Constructing weak simulations from linear implications for processes with private names

Abstract

This paper clarifies that linear implication defines a branching-time preorder, preserved in all contexts, when used to compare embeddings of process in non-commutative logic. The logic considered is a first-order extension of the proof system BV featuring a de Morgan dual pair of nominal quantifiers, called BV1. An embedding of π-calculus processes as formulae in BV1 is defined, and the soundness of linear implication in BV1 with respect to a notion of weak simulation in the π -calculus is established. A novel contribution of this work is that we generalise the notion of a ‘left proof’ to a class of formulae sufficiently large to compare embeddings of processes, from which simulating execution steps are extracted. We illustrate the expressive power of BV1 by demonstrating that results extend to the internal π -calculus, where privacy of inputs is guaranteed. We also remark that linear implication is strictly finer than any interleaving preorder.

An Analytic Propositional Proof System on Graphs

“…In the setting of graphical logics extending GS, non-commutative operators generalise to graphs incorporating directed edges which capture more general patterns of logical time, as explained in [AHMS22]. To make such models useful we also require features such as: additives for choices [TM05,Hor15]; quantifiers for dealing with message passing and private data [HTAC19,HT19]; and even co-inductive proofs [Hor20]. Lifting such features to the setting of graphs seem feasible, but requires a certain amount of future work.…”

Logic Beyond Formulas

“…The notion of formula is central to all applications of logic and proof theory in computer science, ranging from the formal verification of software, where a formula describes a property that the program should satisfy, to logic programming, where a formula represents a program [27,31], and functional programming, where a formula represents a type [25]. Proof theoretical methods are also employed in concurrency theory, where a formula can represent a process whose behaviours may be extracted from a proof of the formula [5,22,23,30]. This formulas-as-processes paradigm is not as well-investigated as the formulas-as-properties, formulas-as-programs and formulas-as-types paradigms mentioned before.…”

An Analytic Propositional Proof System on Graphs

“…The notion of formula is central to all applications of logic and proof theory in computer science, ranging from the formal verification of software, where a formula describes a property that the program should satisfy, to logic programming, where a formula represents a program [MNPS91,KY93], and functional programming, where a formula represents a type [How80]. Proof theoretical methods are also employed in concurrency theory, where a formula can represent a process whose behaviours may be extracted from a proof of the formula [Mil93,Bru02,HT19,Hor19,Hor20]. This formulas-as-processes paradigm is not as well-investigated as the formulas-as-properties, formulas-as-programs and formulas-astypes paradigms mentioned before.…”