6,087 research outputs found
Learning to Prove Theorems via Interacting with Proof Assistants
Humans prove theorems by relying on substantial high-level reasoning and
problem-specific insights. Proof assistants offer a formalism that resembles
human mathematical reasoning, representing theorems in higher-order logic and
proofs as high-level tactics. However, human experts have to construct proofs
manually by entering tactics into the proof assistant. In this paper, we study
the problem of using machine learning to automate the interaction with proof
assistants. We construct CoqGym, a large-scale dataset and learning environment
containing 71K human-written proofs from 123 projects developed with the Coq
proof assistant. We develop ASTactic, a deep learning-based model that
generates tactics as programs in the form of abstract syntax trees (ASTs).
Experiments show that ASTactic trained on CoqGym can generate effective tactics
and can be used to prove new theorems not previously provable by automated
methods. Code is available at https://github.com/princeton-vl/CoqGym.Comment: Accepted to ICML 201
Context Semantics, Linear Logic and Computational Complexity
We show that context semantics can be fruitfully applied to the quantitative
analysis of proof normalization in linear logic. In particular, context
semantics lets us define the weight of a proof-net as a measure of its inherent
complexity: it is both an upper bound to normalization time (modulo a
polynomial overhead, independently on the reduction strategy) and a lower bound
to the number of steps to normal form (for certain reduction strategies).
Weights are then exploited in proving strong soundness theorems for various
subsystems of linear logic, namely elementary linear logic, soft linear logic
and light linear logic.Comment: 22 page
An axiomatic approach for solving geometric problems symbolically
technical reportThis paper describes a new approach for solving geometric constraint problems and problems in geometry theorem proving. We developed a rewrite-rule mechanism operating on geometric predicates. Termination and completeness of the problem solving algorithm can be obtained through well foundedness and confluence of the set of rewrite rules. To guarantee these properties we adapted the Knuth-Bendix completion algorithm to the specific requirements of the geometric problem. A symbolic, geometric solution has the advantage over the usual algebraic approach that it speaks the language of geometry. Therefore, it has the potential to be used in many practical applications in interactive Computer Aided Design
A Science of Reasoning
This paper addresses the question of how we can understand reasoning in general and mathematical proofs in particular. It argues the need for a high-level understanding of proofs to complement the low-level understanding provided by Logic. It proposes a role for computation in providing this high-level understanding, namely by the association of proof plans with proofs. Proof plans are defined and examples are given for two families of proofs. Criteria are given for assessing the association of a proof plan with a proof. 1 Motivation: the understanding of mathematical proofs The understanding of reasoning has interested researchers since, at least, Aristotle. Logic has been proposed by Aristotle, Boole, Frege and others as a way of formalising arguments and understanding their structure. There have also been psychological studies of how people and animals actually do reason. The work on Logic has been especially influential in the automation of reasoning. For instance, resolution..
Globular: an online proof assistant for higher-dimensional rewriting
This article introduces Globular, an online proof assistant for the
formalization and verification of proofs in higher-dimensional category theory.
The tool produces graphical visualizations of higher-dimensional proofs,
assists in their construction with a point-and- click interface, and performs
type checking to prevent incorrect rewrites. Hosted on the web, it has a low
barrier to use, and allows hyperlinking of formalized proofs directly from
research papers. It allows the formalization of proofs from logic, topology and
algebra which are not formalizable by other methods, and we give several
examples
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
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