Unifying Heterogeneous State-Spaces with Lenses

Abstract

Abstract. Most verification approaches embed a model of program state into their semantic treatment. Though a variety of heterogeneous statespace models exists, they all possess common theoretical properties one would like to abstractly capture, such as the laws of programming. In this paper, we propose lenses as a universal state-space modelling solution. Lenses provide an abstract interface for manipulating data types through spatially-separated views. We define a lens algebra that allow their composition and comparison. We show how this algebra can be used to model variables and alphabets in Hoare and He's Unifying Theories of Programming. The combination of lenses and relational algebra gives rise to a model for UTP in which its fundamental laws can be verified. Moreover, we illustrate how they can be used to model more state complex notions such as memory stores and parallel states. We provide a mechanisation in Isabelle/HOL that validates our theory, and facilitates its use in program verification.

UTP by Example: Designs

“…Mechanisation is a key aspect of any formalisation, and UTP has been embedded in a variety of theorem provers, notably in ProofPower-Z and Isabelle [51,50,74,10,26,12,31,33,34]. This allows a theory engineer to mechanically construct UTP theories, experiment with them, prove properties, and eventually deploy them for use in program verification.…”

Towards Verification of Cyber-Physical Systems with UTP and Isabelle/HOL

“…Rather than characterising variables as syntactic entities [16], we instead algebraically characterise the behaviour of variables using lenses [17,14]. Lenses allow us to represent variables as abstract projections on a state space with convenient ways to query and update in a uniform, compositional way.…”

“…All of these operations can be given denotations in terms of the get and put [17]; here we focus on the intuition and algebraic laws.…”

“…Our lenses operations satisfy the following algebraic laws, all of which has been mechanised [17], assuming X , Y , and Z are well-behaved lenses:…”

“…In order to make UTP theories practically applicable to program verification, tool support is needed, and so we are also developing a theorem prover for UTP based on Isabelle/HOL [28], which we call Isabelle/UTP [17,16]. Isabelle is a powerful proof assistant that can be used both for the mechanisation of mathematics, and for the application of such mechanisations to program verification, which is famously illustrated by the seL4 microkernel verification project [26].…”