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For a polynomial P, we consider the sequence of iterated integrals of ln P(x). This sequence is expressed in terms of the zeros of P(x). In the special case of ln(1 + x^2), arithmetic properties of certain coefficients arising are…
There is a strong consensus that combining the versatility of machine learning with the assurances given by formal verification is highly desirable. It is much less clear what verified machine learning should mean exactly. We consider this…
Machine learning has found its way into almost every area of science and engineering, and we are only at the beginning of its exploration across fields. Being a popular, versatile and powerful framework, machine learning has proven most…
A problem based on the Extended Euclidean Algorithm applied to a class of polynomials with many factors is presented and believed to be hard. If so, it is a one-way function well suited for applications in digital signicatures.
In this paper we discuss how semantic annotations can be used to introduce mathematical algorithmic information of the underlying imperative code to enable compilers to produce code transformations that will enable better performance. By…
Proof automation is crucial to large-scale formal mathematics and software/hardware verification projects in ITPs. Sophisticated tools called hammers have been developed to provide general-purpose proof automation in ITPs such as Coq and…
There is a subset of computational problems that are computable in polynomial time for which an existing algorithm may not complete due to a lack of high performance technology on a mission field. We define a subclass of deterministic…
Labeled data for imitation learning of theorem proving in large libraries of formalized mathematics is scarce as such libraries require years of concentrated effort by human specialists to be built. This is particularly challenging when…
This paper compares different representations (in the sense of computable analysis) of a number of function spaces that are of interest in analysis. In particular subspace representations inherited from a larger function space are compared…
Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all…
Program logics are a powerful formal method in the context of program verification. Can we develop a counterpart of program logics in the context of language verification? This paper proposes language logics, which allow for statements of…
Equational reasoning is one of the key features of pure functional languages such as Haskell. To date, however, such reasoning always took place externally to Haskell, either manually on paper, or mechanised in a theorem prover. This…
We first show a simple but striking result in bilevel optimization: unconstrained $C^\infty$ smooth bilevel programming is as hard as general extended-real-valued lower semicontinuous minimization. We then proceed to a worst-case analysis…
In theorem provers based on dependent type theory such as Coq and Lean, induction is a fundamental proof method and induction tactics are omnipresent in proof scripts. Yet the ergonomics of existing induction tactics are not ideal: they do…
The programming language Prolog makes declarative programming possible, at least to a substantial extent. Programs may be written and reasoned about in terms of their declarative semantics. All the advantages of declarative programming are…
We are interested in the fast computation of the exact value of integrals of polynomial functions over convex polyhedra. We present speed ups and extensions of the algorithms presented in previous work. We present the new software…
When implementing regular enough functions (e.g., elementary or special functions) on a computing system, we frequently use polynomial approximations. In most cases, the polynomial that best approximates (for a given distance and in a given…
We consider the problem of constructing semisimple subalgebras of real (semi-) simple Lie algebras. We develop computational methods that help to deal with this problem. Our methods boil down to solving a set of polynomial equations. In…
Runtime efficiency and termination are crucial properties in the studies of program verification. Instead of dealing with these issues in an ad hoc manner, it would be useful to develop a robust framework in which such properties are…
The interactive theorem prover Lean enables the verification of formal mathematical proofs and is backed by an expanding community. Central to this ecosystem is its mathematical library, mathlib4, which lays the groundwork for the…