109,045 research outputs found
Towards Ranking Geometric Automated Theorem Provers
The field of geometric automated theorem provers has a long and rich history,
from the early AI approaches of the 1960s, synthetic provers, to today
algebraic and synthetic provers.
The geometry automated deduction area differs from other areas by the strong
connection between the axiomatic theories and its standard models. In many
cases the geometric constructions are used to establish the theorems'
statements, geometric constructions are, in some provers, used to conduct the
proof, used as counter-examples to close some branches of the automatic proof.
Synthetic geometry proofs are done using geometric properties, proofs that can
have a visual counterpart in the supporting geometric construction.
With the growing use of geometry automatic deduction tools as applications in
other areas, e.g. in education, the need to evaluate them, using different
criteria, is felt. Establishing a ranking among geometric automated theorem
provers will be useful for the improvement of the current
methods/implementations. Improvements could concern wider scope, better
efficiency, proof readability and proof reliability.
To achieve the goal of being able to compare geometric automated theorem
provers a common test bench is needed: a common language to describe the
geometric problems; a comprehensive repository of geometric problems and a set
of quality measures.Comment: In Proceedings ThEdu'18, arXiv:1903.1240
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
An Introduction to Mechanized Reasoning
Mechanized reasoning uses computers to verify proofs and to help discover new
theorems. Computer scientists have applied mechanized reasoning to economic
problems but -- to date -- this work has not yet been properly presented in
economics journals. We introduce mechanized reasoning to economists in three
ways. First, we introduce mechanized reasoning in general, describing both the
techniques and their successful applications. Second, we explain how mechanized
reasoning has been applied to economic problems, concentrating on the two
domains that have attracted the most attention: social choice theory and
auction theory. Finally, we present a detailed example of mechanized reasoning
in practice by means of a proof of Vickrey's familiar theorem on second-price
auctions
Formal representation and proof for cooperative games
In this contribution we present some work we have been doing in representing and proving theorems from the area of economics, and mainly we present work we will do in a project in which we will apply mechanised theorem proving tools to a class of economic problems for which very few general tools currently exist. For mechanised theorem proving, the research introduces the field to a new application domain with a large user base; more specifically, the researchers are collaborating with developers working on state-of-the-art theorem provers. For economics, the research will provide tools for handling a hard class of problems; more generally, as the first application of mechanised theorem proving to centrally involve economic theorists, it aims to properly introduce mechanised theorem proving techniques to the discipline.\u
Designing Normative Theories for Ethical and Legal Reasoning: LogiKEy Framework, Methodology, and Tool Support
A framework and methodology---termed LogiKEy---for the design and engineering
of ethical reasoners, normative theories and deontic logics is presented. The
overall motivation is the development of suitable means for the control and
governance of intelligent autonomous systems. LogiKEy's unifying formal
framework is based on semantical embeddings of deontic logics, logic
combinations and ethico-legal domain theories in expressive classic
higher-order logic (HOL). This meta-logical approach enables the provision of
powerful tool support in LogiKEy: off-the-shelf theorem provers and model
finders for HOL are assisting the LogiKEy designer of ethical intelligent
agents to flexibly experiment with underlying logics and their combinations,
with ethico-legal domain theories, and with concrete examples---all at the same
time. Continuous improvements of these off-the-shelf provers, without further
ado, leverage the reasoning performance in LogiKEy. Case studies, in which the
LogiKEy framework and methodology has been applied and tested, give evidence
that HOL's undecidability often does not hinder efficient experimentation.Comment: 50 pages; 10 figure
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
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