184,874 research outputs found
Automated Reasoning and Presentation Support for Formalizing Mathematics in Mizar
This paper presents a combination of several automated reasoning and proof
presentation tools with the Mizar system for formalization of mathematics. The
combination forms an online service called MizAR, similar to the SystemOnTPTP
service for first-order automated reasoning. The main differences to
SystemOnTPTP are the use of the Mizar language that is oriented towards human
mathematicians (rather than the pure first-order logic used in SystemOnTPTP),
and setting the service in the context of the large Mizar Mathematical Library
of previous theorems,definitions, and proofs (rather than the isolated problems
that are solved in SystemOnTPTP). These differences poses new challenges and
new opportunities for automated reasoning and for proof presentation tools.
This paper describes the overall structure of MizAR, and presents the automated
reasoning systems and proof presentation tools that are combined to make MizAR
a useful mathematical service.Comment: To appear in 10th International Conference on. Artificial
Intelligence and Symbolic Computation AISC 201
Systematic Verification of the Modal Logic Cube in Isabelle/HOL
We present an automated verification of the well-known modal logic cube in
Isabelle/HOL, in which we prove the inclusion relations between the cube's
logics using automated reasoning tools. Prior work addresses this problem but
without restriction to the modal logic cube, and using encodings in first-order
logic in combination with first-order automated theorem provers. In contrast,
our solution is more elegant, transparent and effective. It employs an
embedding of quantified modal logic in classical higher-order logic. Automated
reasoning tools, such as Sledgehammer with LEO-II, Satallax and CVC4, Metis and
Nitpick, are employed to achieve full automation. Though successful, the
experiments also motivate some technical improvements in the Isabelle/HOL tool.Comment: In Proceedings PxTP 2015, arXiv:1507.0837
PyZX: Large Scale Automated Diagrammatic Reasoning
The ZX-calculus is a graphical language for reasoning about ZX-diagrams, a
type of tensor networks that can represent arbitrary linear maps between
qubits. Using the ZX-calculus, we can intuitively reason about quantum theory,
and optimise and validate quantum circuits. In this paper we introduce PyZX, an
open source library for automated reasoning with large ZX-diagrams. We give a
brief introduction to the ZX-calculus, then show how PyZX implements methods
for circuit optimisation, equality validation, and visualisation and how it can
be used in tandem with other software. We end with a set of challenges that
when solved would enhance the utility of automated diagrammatic reasoning.Comment: In Proceedings QPL 2019, arXiv:2004.1475
Learning-Assisted Automated Reasoning with Flyspeck
The considerable mathematical knowledge encoded by the Flyspeck project is
combined with external automated theorem provers (ATPs) and machine-learning
premise selection methods trained on the proofs, producing an AI system capable
of answering a wide range of mathematical queries automatically. The
performance of this architecture is evaluated in a bootstrapping scenario
emulating the development of Flyspeck from axioms to the last theorem, each
time using only the previous theorems and proofs. It is shown that 39% of the
14185 theorems could be proved in a push-button mode (without any high-level
advice and user interaction) in 30 seconds of real time on a fourteen-CPU
workstation. The necessary work involves: (i) an implementation of sound
translations of the HOL Light logic to ATP formalisms: untyped first-order,
polymorphic typed first-order, and typed higher-order, (ii) export of the
dependency information from HOL Light and ATP proofs for the machine learners,
and (iii) choice of suitable representations and methods for learning from
previous proofs, and their integration as advisors with HOL Light. This work is
described and discussed here, and an initial analysis of the body of proofs
that were found fully automatically is provided
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