Modelling contextuality by probabilistic programs with hypergraph semantics

Abstract

Models of a phenomenon are often developed by examining it under different experimental conditions, or measurement contexts. The resultant probabilistic models assume that the underlying random variables, which define a measurable set of outcomes, can be defined independent of the measurement context. The phenomenon is deemed contextual when this assumption fails. Contextuality is an important issue in quantum physics. However, there has been growing speculation that it manifests outside the quantum realm with human cognition being a particularly prominent area of investigation. This article contributes the foundations of a probabilistic programming language that allows convenient exploration of contextuality in wide range of applications relevant to cognitive science and artificial intelligence. Using the style of syntax employed by the probabilistic programming language WebPPL, specific syntax is proposed to allow the specification of "measurement contexts". Each such context delivers a partial model of the phenomenon based on the associated experimental condition described by the measurement context. An important construct in the syntax determines if and how these partial models can be consistently combined into a single model of the phenomenon. The associated semantics are based on hypergraphs in two ways. Firstly, if the schema of random variables of the partial models is acyclic, a hypergraph approach from relational database theory is used to compute a join tree from which the partial models can be combined to form a single joint probability distribution. Secondly, if the schema is cyclic, measurement contexts are mapped to a hypergraph where edges correspond to sets of events denoting outcomes in measurement contexts. Recent theoretical results from the field of quantum physics show that contextuality can be equated with the possibility of constructing a probabilisitic model on the resulting hypergraph. The use of hypergraphs opens the door for a theoretically succinct and efficient computational semantics sensitive to modelling both contextual and non-contextual phenomena. In addition, the hypergraph semantics allow measurement contexts to be combined in various ways. This aspect is exploited to allow the modular specification of experimental designs involving both signalling and no signalling between components of the design. An example is provided as to how the hypergraph semantics may be applied to investigate contextuality in an information fusion setting. Finally, the overarching aim of this article is to raise awareness of contextuality beyond quantum physics and to contribute formal methods to detect its presence by means of probabilistic programming language semantics.

Evaluating probabilistic programming languages for simulating quantum correlations

“…Contextuality scenarios X i , 1 ≤ i ≤ 2 are composed into a composite contextuality scenario X , which is a hypergraph describing the phenomenon P . Composition offers the distinct advantage of allowing experimental designs to be theoretically underpinned by hypergraphs in a modular way [8]. More formally, a contextuality scenario is a hypergraph X = (V, E) such that:…”

“…( f 1 ( 0 , 2 ) − f 1 ( 4 , 6 ) ) + ( f 1 ( 8 , 1 0 ) − f 1 ( 1 2 , 1 4 ) ) ) r e t u r n 2 * ( 1 + d e l t a ) >= abs ( ( v1 * f 2 ( 0 , 3 , 1 , 2 ) ) + ( v2 * f 2 ( 4 , 7 , 5 , 6 ) ) + ( v3 * f 2 ( 8 , 1 1 , 9 , 1 0 ) ) + ( v4 * f 2 ( 1 2 , 1 5 , 1 3 , 1 4 ) ) ) t e s t s = [ e q u a l i t y ( 1 , 1 , 1 , − 1 ) , e q u a l i t y ( 1 , 1 , − 1 , 1 ) , e q u a l i t y ( 1 , − 1 , 1 , 1 ) , e q u a l i t y ( − 1 , 1 , 1 , 1 ) ] . ) + ( f 1 ( 8,9 ) − f 1 ( 12,13 ) ) + ( f 1 ( 0,2 ) − f 1 ( 4,6 ) ) + ( f 1 ( 8,10 ) − f 1 ( 12,14 ) ) ) ( 2 * ( 1 + d e l t a ) ) >= Math . abs ( ( v1 * f 2 ( 0,3,1,2 ) ) + ( v2 * f 2 ( 4,7,5,6 ) ) + ( v3 * f 2 ( 8,11,9,10 ) ) + ( v4 * f 2 ( 12,15,13,14 ) ) ) } val t e s t s = Array [ Boolean ] ( E q u a l i t y ( 1,1,1,-1 ) , E q u a l i t y ( 1,1,-1,1 ) , E q u a l i t y ( 1,-1,1,1 ) , E q u a l i t y ( -1,1,1,1 ) ) .…”

“…+ ( f 1 ( 8,9 ) − f 1 ( 12,13 ) ) + ( f 1 ( 0,2 ) − f 1 ( 4,6 ) ) + ( f 1 ( 8,10 ) − f 1 ( 12,14 ) ) ) ( 2 * ( 1 + d e l t a ) ) >= Math . abs ( ( v1 * f 2 ( 0,3,1,2 ) ) + ( v2 * f 2 ( 4,7,5,6 ) ) + ( v3 * f 2 ( 8,11,9,10 ) ) + ( v4 * f 2 ( 12,15,13,14 ) ) ) } val t e s t s = Array [ Boolean ] ( E q u a l i t y ( 1,1,1,-1 ) , E q u a l i t y ( 1,1,-1,1 ) , E q u a l i t y ( 1,-1,1,1 ) , E q u a l i t y ( -1,1,1,1 ) ) .…”