Tensors, !-graphs, and non-commutative quantum structures (extended version)

次指数非结合非交换线性逻辑探索（扩展版）

Explorations in Subexponential non-associative non-commutative Linear Logic (extended version)

该研究在先前引入的带次指数多模态非结合非交换逻辑基础上，参考相关无指数演算提出其经典单侧多后继版本，并证明直觉主义演算可忠实嵌入该经典片段。

Tensors, !-graphs, and non-commutative quantum structures

Categorical quantum mechanics (CQM) and the theory of quantum groups rely heavily on the use of structures that have both an algebraic and co-algebraic component, making them well-suited for manipulation using diagrammatic techniques. Diagrams allow us to easily form complex compositions of (co)algebraic structures, and prove their equality via graph rewriting. One of the biggest challenges in going beyond simple rewriting-based proofs is designing a graphical language that is expressive enough to prove interesting properties (e.g. normal form results) about not just single diagrams, but entir…

Encoding !-tensors as !-graphs with neighbourhood orders

Diagrammatic reasoning using string diagrams provides an intuitive language for reasoning about morphisms in a symmetric monoidal category. To allow working with infinite families of string diagrams, !-graphs were introduced as a method to mark repeated structure inside a diagram. This led to !-graphs being implemented in the diagrammatic proof assistant Quantomatic. Having a partially automated program for rewriting diagrams has proven very useful, but being based on !-graphs, only commutative theories are allowed. An enriched abstract tensor notation, called !-tensors, has been used to forma…

On embedding Lambek calculus into commutative categorial grammars

We consider tensor grammars, which are an example of \commutative" grammars, based on the classical (rather than intuitionistic) linear logic. They can be seen as a surface representation of abstract categorial grammars ACG in the sense that derivations of ACG translate to derivations of tensor grammars and this translation is isomorphic on the level of string languages. The basic ingredient are tensor terms, which can be seen as encoding and generalizing proof-nets. Using tensor terms makes the syntax extremely simple and a direct geometric meaning becomes transparent. Then we address the pro…

A note on undecidability of propositional non-associative linear logics

We introduce a non-associative and non-commutative version of propositional intuitionistic linear logic, called propositional non-associative non-commutative intuitionistic linear logic (NACILL for short). We prove that NACILL and any of its extensions by the rules of exchange and/or contraction are undecidable. Furthermore, we introduce two types of classical versions of NACILL, i.e., an involutive version of NACILL and a cyclic and involutive version of NACILL. We show that both of these logics are also undecidable.

Non-commutative propositional logic with short-circuited biconditional and NAND

Short-circuit evaluation denotes the semantics of propositional connectives in which the second argument is evaluated only if the first argument does not suffice to determine the value of the expression. In programming, short-circuit evaluation is widely used, with left-sequential conjunction and disjunction as primitive connectives. We consider left-sequential, non-commutative propositional logic, also known as MSCL (memorising short-circuit logic), and start from a previously published, equational axiomatisation. First, we extend this logic with a left-sequential version of the biconditional…