Adding multi-modalities (called subexponentials) to linear logic enhances its power as a logical framework, which has been extensively used in the specification of e.g. proof systems, programming languages and bigraphs. Initially, subexponentials allowed for classical, linear, affine or relevant behaviors. Recently, this framework was enhanced so to allow for commutativity as well. In this work, we close the cycle by considering associativity. We show that the resulting system ($$\mathsf {acLL}_\varSigma $$
acLL
Σ
) admits the (multi)cut rule, and we prove two undecidability results for fragments/variations of $$\mathsf {acLL}_\varSigma $$
acLL
Σ
.
Summary
The authors systematically investigate the class of change‐over designs that are partially balanced for the estimation of the elementary direct and residual effects contrasts, and whose constructions are based on m‐associate PBIB designs. In light of the complicated mathematical structure of the analysis, there is a need to introduce restrictions on the various incidence relations. By means of these assumed constraints the concept of partially balanced change‐over designs based on m‐associate class PBIB designs [PBCOD(m)] is defined. Four new series of these designs are constructed and illustrated.
We show that for n ≥ 3, n = 5, in any partition of P(n), the set of all subsets of [n] = {1, 2, . . . , n}, into 2 n−2 − 1 parts, some part must contain a triangle -three different subsets A, B, C ⊆ [n] such that A ∩ B, A ∩ C, and B ∩ C have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into 2 n−2 triangle-free parts. We also address a more general Ramsey-type problem: for a given graph G, find (estimate) f (n, G), the smallest number of colors needed for a coloring of P(n), such that no color class contains a Berge-G subhypergraph. We give an upper bound for f (n, G) for any connected graph G which is asymptotically sharp (for fixed k) when G = C k , P k , S k , a cycle, path, or star with k edges. Additional bounds are given for G = C 4 and G = S 3 .
SUMMARY
In an earlier paper the authors introduced PBCOD(m) designs, which are change‐over designs that are partially balanced for the estimation of the elementary direct and residual effects contrasts, and whose constructions are based on m‐associate class PBIB designs. In this paper we determine the maximum efficiencies for the estimated direct and residual elementary treatment contrasts of PBCOD(m), and the A‐optimality (Kiefer, 1959) of these designs is explored.
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