The quotient in preorder theories

Seeking the largest solution to an expression of the form Ax 64 B is a common task in several domains of engineering and computer science. This largest solution is commonly called quotient. Across domains, the meanings of the binary operation and the preorder are quite different, yet the syntax for computing the largest solution is remarkably similar. This paper is about finding a common framework to reason about quotients. We only assume we operate on a preorder endowed with an abstract monotonic multiplication and an involution. We provide a condition, called admissibility, which guarantees the existence of the quotient, and which yields its closed form. We call preordered heaps those structures satisfying the admissibility condition. We show that many existing theories in computer science are preordered heaps, and we are thus able to derive a quotient for them, subsuming existing solutions when available in the literature. We introduce the concept of sieved heaps to deal with structures which are given over multiple domains of definition. We show that sieved heaps also have well-defined quotients

The Quotient in Preorder Theories

Seeking the largest solution to an expression of the form A x <= B is a common task in several domains of engineering and computer science. This largest solution is commonly called quotient. Across domains, the meanings of the binary operation and the preorder are quite different, yet the syntax for computing the largest solution is remarkably similar. This paper is about finding a common framework to reason about quotients. We only assume we operate on a preorder endowed with an abstract monotonic multiplication and an involution. We provide a condition, called admissibility, which guarantees the existence of the quotient, and which yields its closed form. We call preordered heaps those structures satisfying the admissibility condition. We show that many existing theories in computer science are preordered heaps, and we are thus able to derive a quotient for them, subsuming existing solutions when available in the literature. We introduce the concept of sieved heaps to deal with structures which are given over multiple domains of definition. We show that sieved heaps also have well-defined quotients.Comment: In Proceedings GandALF 2020, arXiv:2009.0936

Automated synthesis of protocol converters with BALM-II

We address the problem of the automatic design of automata to translatebetween different protocols, and we reduce it to the solution of equationsdefined over regular languages and finite automata (FA)/finite state machines(FSMs).The largest solution of the defined language equations includesall protocol converters that solve the problem;this is a strong advantage over computational techniques that deliver only oneor a few solutions, which might lead to suboptimal implementations(e.g., as sequential circuits).Our model is versatile, because it can handle different topologies andconstraints on the solutions.We propose a fully automatic procedure implemented inside a software packageBALM-II which solves language equations.For illustration we show examples of setting up and solving language equationsfor classical protocol mismatch problems, aiming at the design of protocolconverters to interface an alternating-bit (AB) sender and a non-sequenced(NS) receiver.Our automatic converter synthesis procedure yields a complete solutionfor automata and FSMs, and may serve as a core engineto embed into any full-fledged interface synthesis tool