Pub Date : 2023-05-31DOI: 10.1007/978-3-031-42753-4_8
Robert M Corless, D. J. Jeffrey, A. Shakoori
{"title":"Teaching Linear Algebra in a Mechanized Mathematical Environment","authors":"Robert M Corless, D. J. Jeffrey, A. Shakoori","doi":"10.1007/978-3-031-42753-4_8","DOIUrl":"https://doi.org/10.1007/978-3-031-42753-4_8","url":null,"abstract":"","PeriodicalId":236059,"journal":{"name":"International Conference on Intelligent Computer Mathematics","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134521646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-23DOI: 10.1007/978-3-031-42753-4_1
Lawrence Charles Paulson
{"title":"Large-Scale Formal Proof for the Working Mathematician - Lessons Learnt from the ALEXANDRIA Project","authors":"Lawrence Charles Paulson","doi":"10.1007/978-3-031-42753-4_1","DOIUrl":"https://doi.org/10.1007/978-3-031-42753-4_1","url":null,"abstract":"","PeriodicalId":236059,"journal":{"name":"International Conference on Intelligent Computer Mathematics","volume":"126 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128217673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-01DOI: 10.48550/arXiv.2304.10486
Eric Yeh, B. Hitaj, S. Owre, Maena Quemener, N. Shankar
Interactive Theorem Provers (ITPs) are an indispensable tool in the arsenal of formal method experts as a platform for construction and (formal) verification of proofs. The complexity of the proofs in conjunction with the level of expertise typically required for the process to succeed can often hinder the adoption of ITPs. A recent strain of work has investigated methods to incorporate machine learning models trained on ITP user activity traces as a viable path towards full automation. While a valuable line of investigation, many problems still require human supervision to be completed fully, thus applying learning methods to assist the user with useful recommendations can prove more fruitful. Following the vein of user assistance, we introduce CoProver, a proof recommender system based on transformers, capable of learning from past actions during proof construction, all while exploring knowledge stored in the ITP concerning previous proofs. CoProver employs a neurally learnt sequence-based encoding of sequents, capturing long distance relationships between terms and hidden cues therein. We couple CoProver with the Prototype Verification System (PVS) and evaluate its performance on two key areas, namely: (1) Next Proof Action Recommendation, and (2) Relevant Lemma Retrieval given a library of theories. We evaluate CoProver on a series of well-established metrics originating from the recommender system and information retrieval communities, respectively. We show that CoProver successfully outperforms prior state of the art applied to recommendation in the domain. We conclude by discussing future directions viable for CoProver (and similar approaches) such as argument prediction, proof summarization, and more.
{"title":"CoProver: A Recommender System for Proof Construction","authors":"Eric Yeh, B. Hitaj, S. Owre, Maena Quemener, N. Shankar","doi":"10.48550/arXiv.2304.10486","DOIUrl":"https://doi.org/10.48550/arXiv.2304.10486","url":null,"abstract":"Interactive Theorem Provers (ITPs) are an indispensable tool in the arsenal of formal method experts as a platform for construction and (formal) verification of proofs. The complexity of the proofs in conjunction with the level of expertise typically required for the process to succeed can often hinder the adoption of ITPs. A recent strain of work has investigated methods to incorporate machine learning models trained on ITP user activity traces as a viable path towards full automation. While a valuable line of investigation, many problems still require human supervision to be completed fully, thus applying learning methods to assist the user with useful recommendations can prove more fruitful. Following the vein of user assistance, we introduce CoProver, a proof recommender system based on transformers, capable of learning from past actions during proof construction, all while exploring knowledge stored in the ITP concerning previous proofs. CoProver employs a neurally learnt sequence-based encoding of sequents, capturing long distance relationships between terms and hidden cues therein. We couple CoProver with the Prototype Verification System (PVS) and evaluate its performance on two key areas, namely: (1) Next Proof Action Recommendation, and (2) Relevant Lemma Retrieval given a library of theories. We evaluate CoProver on a series of well-established metrics originating from the recommender system and information retrieval communities, respectively. We show that CoProver successfully outperforms prior state of the art applied to recommendation in the domain. We conclude by discussing future directions viable for CoProver (and similar approaches) such as argument prediction, proof summarization, and more.","PeriodicalId":236059,"journal":{"name":"International Conference on Intelligent Computer Mathematics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129485831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-26DOI: 10.48550/arXiv.2212.13201
Neeraj Gangwar, N. Kani
Operations research deals with modeling and solving real-world problems as mathematical optimization problems. While solving mathematical systems is accomplished by analytical software, formulating a problem as a set of mathematical operations has been typically done manually by domain experts. Recent machine learning methods have shown promise in converting textual problem descriptions to corresponding mathematical formulations. This paper presents an approach that converts linear programming word problems into mathematical formulations. We leverage the named entities in the input and augment the input to highlight these entities. Our approach achieves the highest accuracy among all submissions to the NL4Opt Competition, securing first place in the generation track.
{"title":"Highlighting Named Entities in Input for Auto-Formulation of Optimization Problems","authors":"Neeraj Gangwar, N. Kani","doi":"10.48550/arXiv.2212.13201","DOIUrl":"https://doi.org/10.48550/arXiv.2212.13201","url":null,"abstract":"Operations research deals with modeling and solving real-world problems as mathematical optimization problems. While solving mathematical systems is accomplished by analytical software, formulating a problem as a set of mathematical operations has been typically done manually by domain experts. Recent machine learning methods have shown promise in converting textual problem descriptions to corresponding mathematical formulations. This paper presents an approach that converts linear programming word problems into mathematical formulations. We leverage the named entities in the input and augment the input to highlight these entities. Our approach achieves the highest accuracy among all submissions to the NL4Opt Competition, securing first place in the generation track.","PeriodicalId":236059,"journal":{"name":"International Conference on Intelligent Computer Mathematics","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128494195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-13DOI: 10.48550/arXiv.2208.06642
Elif Deniz, Adnan Rashid, Osman Hasan, S. Tahar
Partial Differential Equations (PDEs) are widely used for modeling the physical phenomena and analyzing the dynamical behavior of many engineering and physical systems. The heat equation is one of the most well-known PDEs that captures the temperature distribution and diffusion of heat within a body. Due to the wider utility of these equations in various safety-critical applications, such as thermal protection systems, a formal analysis of the heat transfer is of utmost importance. In this paper, we propose to use higher-order-logic (HOL) theorem proving for formally analyzing the heat conduction problem in rectangular coordinates. In particular, we formally model the heat transfer as a one-dimensional heat equation for a rectangular slab using the multivariable calculus theories of the HOL Light theorem prover. This requires the formalization of the heat operator and formal verification of its various properties, such as linearity and scaling. Moreover, we use the separation of variables method for formally verifying the solution of the PDEs, which allows modeling the heat transfer in the slab under various initial and boundary conditions using HOL Light.
{"title":"On the Formalization of the Heat Conduction Problem in HOL","authors":"Elif Deniz, Adnan Rashid, Osman Hasan, S. Tahar","doi":"10.48550/arXiv.2208.06642","DOIUrl":"https://doi.org/10.48550/arXiv.2208.06642","url":null,"abstract":"Partial Differential Equations (PDEs) are widely used for modeling the physical phenomena and analyzing the dynamical behavior of many engineering and physical systems. The heat equation is one of the most well-known PDEs that captures the temperature distribution and diffusion of heat within a body. Due to the wider utility of these equations in various safety-critical applications, such as thermal protection systems, a formal analysis of the heat transfer is of utmost importance. In this paper, we propose to use higher-order-logic (HOL) theorem proving for formally analyzing the heat conduction problem in rectangular coordinates. In particular, we formally model the heat transfer as a one-dimensional heat equation for a rectangular slab using the multivariable calculus theories of the HOL Light theorem prover. This requires the formalization of the heat operator and formal verification of its various properties, such as linearity and scaling. Moreover, we use the separation of variables method for formally verifying the solution of the PDEs, which allows modeling the heat transfer in the slab under various initial and boundary conditions using HOL Light.","PeriodicalId":236059,"journal":{"name":"International Conference on Intelligent Computer Mathematics","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132374594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-26DOI: 10.48550/arXiv.2207.12742
S. Gouëzel
We report on a formalization of the change of variables formula in integrals, in the mathlib library for Lean. Our version of this theorem is extremely general, and builds on developments in linear algebra, analysis, measure theory and descriptive set theory. The interplay between these domains is transparent thanks to the highly integrated development model of mathlib.
{"title":"A formalization of the change of variables formula for integrals in mathlib","authors":"S. Gouëzel","doi":"10.48550/arXiv.2207.12742","DOIUrl":"https://doi.org/10.48550/arXiv.2207.12742","url":null,"abstract":"We report on a formalization of the change of variables formula in integrals, in the mathlib library for Lean. Our version of this theorem is extremely general, and builds on developments in linear algebra, analysis, measure theory and descriptive set theory. The interplay between these domains is transparent thanks to the highly integrated development model of mathlib.","PeriodicalId":236059,"journal":{"name":"International Conference on Intelligent Computer Mathematics","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116797639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-25DOI: 10.48550/arXiv.2207.12039
Ciarán Dunne, J. Wells
A generalized set theory (GST) is like a standard set theory but also can have non-set structured objects that can contain other structured objects including sets. This paper presents Isabelle/HOL support for GSTs, which are treated as type classes that combine features that specify kinds of mathematical objects, e.g., sets, ordinal numbers, functions, etc. GSTs can have an exception feature that eases representing partial functions and undefinedness. When assembling a GST, extra axioms are generated following a user-modifiable policy to fill specification gaps. Specialized type-like predicates called soft types are used extensively. Although a GST can be used without a model, for confidence in its consistency we build a model for each GST from components that specify each feature's contribution to each tier of a von-Neumann-style cumulative hierarchy defined via ordinal recursion, and we then connect the model to a separate type which the GST occupies.
{"title":"Isabelle/HOL/GST: A Formal Proof Environment for Generalized Set Theories","authors":"Ciarán Dunne, J. Wells","doi":"10.48550/arXiv.2207.12039","DOIUrl":"https://doi.org/10.48550/arXiv.2207.12039","url":null,"abstract":"A generalized set theory (GST) is like a standard set theory but also can have non-set structured objects that can contain other structured objects including sets. This paper presents Isabelle/HOL support for GSTs, which are treated as type classes that combine features that specify kinds of mathematical objects, e.g., sets, ordinal numbers, functions, etc. GSTs can have an exception feature that eases representing partial functions and undefinedness. When assembling a GST, extra axioms are generated following a user-modifiable policy to fill specification gaps. Specialized type-like predicates called soft types are used extensively. Although a GST can be used without a model, for confidence in its consistency we build a model for each GST from components that specify each feature's contribution to each tier of a von-Neumann-style cumulative hierarchy defined via ordinal recursion, and we then connect the model to a separate type which the GST occupies.","PeriodicalId":236059,"journal":{"name":"International Conference on Intelligent Computer Mathematics","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123971020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-26DOI: 10.1007/978-3-030-81097-9_12
S. Arndt, P. Ion, M. Runnwerth, M. Schubotz, O. Teschke
{"title":"10 Years Later: The Mathematics Subject Classification and Linked Open Data","authors":"S. Arndt, P. Ion, M. Runnwerth, M. Schubotz, O. Teschke","doi":"10.1007/978-3-030-81097-9_12","DOIUrl":"https://doi.org/10.1007/978-3-030-81097-9_12","url":null,"abstract":"","PeriodicalId":236059,"journal":{"name":"International Conference on Intelligent Computer Mathematics","volume":"91 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133380898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-28DOI: 10.4230/LIPIcs.ITP.2022.11
C. Edmonds, Lawrence Charles Paulson
Combinatorial design theory studies set systems with certain balance and symmetry properties and has applications to computer science and elsewhere. This paper presents a modular approach to formalising designs for the first time using Isabelle and assesses the usability of a locale-centric approach to formalisations of mathematical structures. We demonstrate how locales can be used to specify numerous types of designs and their hierarchy. The resulting library, which is concise and adaptable, includes formal definitions and proofs for many key properties, operations, and theorems on the construction and existence of designs.
{"title":"A Modular First Formalisation of Combinatorial Design Theory","authors":"C. Edmonds, Lawrence Charles Paulson","doi":"10.4230/LIPIcs.ITP.2022.11","DOIUrl":"https://doi.org/10.4230/LIPIcs.ITP.2022.11","url":null,"abstract":"Combinatorial design theory studies set systems with certain balance and symmetry properties and has applications to computer science and elsewhere. This paper presents a modular approach to formalising designs for the first time using Isabelle and assesses the usability of a locale-centric approach to formalisations of mathematical structures. We demonstrate how locales can be used to specify numerous types of designs and their hierarchy. The resulting library, which is concise and adaptable, includes formal definitions and proofs for many key properties, operations, and theorems on the construction and existence of designs.","PeriodicalId":236059,"journal":{"name":"International Conference on Intelligent Computer Mathematics","volume":"299 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128617944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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