Canonical Proof nets for Classical Logic

Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proof-net, they are in essence the same proof. Providing a convincing proof-net counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs. By convincing, we mean that (a) there should be a canonical function from sequent proofs to proof nets, (b) it should be possible to check the correctness of a net in polynomial time, (c) every correct net should be obtainable from a sequent calculus proof, and (d) there should be a cut-elimination procedure which preserves correctness. Previous attempts to give proof-net-like objects for propositional classical logic have failed at least one of the above conditions. In [23], the author presented a calculus of proof nets (expansion nets) satisfying (a) and (b); the paper defined a sequent calculus corresponding to expansion nets but gave no explicit demonstration of (c). That sequent calculus, called LK\ast in this paper, is a novel one-sided sequent calculus with both additively and multiplicatively formulated disjunction rules. In this paper (a self-contained extended version of [23]), we give a full proof of (c) for expansion nets with respect to LK\ast, and in addition give a cut-elimination procedure internal to expansion nets - this makes expansion nets the first notion of proof-net for classical logic satisfying all four criteria.Comment: Accepted for publication in APAL (Special issue, Classical Logic and Computation

A proof-theoretic analysis of the classical propositional matrix method

The matrix method, due to Bibel and Andrews, is a proof procedure designed for automated theorem-proving. We show that underlying this method is a fully structured combinatorial model of conventional classical proof theory. © 2012 The Author, 2012. Published by Oxford University Press

Neural Mechanism of Language

This paper is based on our previous work on neural coding. It is a self-organized model supported by existing evidences. Firstly, we briefly introduce this model in this paper, and then we explain the neural mechanism of language and reasoning with it. Moreover, we find that the position of an area determines its importance. Specifically, language relevant areas are in the capital position of the cortical kingdom. Therefore they are closely related with autonomous consciousness and working memories. In essence, language is a miniature of the real world. Briefly, this paper would like to bridge the gap between molecule mechanism of neurons and advanced functions such as language and reasoning.Comment: 6 pages, 3 figure

Open Graphs and Monoidal Theories

String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to vertices at both ends, and these unconnected ends can be interpreted as the inputs and outputs of a diagram. In this paper, we give a concrete construction for string diagrams using a special kind of typed graph called an open-graph. While the category of open-graphs is not itself adhesive, we introduce the notion of a selective adhesive functor, and show that such a functor embeds the category of open-graphs into the ambient adhesive category of typed graphs. Using this functor, the category of open-graphs inherits "enough adhesivity" from the category of typed graphs to perform double-pushout (DPO) graph rewriting. A salient feature of our theory is that it ensures rewrite systems are "type-safe" in the sense that rewriting respects the inputs and outputs. This formalism lets us safely encode the interesting structure of a computational model, such as evaluation dynamics, with succinct, explicit rewrite rules, while the graphical representation absorbs many of the tedious details. Although topological formalisms exist for string diagrams, our construction is discreet, finitary, and enjoys decidable algorithms for composition and rewriting. We also show how open-graphs can be parametrised by graphical signatures, similar to the monoidal signatures of Joyal and Street, which define types for vertices in the diagrammatic language and constraints on how they can be connected. Using typed open-graphs, we can construct free symmetric monoidal categories, PROPs, and more general monoidal theories. Thus open-graphs give us a handle for mechanised reasoning in monoidal categories.Comment: 31 pages, currently technical report, submitted to MSCS, waiting review

Against the Virtual: Kleinherenbrink’s Externality Thesis and Deleuze’s Machine Ontology

Drawing from Arjen Kleinherenbrink's recent book, Against Continuity: Gilles Deleuze's Speculative Realism (2019), this paper undertakes a detailed review of Kleinherenbrink's fourfold "externality thesis" vis-à-vis Deleuze's machine ontology. Reading Deleuze as a philosopher of the actual, this paper renders Deleuzean syntheses as passive contemplations, pulling other (passive) entities into an (active) experience and designating relations as expressed through contraction. In addition to reviewing Kleinherenbrink's book (which argues that the machine ontology is a guiding current that emerges in Deleuze's work after Difference and Repetition) alongside much of Deleuze's oeuvre, we relate and juxtapose Deleuze's machine ontology to positions concerning externality held by a host of speculative realists. Arguing that the machine ontology has its own account of interaction, change, and novelty, we ultimately set to prove that positing an ontological "cut" on behalf of the virtual realm is unwarranted because, unlike the realm of actualities, it is extraneous to the structure of becoming-that is, because it cannot be homogenous, any theory of change vis-à-vis the virtual makes it impossible to explain how and why qualitatively different actualities are produced

Dual-Context Calculi for Modal Logic

We present natural deduction systems and associated modal lambda calculi for the necessity fragments of the normal modal logics K, T, K4, GL and S4. These systems are in the dual-context style: they feature two distinct zones of assumptions, one of which can be thought as modal, and the other as intuitionistic. We show that these calculi have their roots in in sequent calculi. We then investigate their metatheory, equip them with a confluent and strongly normalizing notion of reduction, and show that they coincide with the usual Hilbert systems up to provability. Finally, we investigate a categorical semantics which interprets the modality as a product-preserving functor.Comment: Full version of article previously presented at LICS 2017 (see arXiv:1602.04860v4 or doi: 10.1109/LICS.2017.8005089