72 research outputs found
Calculational Proofs in ACL2s
Teaching college students how to write rigorous proofs is a critical
objective in courses that introduce formal reasoning. Over the course of
several years, we have developed a mechanically-checkable style of
calculational reasoning that we used to teach over a thousand freshman-level
undergraduate students how to reason about computation in our "Logic and
Computation" class at Northeastern University. We were inspired by Dijkstra,
who advocated the use of calculational proofs, writing "calculational proofs
are almost always more effective than all informal alternatives, ..., the
design of calculational proofs seems much more teachable than the elusive art
of discovering an informal proof." Our calculational proof checker is
integrated into ACL2s and is available as an Eclipse IDE plugin, via a Web
interface, and as a stand-alone tool. It automatically checks proofs for
correctness and provides useful feedback. We describe the architecture of the
checker, its proof format, its underlying algorithms, its correctness and
provide examples using proofs from our undergraduate class and from Dijkstra.
We also describe our experiences using the proof checker to teach
undergraduates how to formally reason about computation
Making Presentation Math Computable
This Open-Access-book addresses the issue of translating mathematical expressions from LaTeX to the syntax of Computer Algebra Systems (CAS). Over the past decades, especially in the domain of Sciences, Technology, Engineering, and Mathematics (STEM), LaTeX has become the de-facto standard to typeset mathematical formulae in publications. Since scientists are generally required to publish their work, LaTeX has become an integral part of today's publishing workflow. On the other hand, modern research increasingly relies on CAS to simplify, manipulate, compute, and visualize mathematics. However, existing LaTeX import functions in CAS are limited to simple arithmetic expressions and are, therefore, insufficient for most use cases. Consequently, the workflow of experimenting and publishing in the Sciences often includes time-consuming and error-prone manual conversions between presentational LaTeX and computational CAS formats. To address the lack of a reliable and comprehensive translation tool between LaTeX and CAS, this thesis makes the following three contributions. First, it provides an approach to semantically enhance LaTeX expressions with sufficient semantic information for translations into CAS syntaxes. Second, it demonstrates the first context-aware LaTeX to CAS translation framework LaCASt. Third, the thesis provides a novel approach to evaluate the performance for LaTeX to CAS translations on large-scaled datasets with an automatic verification of equations in digital mathematical libraries. This is an open access book
Admissible types-to-PERs relativization in higher-order logic
Relativizing statements in Higher-Order Logic (HOL) from types to sets is useful for improving productivity when working with HOL-based interactive theorem provers such as HOL4, HOL Light and Isabelle/HOL. This paper provides the first comprehensive definition and study of types-to-sets relativization in HOL, done in the more general form of types-to-PERs (partial equivalence relations). We prove that, for a large practical fragment of HOL which includes container types such as datatypes and codatatypes, types-to-PERs relativization is admissible, in that the provability of the original, type-based statement implies the provability of its relativized, PER-based counterpart. Our results also imply the admissibility of a previously proposed axiomatic extension of HOL with local type definitions. We have implemented types-to-PERs relativization as an Isabelle tool that performs relativization of HOL theorems on demand
A Survey of Deep Learning for Mathematical Reasoning
Mathematical reasoning is a fundamental aspect of human intelligence and is
applicable in various fields, including science, engineering, finance, and
everyday life. The development of artificial intelligence (AI) systems capable
of solving math problems and proving theorems has garnered significant interest
in the fields of machine learning and natural language processing. For example,
mathematics serves as a testbed for aspects of reasoning that are challenging
for powerful deep learning models, driving new algorithmic and modeling
advances. On the other hand, recent advances in large-scale neural language
models have opened up new benchmarks and opportunities to use deep learning for
mathematical reasoning. In this survey paper, we review the key tasks,
datasets, and methods at the intersection of mathematical reasoning and deep
learning over the past decade. We also evaluate existing benchmarks and
methods, and discuss future research directions in this domain.Comment: Accepted to ACL 2023. The repository is available at
https://github.com/lupantech/dl4mat
Mechanising Euler's use of infinitesimals in the proof of the Basel problem
In 1736 Euler published a proof of an astounding relation between π and the reciprocals of the squares.
π²/6 = 1+ 1/4+ 1/9 + 1/25 …
Until this point, π had not been part of any mathematical relation outside of geometry. This relation would have had an almost supernatural significance to the mathematicians of the time. But even more amazing is Euler's proof. He factorises a transcendental function as if it were a polynomial of infinite degree. He discards infinitely-many infinitely-small numbers. He substitutes 1 for the ratio of two distinct infinite numbers.
Nowadays Euler's proof is held up as an example of both genius intuition and flagrantly unrigorous method. In this thesis we describe how, with the aid of nonstandard analysis, which gives a consistent formal theory of infinitely-small and large numbers, and the proof assistant Isabelle, we construct a partial formal proof of the Basel problem which follows the method of Euler's proof from his 'Introductio in Analysin Infinitorum'. We use our proof to demonstrate that Euler was systematic in his use of infinitely-large and infinitely-small numbers and did not make unjustified leaps of intuition. The concept of 'hidden lemmas' was developed by McKinzie and Tuckey based on Lakatos and Laugwitz to represent general principles which Euler's proof followed. We develop a theory of infinite 'hyperpolynomials' in Isabelle in order to formalise these hidden lemmas and we find that formal reconstruction of his proof using hidden lemmas is an effective way to discover the nuances in Euler's reasoning and demystify the controversial points. In conclusion, we find that Euler's reasoning was consistent and insightful, and yet has some distinct methodology to modern deductive proof
LEGO-Prover: Neural Theorem Proving with Growing Libraries
Despite the success of large language models (LLMs), the task of theorem
proving still remains one of the hardest reasoning tasks that is far from being
fully solved. Prior methods using language models have demonstrated promising
results, but they still struggle to prove even middle school level theorems.
One common limitation of these methods is that they assume a fixed theorem
library during the whole theorem proving process. However, as we all know,
creating new useful theorems or even new theories is not only helpful but
crucial and necessary for advancing mathematics and proving harder and deeper
results. In this work, we present LEGO-Prover, which employs a growing skill
library containing verified lemmas as skills to augment the capability of LLMs
used in theorem proving. By constructing the proof modularly, LEGO-Prover
enables LLMs to utilize existing skills retrieved from the library and to
create new skills during the proving process. These skills are further evolved
(by prompting an LLM) to enrich the library on another scale. Modular and
reusable skills are constantly added to the library to enable tackling
increasingly intricate mathematical problems. Moreover, the learned library
further bridges the gap between human proofs and formal proofs by making it
easier to impute missing steps. LEGO-Prover advances the state-of-the-art pass
rate on miniF2F-valid (48.0% to 57.0%) and miniF2F-test (45.5% to 47.1%).
During the proving process, LEGO-Prover also manages to generate over 20,000
skills (theorems/lemmas) and adds them to the growing library. Our ablation
study indicates that these newly added skills are indeed helpful for proving
theorems, resulting in an improvement from a success rate of 47.1% to 50.4%. We
also release our code and all the generated skills
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
An Implementation of Set Theory with Pointed Graphs in Dedukti
International audienceDEDUKTI is a type-checker for the λ Π-calculus modulo theory, a logical framework that allows the extension of conversion with user-defined rewrite rules. In this paper, we present the implementation of a version of Dowek-Miquel's intuitionistic set theory in DEDUKTI. To do so, we adapt this theory-based on the concept of pointed graphs-from Deduction modulo theory to λ Π-calculus modulo theory, and we formally write the proofs in DEDUKTI. In particular, this implementation requires the definition of a deep embedding of a certain class of formulas, as well as its interpretation in the theory
Mechanised metamathematics : an investigation of first-order logic and set theory in constructive type theory
In this thesis, we investigate several key results in the canon of metamathematics, applying the contemporary perspective of formalisation in constructive type theory and mechanisation in the Coq proof assistant. Concretely, we consider the central completeness, undecidability, and incompleteness theorems of first-order logic as well as properties of the axiom of choice and the continuum hypothesis in axiomatic set theory. Due to their fundamental role in the foundations of mathematics and their technical intricacies, these results have a long tradition in the codification as standard literature and, in more recent investigations, increasingly serve as a benchmark for computer mechanisation. With the present thesis, we continue this tradition by uniformly analysing the aforementioned cornerstones of metamathematics in the formal framework of constructive type theory. This programme offers novel insights into the constructive content of completeness, a synthetic approach to undecidability and incompleteness that largely eliminates the notorious tedium obscuring the essence of their proofs, as well as natural representations of set theory in the form of a second-order axiomatisation and of a fully type-theoretic account. The mechanisation concerning first-order logic is organised as a comprehensive Coq library open to usage and contribution by external users.In dieser Doktorarbeit werden einige Schlüsselergebnisse aus dem Kanon der Metamathematik untersucht, unter Verwendung der zeitgenössischen Perspektive von Formalisierung in konstruktiver Typtheorie und Mechanisierung mit Hilfe des Beweisassistenten Coq. Konkret werden die zentralen Vollständigkeits-, Unentscheidbarkeits- und Unvollständigkeitsergebnisse der Logik erster Ordnung sowie Eigenschaften des Auswahlaxioms und der Kontinuumshypothese in axiomatischer Mengenlehre betrachtet. Aufgrund ihrer fundamentalen Rolle in der Fundierung der Mathematik und ihrer technischen Schwierigkeiten, besitzen diese Ergebnisse eine lange Tradition der Kodifizierung als Standardliteratur und, besonders in jüngeren Untersuchungen, eine zunehmende Bedeutung als Maßstab für Mechanisierung mit Computern. Mit der vorliegenden Doktorarbeit wird diese Tradition fortgeführt, indem die zuvorgenannten Grundpfeiler der Methamatematik uniform im formalen Rahmen der konstruktiven Typtheorie analysiert werden. Dieses Programm ermöglicht neue Einsichten in den konstruktiven Gehalt von Vollständigkeit, einen synthetischen Ansatz für Unentscheidbarkeit und Unvollständigkeit, der großteils den berüchtigten, die Essenz der Beweise verdeckenden, technischen Aufwand eliminiert, sowie natürliche Repräsentationen von Mengentheorie in Form einer Axiomatisierung zweiter Ordnung und einer vollkommen typtheoretischen Darstellung. Die Mechanisierung zur Logik erster Ordnung ist als eine umfassende Coq-Bibliothek organisiert, die offen für Nutzung und Beiträge externer Anwender ist
Standard Formalization
A \emph{standard formalization} of a scientific theory is a system of axioms for that theory in a first-order language (possibly many-sorted; possibly with the membership primitive ). Patrick Suppes (\cite{sup92}) expressed skepticism about whether there is a ``simple or elegant method'' for presenting mathematicized scientific theories in such a standard formalization, because they ``assume a great deal of mathematics as part of their substructure''.
The major difficulties amount to these. First, as the theories of interest are \emph{mathematicized}, one must specify the underlying \emph{applied mathematics base theory}, which the physical axioms live on top of. Second, such theories are typically \emph{geometric}, concerning quantities or trajectories in space/time: so, one must specify the underlying \emph{physical geometry}. Third, the differential equations involved generally refer to \emph{coordinate representations} of these physical quantities with respect to some implicit coordinate chart, not to the original quantities.
These issues may be resolved. Once this is done, constructing standard formalizations is not so difficult---at least for the theories where the mathematics has been worked out rigorously. Here we give what may be claimed to be a simple and elegant means of doing that. This is for mathematicized scientific theories comprising differential equations for -valued quantities (that is, scalar fields), defined on (``spatial'' or ``temporal'') dimensions, taken to be isomorphic to the usual Euclidean space . For illustration, I give standard (in a sense, ``text-book'') formalizations: for the simple harmonic oscillator equation in one-dimension and for the Laplace equation in two dimensions
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