Decidable Exponentials in Nonassociative Noncommutative Linear Logic

The use of exponentials in linear logic greatly enhances its expressive power. In this paper we focus on nonassociative noncommutative multiplicative linear logic, and systematically explore modal axioms K, T, and 4 as well as the structural rules of contraction and weakening. We give sequent systems for each subset of these axioms; these enjoy cut elimination and have analogues in more structural logics. We then appeal to work of Bulinska extending work of Buszkowski to show that several of these logics are PTIME decidable and generate context free languages as categorial grammars. This contrasts associative systems where similar logics are known to generate all recursively enumerable languages, and are thus in particular undecidable

Extensions of the Morse-Hedlund Theorem

Bi-infinite words are sequences of characters that are infinite forwards and backwards; for example ...ababababab... . The Morse-Hedlund theorem says that a bi-infinite word f repeats itself, in at most n letters, if and only if the number of distinct subwords of length n is at most n. Using the example, ...ababababab... , there are 2 subwords of length 3, namely aba and bab . Since 2 is less than 3, we must have that ...ababababab... repeats itself after at most 3 letters. In fact it does repeat itself every two letters. Interestingly, there are many extensions of this theorem to multiple dimensions and beyond. We prove a few results in two-dimensions, including a specific partial result of a question known as the Nivat conjecture. We also consider a novel extension to the more general setting of \u27group actions\u27, and we prove an optimal analogue of the Morse-Hedlund theorem in this setting

Partitioning the power set of [n][n] into CkC_k-free parts

We show that for $n \geq 3, n\ne 5$, in any partition of $\mathcal{P}(n)$, the set of all subsets of $[n]=\{1,2,\dots,n\}$, into $2^{n-2}-1$ parts, some part must contain a triangle --- three different subsets $A,B,C\subseteq [n]$ such that $A\cap B$, $A\cap C$, and $B\cap C$ 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 $\mathcal{P}(n)$, 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 $G=S_3$.Comment: 12 page

Explorations in Subexponential non-associative non-commutative Linear Logic

In a previous work we introduced a non-associative non-commutative logic extended by multimodalities, called subexponentials, licensing local application of structural rules. Here, we further explore this system, considering a classical one-sided multi-succedent classical version of the system, following the exponential-free calculi of Buszkowski's and de Groote and Lamarche's works, where the intuitionistic calculus is shown to embed faithfully into the classical fragment

Explorations in Subexponential Non-associative Non-commutative Linear Logic

In a previous work we introduced a non-associative non-commutative logic extended by multimodalities, called subexponentials, licensing local application of structural rules. Here, we further explore this system, exhibiting a classical one-sided multi-succedent classical analogue of our intuitionistic system, following the exponential-free calculi of Buszkowski, and de Groote, Lamarche. A large fragment of the intuitionistic calculus is shown to embed faithfully into the classical fragment

Partitioning the power set of [n] into Ck-free parts

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 2n−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 2n−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=Ck,Pk,Sk, a cycle, path, or star with k edges. Additional bounds are given for G=C4 and G=S3