Comparator automata in quantitative verification

The notion of comparison between system runs is fundamental in formal verification. This concept is implicitly present in the verification of qualitative systems, and is more pronounced in the verification of quantitative systems. In this work, we identify a novel mode of comparison in quantitative systems: the online comparison of the aggregate values of two sequences of quantitative weights. This notion is embodied by {\em comparator automata} ({\em comparators}, in short), a new class of automata that read two infinite sequences of weights synchronously and relate their aggregate values. We show that {aggregate functions} that can be represented with B\"uchi automaton result in comparators that are finite-state and accept by the B\"uchi condition as well. Such {\em $\omega$-regular comparators} further lead to generic algorithms for a number of well-studied problems, including the quantitative inclusion and winning strategies in quantitative graph games with incomplete information, as well as related non-decision problems, such as obtaining a finite representation of all counterexamples in the quantitative inclusion problem. We study comparators for two aggregate functions: discounted-sum and limit-average. We prove that the discounted-sum comparator is $\omega$-regular iff the discount-factor is an integer. Not every aggregate function, however, has an $\omega$-regular comparator. Specifically, we show that the language of sequence-pairs for which limit-average aggregates exist is neither $\omega$-regular nor $\omega$-context-free. Given this result, we introduce the notion of {\em prefix-average} as a relaxation of limit-average aggregation, and show that it admits $\omega$-context-free comparators

A Semantics for Approximate Program Transformations

An approximate program transformation is a transformation that can change the semantics of a program within a specified empirical error bound. Such transformations have wide applications: they can decrease computation time, power consumption, and memory usage, and can, in some cases, allow implementations of incomputable operations. Correctness proofs of approximate program transformations are by definition quantitative. Unfortunately, unlike with standard program transformations, there is as of yet no modular way to prove correctness of an approximate transformation itself. Error bounds must be proved for each transformed program individually, and must be re-proved each time a program is modified or a different set of approximations are applied. In this paper, we give a semantics that enables quantitative reasoning about a large class of approximate program transformations in a local, composable way. Our semantics is based on a notion of distance between programs that defines what it means for an approximate transformation to be correct up to an error bound. The key insight is that distances between programs cannot in general be formulated in terms of metric spaces and real numbers. Instead, our semantics admits natural notions of distance for each type construct; for example, numbers are used as distances for numerical data, functions are used as distances for functional data, an polymorphic lambda-terms are used as distances for polymorphic data. We then show how our semantics applies to two example approximations: replacing reals with floating-point numbers, and loop perforation

Euler: A System for Numerical Optimization of Programs

We give a tutorial introduction to Euler, a system for solving difficult optimization problems involving programs.National Science Foundation (U.S.) (Award 1156059)National Science Foundation (U.S.) (Award 1116362

Symbolic Quantum Simulation with Quasimodo

The simulation of quantum circuits on classical computers is an important problem in quantum computing. Such simulation requires representations of distributions over very large sets of basis vectors, and recent work has used symbolic data-structures such as Binary Decision Diagrams (BDDs) for this purpose. In this tool paper, we present Quasimodo, an extensible, open-source Python library for symbolic simulation of quantum circuits. Quasimodo is specifically designed for easy extensibility to other backends. Quasimodo allows simulations of quantum circuits, checking properties of the outputs of quantum circuits, and debugging quantum circuits. It also allows the user to choose from among several symbolic data-structures -- both unweighted and weighted BDDs, and a recent structure called Context-Free-Language Ordered Binary Decision Diagrams (CFLOBDDs) -- and can be easily extended to support other symbolic data-structures.Comment: 15 pages; 35th International Conference on Computer Aided Verification (CAV 2023