400 research outputs found
Learning to Prove Trigonometric Identities
Automatic theorem proving with deep learning methods has attracted attentions
recently. In this paper, we construct an automatic proof system for
trigonometric identities. We define the normalized form of trigonometric
identities, design a set of rules for the proof and put forward a method which
can generate theoretically infinite trigonometric identities. Our goal is not
only to complete the proof, but to complete the proof in as few steps as
possible. For this reason, we design a model to learn proof data generated by
random BFS (rBFS), and it is proved theoretically and experimentally that the
model can outperform rBFS after a simple imitation learning. After further
improvement through reinforcement learning, we get AutoTrig, which can give
proof steps for identities in almost as short steps as BFS (theoretically
shortest method), with a time cost of only one-thousandth. In addition,
AutoTrig also beats Sympy, Matlab and human in the synthetic dataset, and
performs well in many generalization tasks
FunGrim: a symbolic library for special functions
We present the Mathematical Functions Grimoire (FunGrim), a website and
database of formulas and theorems for special functions. We also discuss the
symbolic computation library used as the backend and main development tool for
FunGrim, and the Grim formula language used in these projects to represent
mathematical content semantically
Automated Generation of User Guidance by Combining Computation and Deduction
Herewith, a fairly old concept is published for the first time and named
"Lucas Interpretation". This has been implemented in a prototype, which has
been proved useful in educational practice and has gained academic relevance
with an emerging generation of educational mathematics assistants (EMA) based
on Computer Theorem Proving (CTP).
Automated Theorem Proving (ATP), i.e. deduction, is the most reliable
technology used to check user input. However ATP is inherently weak in
automatically generating solutions for arbitrary problems in applied
mathematics. This weakness is crucial for EMAs: when ATP checks user input as
incorrect and the learner gets stuck then the system should be able to suggest
possible next steps.
The key idea of Lucas Interpretation is to compute the steps of a calculation
following a program written in a novel CTP-based programming language, i.e.
computation provides the next steps. User guidance is generated by combining
deduction and computation: the latter is performed by a specific language
interpreter, which works like a debugger and hands over control to the learner
at breakpoints, i.e. tactics generating the steps of calculation. The
interpreter also builds up logical contexts providing ATP with the data
required for checking user input, thus combining computation and deduction.
The paper describes the concepts underlying Lucas Interpretation so that open
questions can adequately be addressed, and prerequisites for further work are
provided.Comment: In Proceedings THedu'11, arXiv:1202.453
SBML models and MathSBML
MathSBML is an open-source, freely-downloadable Mathematica package that facilitates working with Systems Biology Markup Language (SBML) models. SBML is a toolneutral,computer-readable format for representing models of biochemical reaction networks, applicable to metabolic networks, cell-signaling pathways, genomic regulatory networks, and other modeling problems in systems biology that is widely supported by the systems biology community. SBML is based on XML, a standard medium for representing and transporting data that is widely supported on the internet as well as in computational biology and bioinformatics. Because SBML is tool-independent, it enables model transportability, reuse, publication and survival. In addition to MathSBML, a number of other tools that support SBML model examination and manipulation are provided on the sbml.org website, including libSBML, a C/C++ library for reading SBML models; an SBML Toolbox for MatLab; file conversion programs; an SBML model validator and visualizer; and SBML specifications and schemas. MathSBML enables SBML file import to and export from Mathematica as well as providing an API for model manipulation and simulation
Using features for automated problem solving
We motivate and present an architecture for problem solving where an abstraction
layer of "features" plays the key role in determining methods to apply. The system
is presented in the context of theorem proving with Isabelle, and we demonstrate
how this approach to encoding control knowledge is expressively different to
other common techniques. We look closely at two areas where the feature
layer may offer benefits to theorem proving — semi-automation and learning
—
and find strong evidence that in these particular domains, the approach shows
compelling promise. The system includes a graphical theorem-proving user
interface for Eclipse ProofGeneral and is available from the project web page,
http://feasch.heneveld.org
FunGrim: a symbolic library for special functions
We present the Mathematical Functions Grimoire (FunGrim), a website and database of formulas and theorems for special functions. We also discuss the symbolic computation library used as the backend and main development tool for FunGrim, and the Grim formula language used in these projects to represent mathematical content semantically
Automated Deduction – CADE 28
This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions
Comparative Verification of the Digital Library of Mathematical Functions and Computer Algebra Systems
Digital mathematical libraries assemble the knowledge of years of
mathematical research. Numerous disciplines (e.g., physics, engineering, pure
and applied mathematics) rely heavily on compendia gathered findings. Likewise,
modern research applications rely more and more on computational solutions,
which are often calculated and verified by computer algebra systems. Hence, the
correctness, accuracy, and reliability of both digital mathematical libraries
and computer algebra systems is a crucial attribute for modern research.
In this paper, we present a novel approach to verify a digital mathematical
library and two computer algebra systems with one another by converting
mathematical expressions from one system to the other. We use our previously
eveloped conversion tool (referred to as LaCASt) to translate formulae from the
NIST Digital Library of Mathematical Functions to the computer algebra systems
Maple and Mathematica. The contributions of our presented work are as follows:
(1) we present the most comprehensive verification of computer algebra systems
and digital mathematical libraries with one another; (2) we significantly
enhance the performance of the underlying translator in terms of coverage and
accuracy; and (3) we provide open access to translations for Maple and
Mathematica of the formulae in the NIST Digital Library of Mathematical
Functions
Formalization of Real Analysis: A Survey of Proof Assistants and Libraries
International audienceIn the recent years, numerous proof systems have improved enough to be used for formally verifying non-trivial mathematical results. They, however, have different purposes and it is not always easy to choose which one is adapted to undertake a formalization effort. In this survey, we focus on properties related to real analysis: real numbers, arithmetic operators, limits, differentiability, integrability, and so on. We have chosen to look into the formalizations provided in standard by the following systems: Coq, HOL4, HOL Light, Isabelle/HOL, Mizar, ProofPower-HOL, and PVS. We have also accounted for large developments that play a similar role or extend standard libraries: ACL2(r) for ACL2, C-CoRN/MathClasses for Coq, and the NASA PVS library. This survey presents how real numbers have been defined in these various provers and how the notions of real analysis described above have been formalized. We also look at the methods of automation these systems provide for real analysis
- …