Hypergraph rewriting and Causal structure of $λ-$calculus

TL;DR

This paper first study hypergraph rewriting in categorical terms in an attempt to define the notion of events and develop foundations of causality in graph rewriting and attempts to extend this definition to arbitrary $\lambda-$expressions.

Abstract

In this paper, we first study hypergraph rewriting in categorical terms in an attempt to define the notion of events and develop foundations of causality in graph rewriting. We introduce novel concepts within the framework of double-pushout rewriting in adhesive categories. Secondly, we will study the notion of events in $λ-$calculus, wherein we construct an algorithm to determine causal relations between events following the evaluation of a $λ-$expression satisfying certain conditions. Lastly, we attempt to extend this definition to arbitrary $λ-$expressions.

Figures (6)

• Figure 1: The edge lists are {(1,2),(2,3),(1,3)} and {(1,2,3),(3,4,5)} respectively
• Figure 2: Example taken from class. Here, the vertex which is shaded light blue on the right hand side graph is newly created
• Figure 3: The dotted edges represent the part of the graph being removed. And the red edges represent the new edges added
• Figure 4: 2 events that can happen together
• Figure 5: The causal graph for the expression $(((\lambda x.\lambda y. (x) y)\lambda z.z)\lambda v.v)b$

Theorems & Definitions (61)

• Definition : Directed Multihypergraph
• Definition : Update rule
• Definition : Morphism
• Definition : Rewrite rule
• Definition : Deletion of vertices
• Definition : Cut graph
• Proposition
• proof
• Proposition
• proof