Graph-theoretic proof of a network theorem and some consequences

This paper generalizes the Huet and Lévy theory of normalization by neededness to an abstract setting. We define Stable Deterministic Residual Structures (SDRS) and Deterministic Family Structures (DFS) by axiomatizing some properties of the residual relation and the family relation on redexes in an Abstract Rewriting System. We present two proofs of the Relative Normalization Theorem, one for SDRSs for regular stable sets, and another for DFSs for all stable sets of desirable ‘normal forms’. We further prove the Relative Optimality Theorem for DFSs. We extend this result to deterministic Computation Structures which are deterministic Event Structures with an extra relation expressing self-essentiality.

The paper is introducing an economic method (interindustry relations analysis) into studies of autopoietic systems and shows its application to scientometrics, which can also be regarded as the analysis of autopoietic systems. The merit of the application is discussed, and the outline of the proof of a related theorem is suggested in the appendix.

One of the early partition relation theorems which include ordinals was the observation of Erdös and Rado [7] that if κ = cf(κ) > ω then the Dushnik–Miller theorem can be sharpened to κ→(κ, ω + 1)2. The question on the possible further extension of this result was answered by Hajnal who in [8] proved that the continuum hypothesis implies ω1 ↛ (ω1, ω + 2)2. He actually proved the stronger result ω1 ↛ (ω: 2))2. The consistency of the relation κ↛(κ, (ω: 2))2 was later extensively studied. Baumgartner [1] proved it for every κ which is the successor of a regular cardinal. Laver [9] showed that if κ is Mahlo there is a forcing notion which adds a witness for κ↛ (κ, (ω: 2))2 and preserves Mahloness, ω-Mahloness of κ, etc. We notice in connection with these results that λ→(λ, (ω: 2))2 holds if λ is singular, in fact λ→(λ, (μ: n))2 for n < ω, μ < λ (Theorem 4).In [11] Todorčević proved that if cf(λ) > ω then a ccc forcing can add a counter-example to λ→(λ, ω + 2)2. We give an alternative proof of this (Theorem 5) and extend it to larger cardinals: if GCH holds, cf (λ) > κ = cf (κ) then < κ-closed, κ+-c.c. forcing adds a counter-example to λ→(λ, κ + 2)2 (Theorem 6).Erdös and Hajnal remarked in their problem paper [5] that Galvin had proved ω2→(ω1, ω + 2)2 and he had also asked if ω2→(ω1, ω + 3)2 is true. We show in Theorem 1 that the negative relation is consistent.

Dijkstra has given a derivation of an efficient algorithm for a problem concerning monotonic subsequences, and extracted a proof of a related theorem from the algorithm. Here it is shown that a careful separation of concerns can lead to a beautiful conventional proof, a very different derivation of Dijkstra's algorithm, a more elegant proof from the algorithm, and the discovery of a duality property.