Related papers: Computer-Assisted Verification of Four Interval Ar…
Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly…
Interval arithmetic is a simple way to compute a mathematical expression to an arbitrary accuracy, widely used for verifying floating-point computations. Yet this simplicity belies challenges. Some inputs violate preconditions or cause…
Interval computation is widely used to certify computations that use floating point operations to avoid pitfalls related to rounding error introduced by inaccurate operations. Despite its popularity and practical benefits, support for…
Most existing implementations of multiple precision arithmetic demand that the user sets the precision {\em a priori}. Some libraries are said adaptable in the sense that they dynamically change the precision of each intermediate operation…
Gappa uses interval arithmetic to certify bounds on mathematical expressions that involve rounded as well as exact operators. Gappa generates a theorem with its proof for each bound treated. The proof can be checked with a higher order…
If the non-zero finite floating-point numbers are interpreted as point intervals, then the effect of rounding can be interpreted as computing one of the bounds of the result according to interval arithmetic. We give an interval…
We give a process for verifying numerical programs against their functional specifications. Our implementation is capable of automatically verifying programs against tight error bounds featuring common elementary functions. We demonstrate…
System identification is an important area of science, which aims to describe the characteristics of the system, representing them by mathematical models. Since many of these models can be seen as recursive functions, it is extremely…
In this paper, we present a toolbox for interval analysis in numpy, with an application to formal verification of neural network controlled systems. Using the notion of natural inclusion functions, we systematically construct interval…
We consider the prospect of a processor that can perform interval arithmetic at the same speed as conventional floating-point arithmetic. This makes it possible for all arithmetic to be performed with the superior security of interval…
Formal proof checkers such as Coq are capable of validating proofs of correction of algorithms for finite field arithmetics but they require extensive training from potential users. The delayed solution of a triangular system over a finite…
Verification of programs using floating-point arithmetic is challenging on several accounts. One of the difficulties of reasoning about such programs is due to the peculiarities of floating-point arithmetic: rounding errors, infinities,…
Floating-point arithmetic (FPA) is a mechanical representation of real arithmetic (RA), where each operation is replaced with a rounded counterpart. Various numerical properties can be verified by using SMT solvers that support the logic of…
Inspired by computer assisted proofs in analysis, we present an interval approach to real-number computations.
Verification of C++ programs has seen considerable progress in several areas, but not for programs that use these languages' mathematical libraries. The reason is that all libraries in widespread use come with no guarantees about the…
Floating point operations are fast, but require continuous effort on the part of the user in order to ensure that the results are correct. This burden can be shifted away from the user by providing a library of exact analysis in which the…
Reasoning about real number expressions in a proof assistant is challenging. Several problems in theorem proving can be solved by using exact real number computation. I have implemented a library for reasoning and computing with complete…
High confidence in floating-point programs requires proving numerical properties of final and intermediate values. One may need to guarantee that a value stays within some range, or that the error relative to some ideal value is well…
This paper analyzes the computational complexity of validated interval methods for uncertain nonlinear systems and steady-state enclosure. Interval analysis produces guaranteed enclosures that account for uncertainty and round-off, but its…
As developers of libraries implementing interval arithmetic, we faced the same difficulties when it came to testing our libraries. What must be tested? How can we devise relevant test cases for unit testing? How can we ensure a high (and…