Enumerating Independent Linear Inferences

A linear inference is a valid inequality of Boolean algebra in which each variable occurs at most once on each side. Equivalently, it is a linear rewrite rule on Boolean terms that constitutes a valid implication. Linear inferences have played a significant role in structural proof theory, in particular in models of substructural logics and in normalisation arguments for deep inference proof systems. Systems of linear logic and, later, deep inference are founded upon two particular linear inferences, switch : x (y z) → (x y) z, and medial : (w x) (y z) → (w y) (x z). It is well-known that these two are not enough to derive all linear inferences (even modulo all valid linear equations), but beyond this little more is known about the structure of linear inferences in general. In particular despite recurring attention in the literature, the smallest linear inference not derivable under switch and medial (`switch-medial-independent') was not previously known. In this work we leverage recently developed graphical representations of linear formulae to build an implementation that is capable of more efficiently searching for switch-medial-independent inferences. We use it to find four `minimal' 8-variable independent inferences and also prove that no smaller ones exist; in contrast, a previous approach based directly on formulae reached computational limits already at 7 variables. Two of these new inferences derive some previously found independent linear inferences. The other two (which are dual) exhibit structure seemingly beyond the scope of previous approaches we are aware of; in particular, their existence contradicts a conjecture of Das and Strassburger.

• 15 publications
• 3 publications
research
08/14/2019

The lexical and grammatical sources of neg-raising inferences

We investigate neg(ation)-raising inferences, wherein negation on a pred...
research
01/06/2018

On the inherent competition between valid and spurious inductive inferences in Boolean data

Inductive inference is the process of extracting general rules from spec...
research
05/16/2023

Prawitz's Conjecture is False; So What?

Several recent results bring into focus the superintuitionistic nature o...
research
01/11/2023

RobustIV and controlfunctionIV: Causal Inference for Linear and Nonlinear Models with Invalid Instrumental Variables

We present R software packages RobustIV and controlfunctionIV for causal...
research
12/14/2021

Representing Inferences and their Lexicalization

We have recently begun a project to develop a more effective and efficie...
research
07/06/2020

Inferences and Modal Vocabulary

Deduction is the one of the major forms of inferences and commonly used ...
research
08/31/2023

Detecting Evidence of Organization in groups by Trajectories

Effective detection of organizations is essential for fighting crime and...