary fgraph ‘ TameEnum ⊆ Archive follows by correctness of tameEnum (Theorem 4) . We cannot detail the deﬁnition of same (or its correctness theorem) but we should point out that it is a potential bottleneck: for p = 2 we need to check 15000 graphs for inclusion in an archive of 1500 graphs — modulo graph isomorphism!
Advanced Research in VLSI: Proceedings of the 6th MIT Conference by William J. Dally