An Interpreter for Quantum Circuits

This paper describes an ACL2 interpreter for "netlists" describing quantum circuits. Several quantum gates are implemented, including the Hadamard gate H, which rotates vectors by 45 degrees, necessitating the use of irrational numbers, at least at the logical level. Quantum measurement presents an especially difficult challenge, because it requires precise comparisons of irrational numbers and the use of random numbers. This paper does not address computation with irrational numbers or the generation of random numbers, although future work includes the development of pseudo-random generators for ACL2.Comment: In Proceedings ACL2 2013, arXiv:1304.712

Formal Verification of Medina's Sequence of Polynomials for Approximating Arctangent

The verification of many algorithms for calculating transcendental functions is based on polynomial approximations to these functions, often Taylor series approximations. However, computing and verifying approximations to the arctangent function are very challenging problems, in large part because the Taylor series converges very slowly to arctangent-a 57th-degree polynomial is needed to get three decimal places for arctan(0.95). Medina proposed a series of polynomials that approximate arctangent with far faster convergence-a 7th-degree polynomial is all that is needed to get three decimal places for arctan(0.95). We present in this paper a proof in ACL2(r) of the correctness and convergence rate of this sequence of polynomials. The proof is particularly beautiful, in that it uses many results from real analysis. Some of these necessary results were proven in prior work, but some were proven as part of this effort.Comment: In Proceedings ACL2 2014, arXiv:1406.123

Fourier Series Formalization in ACL2(r)

We formalize some basic properties of Fourier series in the logic of ACL2(r), which is a variant of ACL2 that supports reasoning about the real and complex numbers by way of non-standard analysis. More specifically, we extend a framework for formally evaluating definite integrals of real-valued, continuous functions using the Second Fundamental Theorem of Calculus. Our extended framework is also applied to functions containing free arguments. Using this framework, we are able to prove the orthogonality relationships between trigonometric functions, which are the essential properties in Fourier series analysis. The sum rule for definite integrals of indexed sums is also formalized by applying the extended framework along with the First Fundamental Theorem of Calculus and the sum rule for differentiation. The Fourier coefficient formulas of periodic functions are then formalized from the orthogonality relations and the sum rule for integration. Consequently, the uniqueness of Fourier sums is a straightforward corollary. We also present our formalization of the sum rule for definite integrals of infinite series in ACL2(r). Part of this task is to prove the Dini Uniform Convergence Theorem and the continuity of a limit function under certain conditions. A key technique in our proofs of these theorems is to apply the overspill principle from non-standard analysis.Comment: In Proceedings ACL2 2015, arXiv:1509.0552

Formalized linear algebra over Elementary Divisor Rings in Coq

This paper presents a Coq formalization of linear algebra over elementary divisor rings, that is, rings where every matrix is equivalent to a matrix in Smith normal form. The main results are the formalization that these rings support essential operations of linear algebra, the classification theorem of finitely presented modules over such rings and the uniqueness of the Smith normal form up to multiplication by units. We present formally verified algorithms computing this normal form on a variety of coefficient structures including Euclidean domains and constructive principal ideal domains. We also study different ways to extend B\'ezout domains in order to be able to compute the Smith normal form of matrices. The extensions we consider are: adequacy (i.e. the existence of a gdco operation), Krull dimension $\leq 1$ and well-founded strict divisibility

Formalization of Transform Methods using HOL Light

Transform methods, like Laplace and Fourier, are frequently used for analyzing the dynamical behaviour of engineering and physical systems, based on their transfer function, and frequency response or the solutions of their corresponding differential equations. In this paper, we present an ongoing project, which focuses on the higher-order logic formalization of transform methods using HOL Light theorem prover. In particular, we present the motivation of the formalization, which is followed by the related work. Next, we present the task completed so far while highlighting some of the challenges faced during the formalization. Finally, we present a roadmap to achieve our objectives, the current status and the future goals for this project.Comment: 15 Pages, CICM 201

Applying ACL2 to the Formalization of Algebraic Topology: Simplicial Polynomials

In this paper we present a complete formalization, using the ACL2 theorem prover, of the Normalization Theorem, a result in Algebraic Simplicial Topology stating that there exists a homotopy equivalence between the chain complex of a simplicial set, and a smaller chain complex for the same simplicial set, called the normalized chain complex. The interest of this work stems from three sources. First, the normalization theorem is the basis for some design decisions in the Kenzo computer algebra system, a program for computing in Algebraic Topology. Second, our proof of the theorem is new and shows the correctness of some formulas found experimentally, giving explicit expressions for the above-mentioned homotopy equivalence. And third, it demonstrates that the ACL2 theorem prover can be effectively used to formalize mathematics, even in areas where higher-order tools could be thought to be more appropriate.Ministerio de Ciencia e Innovación MTM2009-13842European Commission FP7 STREP project ForMath n. 24384

Verified Computer Algebra in ACL2 (Gröbner Bases Computation)

In this paper, we present the formal verification of a Common Lisp implementation of Buchberger’s algorithm for computing Gröbner bases of polynomial ideals. This work is carried out in the Acl2 system and shows how verified Computer Algebra can be achieved in an executable logic