Weak Typed Boehm Theorem on IMLL

In the Boehm theorem workshop on Crete island, Zoran Petric called Statman's ``Typical Ambiguity theorem'' typed Boehm theorem. Moreover, he gave a new proof of the theorem based on set-theoretical models of the simply typed lambda calculus. In this paper, we study the linear version of the typed Boehm theorem on a fragment of Intuitionistic Linear Logic. We show that in the multiplicative fragment of intuitionistic linear logic without the multiplicative unit 1 (for short IMLL) weak typed Boehm theorem holds. The system IMLL exactly corresponds to the linear lambda calculus without exponentials, additives and logical constants. The system IMLL also exactly corresponds to the free symmetric monoidal closed category without the unit object. As far as we know, our separation result is the first one with regard to these systems in a purely syntactical manner.Comment: a few minor correction

A New Proof of P-time Completeness of Linear Lambda Calculus

We give a new proof of P-time completeness of Linear Lambda Calculus, which was originally given by H. Mairson in 2003. Our proof uses an essentially different Boolean type from the type Mairson used. Moreover the correctness of our proof can be machined-checked using an implementation of Standard ML

A type-assignment of linear erasure and duplication

We introduce $\mathsf{LEM}$, a type-assignment system for the linear $\lambda$-calculus that extends second-order $\mathsf{IMLL}_2$, i.e., intuitionistic multiplicative Linear Logic, by means of logical rules that weaken and contract assumptions, but in a purely linear setting. $\mathsf{LEM}$ enjoys both a mildly weakened cut-elimination, whose computational cost is cubic, and Subject reduction. A translation of $\mathsf{LEM}$ into $\mathsf{IMLL}_2$ exists such that the derivations of the former can exponentially compress the dimension of the derivations in the latter. $\mathsf{LEM}$ allows for a modular and compact representation of boolean circuits, directly encoding the fan-out nodes, by contraction, and disposing garbage, by weakening. It can also represent natural numbers with terms very close to standard Church numerals which, moreover, apply to Hereditarily Finite Permutations, i.e. a group structure that exists inside the linear $\lambda$-calculus.Comment: 43 pages (10 pages of technical appendix). The final version will appear on Theoretical Computer Science https://doi.org/10.1016/j.tcs.2020.05.00

A Coding Theoretic Study on MLL proof nets

Coding theory is very useful for real world applications. A notable example is digital television. Basically, coding theory is to study a way of detecting and/or correcting data that may be true or false. Moreover coding theory is an area of mathematics, in which there is an interplay between many branches of mathematics, e.g., abstract algebra, combinatorics, discrete geometry, information theory, etc. In this paper we propose a novel approach for analyzing proof nets of Multiplicative Linear Logic (MLL) by coding theory. We define families of proof structures and introduce a metric space for each family. In each family, 1. an MLL proof net is a true code element; 2. a proof structure that is not an MLL proof net is a false (or corrupted) code element. The definition of our metrics reflects the duality of the multiplicative connectives elegantly. In this paper we show that in the framework one error-detecting is possible but one error-correcting not. Our proof of the impossibility of one error-correcting is interesting in the sense that a proof theoretical property is proved using a graph theoretical argument. In addition, we show that affine logic and MLL + MIX are not appropriate for this framework. That explains why MLL is better than such similar logics.Comment: minor modification