15 research outputs found
On the Herbrand content of LK
We present a structural representation of the Herbrand content of LK-proofs
with cuts of complexity prenex Sigma-2/Pi-2. The representation takes the form
of a typed non-deterministic tree grammar of order 2 which generates a finite
language of first-order terms that appear in the Herbrand expansions obtained
through cut-elimination. In particular, for every Gentzen-style reduction
between LK-proofs we study the induced grammars and classify the cases in which
language equality and inclusion hold.Comment: In Proceedings CL&C 2016, arXiv:1606.0582
Introducing Quantified Cuts in Logic with Equality
Cut-introduction is a technique for structuring and compressing formal
proofs. In this paper we generalize our cut-introduction method for the
introduction of quantified lemmas of the form (for
quantifier-free ) to a method generating lemmas of the form . Moreover, we extend the original method to predicate
logic with equality. The new method was implemented and applied to the TSTP
proof database. It is shown that the extension of the method to handle equality
and quantifier-blocks leads to a substantial improvement of the old algorithm
Project Presentation: Algorithmic Structuring and Compression of Proofs (ASCOP)
International audienceComputer-generated proofs are typically analytic, i.e. they essentially consist only of formulas which are present in the theorem that is shown. In contrast, mathematical proofs written by humans almost never are: they are highly structured due to the use of lemmas. The ASCOP-project aims at developing algorithms and software which structure and abbreviate analytic proofs by computing useful lemmas. These algorithms will be based on recent groundbreaking results establishing a new connection between proof theory and formal language theory. This connection allows the application of e cient algorithms based on formal grammars to structure and compress proofs
Project Presentation: Algorithmic Structuring and Compression of Proofs (ASCOP)
International audienceComputer-generated proofs are typically analytic, i.e. they essentially consist only of formulas which are present in the theorem that is shown. In contrast, mathematical proofs written by humans almost never are: they are highly structured due to the use of lemmas. The ASCOP-project aims at developing algorithms and software which structure and abbreviate analytic proofs by computing useful lemmas. These algorithms will be based on recent groundbreaking results establishing a new connection between proof theory and formal language theory. This connection allows the application of e cient algorithms based on formal grammars to structure and compress proofs
Lemmas: Generation, Selection, Application
Noting that lemmas are a key feature of mathematics, we engage in an
investigation of the role of lemmas in automated theorem proving. The paper
describes experiments with a combined system involving learning technology that
generates useful lemmas for automated theorem provers, demonstrating
improvement for several representative systems and solving a hard problem not
solved by any system for twenty years. By focusing on condensed detachment
problems we simplify the setting considerably, allowing us to get at the
essence of lemmas and their role in proof search
REFACTOR: Learning to Extract Theorems from Proofs
Human mathematicians are often good at recognizing modular and reusable
theorems that make complex mathematical results within reach. In this paper, we
propose a novel method called theoREm-from-prooF extrACTOR (REFACTOR) for
training neural networks to mimic this ability in formal mathematical theorem
proving. We show on a set of unseen proofs, REFACTOR is able to extract 19.6%
of the theorems that humans would use to write the proofs. When applying the
model to the existing Metamath library, REFACTOR extracted 16 new theorems.
With newly extracted theorems, we show that the existing proofs in the MetaMath
database can be refactored. The new theorems are used very frequently after
refactoring, with an average usage of 733.5 times, and help shorten the proof
lengths. Lastly, we demonstrate that the prover trained on the new-theorem
refactored dataset proves more test theorems and outperforms state-of-the-art
baselines by frequently leveraging a diverse set of newly extracted theorems.
Code can be found at https://github.com/jinpz/refactor.Comment: ICLR 202