The sequent calculus is a formalism for proving validity of statements formulated in First-Order Logic. It is routinely used in computer science modules on mathematical logic. Formal proofs in the sequent calculus are finite trees obtained by successively applying proof rules to formulas, thus simplifying them step-by-step. Students often struggle with the mathematical formalities and the level of abstraction that topics like formal logic and formal proofs involve. The difficulties can be categorised as syntactic or semantic. On the syntactic level, students need to understand what a correctly formed proof is, how rules can be applied (on paper for instance) without leaving the mathematical framework of the sequent calculus, and so on. Beyond this, on the semantic level, students need to acquire strategies that let them find the right proof. The Sequent Calculus Trainer is a tool that is designed to aid students in learning the techniques of proving given statements formally. In this paper we describe the didactical motivation behind the tool and the techniques used to address issues on the syntactic as well as on the semantic level.
{"title":"The Sequent Calculus Trainer with Automated Reasoning - Helping Students to Find Proofs","authors":"Arno Ehle, Norbert Hundeshagen, M. Lange","doi":"10.4204/EPTCS.267.2","DOIUrl":"https://doi.org/10.4204/EPTCS.267.2","url":null,"abstract":"The sequent calculus is a formalism for proving validity of statements formulated in First-Order Logic. It is routinely used in computer science modules on mathematical logic. Formal proofs in the sequent calculus are finite trees obtained by successively applying proof rules to formulas, thus simplifying them step-by-step. \u0000Students often struggle with the mathematical formalities and the level of abstraction that topics like formal logic and formal proofs involve. The difficulties can be categorised as syntactic or semantic. On the syntactic level, students need to understand what a correctly formed proof is, how rules can be applied (on paper for instance) without leaving the mathematical framework of the sequent calculus, and so on. Beyond this, on the semantic level, students need to acquire strategies that let them find the right proof. \u0000The Sequent Calculus Trainer is a tool that is designed to aid students in learning the techniques of proving given statements formally. In this paper we describe the didactical motivation behind the tool and the techniques used to address issues on the syntactic as well as on the semantic level.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"33 1","pages":"19-37"},"PeriodicalIF":0.0,"publicationDate":"2018-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75911588","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}
The 6th International Workshop on Theorem proving components for Educational software (ThEdu'17) was held in Gothenburg, Sweden, on 6 Aug 2017. It was associated to the conference CADE26. Topics of interest include: methods of automated deduction applied to checking students' input; methods of automated deduction applied to prove post-conditions for particular problem solutions; combinations of deduction and computation enabling systems to propose next steps; automated provers specific for dynamic geometry systems; proof and proving in mathematics education. ThEdu'17 was a vibrant workshop, with one invited talk and eight contributions. It triggered the post-proceedings at hand.
{"title":"Proceedings 6th International Workshop on Theorem proving components for Educational software, ThEdu@CADE 2017, Gothenburg, Sweden, 6 Aug 2017","authors":"P. Quaresma, Walther Neuper","doi":"10.4204/EPTCS.267","DOIUrl":"https://doi.org/10.4204/EPTCS.267","url":null,"abstract":"The 6th International Workshop on Theorem proving components for Educational software (ThEdu'17) was held in Gothenburg, Sweden, on 6 Aug 2017. It was associated to the conference CADE26. Topics of interest include: methods of automated deduction applied to checking students' input; methods of automated deduction applied to prove post-conditions for particular problem solutions; combinations of deduction and computation enabling systems to propose next steps; automated provers specific for dynamic geometry systems; proof and proving in mathematics education. \u0000ThEdu'17 was a vibrant workshop, with one invited talk and eight contributions. It triggered the post-proceedings at hand.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89818521","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}
This paper presents an overview of Carnap, a free and open framework for the development of formal reasoning applications. Carnap's design emphasizes flexibility, extensibility, and rapid prototyping. Carnap-based applications are written in Haskell, but can be compiled to JavaScript to run in standard web browsers. This combination of features makes Carnap ideally suited for educational applications, where ease-of-use is crucial for students and adaptability to different teaching strategies and classroom needs is crucial for instructors. The paper describes Carnap's implementation, along with its current and projected pedagogical applications.
{"title":"Carnap: An Open Framework for Formal Reasoning in the Browser","authors":"Graham Leach-Krouse","doi":"10.4204/EPTCS.267.5","DOIUrl":"https://doi.org/10.4204/EPTCS.267.5","url":null,"abstract":"This paper presents an overview of Carnap, a free and open framework for the development of formal reasoning applications. Carnap's design emphasizes flexibility, extensibility, and rapid prototyping. Carnap-based applications are written in Haskell, but can be compiled to JavaScript to run in standard web browsers. This combination of features makes Carnap ideally suited for educational applications, where ease-of-use is crucial for students and adaptability to different teaching strategies and classroom needs is crucial for instructors. The paper describes Carnap's implementation, along with its current and projected pedagogical applications.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"2006 1","pages":"70-88"},"PeriodicalIF":0.0,"publicationDate":"2018-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82559196","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}
The Web Geometry Laboratory (WGL) is a collaborative and adaptive e-learning Web platform integrating a well known dynamic geometry system. Thousands of Geometric problems for Geometric Theorem Provers (TGTP) is a Web-based repository of geometric problems to support the testing and evaluation of geometric automated theorem proving systems. The users of these systems should be able to profit from each other. The TGTP corpus must be made available to the WGL user, allowing, in this way, the exploration of TGTP problems and their proofs. On the other direction TGTP could gain by the possibility of a wider users base submitting new problems. Such information exchange between clients (e.g. WGL) and servers (e.g. TGTP) raises many issues: geometric search - someone, working in a geometric problem, must be able to ask for more information regarding that construction; levels of geometric knowledge and interest - the problems in the servers must be classified in such a way that, in response to a client query, only the problems in the user's level and/or interest are returned; different aims of each tool - e.g. WGL is about secondary school geometry, TGTP is about formal proofs in semi-analytic and algebraic proof methods, not a perfect match indeed; localisation issues, e.g. a Portuguese user obliged to make the query and process the answer in English; technical issues-many technical issues need to be addressed to make this exchange of geometric information possible and useful. Instead of a giant (difficult to maintain) tool, trying to cover all, the interconnection of specialised tools seems much more promising. The challenges to make that connection work are many and difficult, but, it is the authors impression, not insurmountable.
{"title":"Exchange of Geometric Information Between Applications","authors":"P. Quaresma, Vanda Santos, Nuno Baeta","doi":"10.4204/EPTCS.267.7","DOIUrl":"https://doi.org/10.4204/EPTCS.267.7","url":null,"abstract":"The Web Geometry Laboratory (WGL) is a collaborative and adaptive e-learning Web platform integrating a well known dynamic geometry system. Thousands of Geometric problems for Geometric Theorem Provers (TGTP) is a Web-based repository of geometric problems to support the testing and evaluation of geometric automated theorem proving systems. \u0000The users of these systems should be able to profit from each other. The TGTP corpus must be made available to the WGL user, allowing, in this way, the exploration of TGTP problems and their proofs. On the other direction TGTP could gain by the possibility of a wider users base submitting new problems. \u0000Such information exchange between clients (e.g. WGL) and servers (e.g. TGTP) raises many issues: geometric search - someone, working in a geometric problem, must be able to ask for more information regarding that construction; levels of geometric knowledge and interest - the problems in the servers must be classified in such a way that, in response to a client query, only the problems in the user's level and/or interest are returned; different aims of each tool - e.g. WGL is about secondary school geometry, TGTP is about formal proofs in semi-analytic and algebraic proof methods, not a perfect match indeed; localisation issues, e.g. a Portuguese user obliged to make the query and process the answer in English; technical issues-many technical issues need to be addressed to make this exchange of geometric information possible and useful. \u0000Instead of a giant (difficult to maintain) tool, trying to cover all, the interconnection of specialised tools seems much more promising. The challenges to make that connection work are many and difficult, but, it is the authors impression, not insurmountable.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"45 1","pages":"108-119"},"PeriodicalIF":0.0,"publicationDate":"2018-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75795972","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}
"Systems that Explain Themselves" appears a provocative wording, in particular in the context of mathematics education -- it is as provocative as the idea of building educational software upon technology from computer theorem proving. In spite of recent success stories like the proofs of the Four Colour Theorem or the Kepler Conjecture, mechanised proof is still considered somewhat esoteric by mainstream mathematics. This paper describes the process of prototyping in the ISAC project from a technical perspective. This perspective depends on two moving targets: On the one side the rapidly increasing power and coverage of computer theorem provers and their user interfaces, and on the other side potential users: What can students and teachers request from educational systems based on technology and concepts from computer theorem proving, now and then? By the way of describing the process of prototyping the first comprehensive survey on the state of the ISAC prototype is given as a side effect, made precise by pointers to the code and by citation of all contributing theses.
{"title":"Prototyping \"Systems that Explain Themselves\" for Education","authors":"Alan Krempler, Walther Neuper","doi":"10.4204/EPTCS.267.6","DOIUrl":"https://doi.org/10.4204/EPTCS.267.6","url":null,"abstract":"\"Systems that Explain Themselves\" appears a provocative wording, in particular in the context of mathematics education -- it is as provocative as the idea of building educational software upon technology from computer theorem proving. In spite of recent success stories like the proofs of the Four Colour Theorem or the Kepler Conjecture, mechanised proof is still considered somewhat esoteric by mainstream mathematics. This paper describes the process of prototyping in the ISAC project from a technical perspective. This perspective depends on two moving targets: On the one side the rapidly increasing power and coverage of computer theorem provers and their user interfaces, and on the other side potential users: What can students and teachers request from educational systems based on technology and concepts from computer theorem proving, now and then? By the way of describing the process of prototyping the first comprehensive survey on the state of the ISAC prototype is given as a side effect, made precise by pointers to the code and by citation of all contributing theses.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"49 1","pages":"89-107"},"PeriodicalIF":0.0,"publicationDate":"2018-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84740508","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}
We have developed an alternative approach to teaching computer science students how to prove. First, students are taught how to prove theorems with the Coq proof assistant. In a second, more difficult, step students will transfer their acquired skills to the area of textbook proofs. In this article we present a realisation of the second step. Proofs in Coq have a high degree of formality while textbook proofs have only a medium one. Therefore our key idea is to reduce the degree of formality from the level of Coq to textbook proofs in several small steps. For that purpose we introduce three proof styles between Coq and textbook proofs, called line by line comments, weakened line by line comments, and structure faithful proofs. While this article is mostly conceptional we also report on experiences with putting our approach into practise.
{"title":"Learning how to Prove: From the Coq Proof Assistant to Textbook Style","authors":"Sebastian Böhne, C. Kreitz","doi":"10.4204/EPTCS.267.1","DOIUrl":"https://doi.org/10.4204/EPTCS.267.1","url":null,"abstract":"We have developed an alternative approach to teaching computer science students how to prove. First, students are taught how to prove theorems with the Coq proof assistant. In a second, more difficult, step students will transfer their acquired skills to the area of textbook proofs. In this article we present a realisation of the second step. \u0000Proofs in Coq have a high degree of formality while textbook proofs have only a medium one. Therefore our key idea is to reduce the degree of formality from the level of Coq to textbook proofs in several small steps. For that purpose we introduce three proof styles between Coq and textbook proofs, called line by line comments, weakened line by line comments, and structure faithful proofs. \u0000While this article is mostly conceptional we also report on experiences with putting our approach into practise.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"24 1","pages":"1-18"},"PeriodicalIF":0.0,"publicationDate":"2018-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87853373","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}
Previously we have shown that the topos approach to quantum theory of Doering and Isham can be generalised to a class of categories typically studied within the monoidal approach to quantum theory of Abramsky and Coecke. In the monoidal approach to quantum theory H*-algebras provide an axiomatisation of states and observables. Here we show that H*-algebras naturally correspond with the notions of states and observables in the generalised topos approach to quantum theory. We then combine these results with the dagger-kernel approach to quantumlogic of Heunen and Jacobs, which we use to prove a structure theorem for H*-algebras. This structure theorem is a generalisation of the structure theorem of Ambrose for H*-algebras the category of Hilbert spaces.
{"title":"On the Structure of Abstract H*-Algebras","authors":"Kevin Dunne","doi":"10.4204/EPTCS.266.13","DOIUrl":"https://doi.org/10.4204/EPTCS.266.13","url":null,"abstract":"Previously we have shown that the topos approach to quantum theory of Doering and Isham can be generalised to a class of categories typically studied within the monoidal approach to quantum theory of Abramsky and Coecke. In the monoidal approach to quantum theory H*-algebras provide an axiomatisation of states and observables. Here we show that H*-algebras naturally correspond with the notions of states and observables in the generalised topos approach to quantum theory. We then combine these results with the dagger-kernel approach to quantumlogic of Heunen and Jacobs, which we use to prove a structure theorem for H*-algebras. This structure theorem is a generalisation of the structure theorem of Ambrose for H*-algebras the category of Hilbert spaces.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"30 1","pages":"197-208"},"PeriodicalIF":0.0,"publicationDate":"2018-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86811893","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}
In the topos approach to quantum theory of Doering and Isham the Kochen-Specker Theorem, which asserts the contextual nature of quantum theory, can be reformulated in terms of the global sections of a presheaf characterised by the Gelfand spectrum of a commutativeC-Algebra. In previous work we showed how this topos perspective can be generalised to a class of categories typically studied within the monoidal approach to quantum theory of Abramsky and Coecke, and in particular how one can generalise the Gelfand spectrum. Here we study the Gelfand spectrum presheaf for categories of quantale-valued relations, and by considering its global sections we give a non-contextuality result for these categories. We also show that the Gelfand spectrum comes equipped with a topology which has a natural interpretation when thinking of these structures as representing physical theories.
{"title":"Spectral Presheaves, Kochen-Specker Contextuality, and Quantale-Valued Relations","authors":"Kevin Dunne","doi":"10.4204/EPTCS.266.24","DOIUrl":"https://doi.org/10.4204/EPTCS.266.24","url":null,"abstract":"In the topos approach to quantum theory of Doering and Isham the Kochen-Specker Theorem, which asserts the contextual nature of quantum theory, can be reformulated in terms of the global sections of a presheaf characterised by the Gelfand spectrum of a commutativeC-Algebra. In previous work we showed how this topos perspective can be generalised to a class of categories typically studied within the monoidal approach to quantum theory of Abramsky and Coecke, and in particular how one can generalise the Gelfand spectrum. Here we study the Gelfand spectrum presheaf for categories of quantale-valued relations, and by considering its global sections we give a non-contextuality result for these categories. We also show that the Gelfand spectrum comes equipped with a topology which has a natural interpretation when thinking of these structures as representing physical theories.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"67 1","pages":"386-398"},"PeriodicalIF":0.0,"publicationDate":"2018-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83855744","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}
A proof tableau of Hoare logic is an annotated program with pre- and post-conditions, which corresponds to an inference tree of Hoare logic. In this paper, we show that a proof tableau for partial correctness can be transformed into an inference sequence of rewriting induction for constrained rewriting. We also show that the resulting sequence is a valid proof for an inductive theorem corresponding to the Hoare triple if the constrained rewriting system obtained from the program is terminating. Such a valid proof with termination of the constrained rewriting system implies total correctness of the program w.r.t. the Hoare triple. The transformation enables us to apply techniques for proving termination of constrained rewriting to proving total correctness of programs together with proof tableaux for partial correctness.
{"title":"Transforming Proof Tableaux of Hoare Logic into Inference Sequences of Rewriting Induction","authors":"Shinnosuke Mizutani, Naoki Nishida","doi":"10.4204/EPTCS.265.4","DOIUrl":"https://doi.org/10.4204/EPTCS.265.4","url":null,"abstract":"A proof tableau of Hoare logic is an annotated program with pre- and post-conditions, which corresponds to an inference tree of Hoare logic. In this paper, we show that a proof tableau for partial correctness can be transformed into an inference sequence of rewriting induction for constrained rewriting. We also show that the resulting sequence is a valid proof for an inductive theorem corresponding to the Hoare triple if the constrained rewriting system obtained from the program is terminating. Such a valid proof with termination of the constrained rewriting system implies total correctness of the program w.r.t. the Hoare triple. The transformation enables us to apply techniques for proving termination of constrained rewriting to proving total correctness of programs together with proof tableaux for partial correctness.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"12 1","pages":"35-51"},"PeriodicalIF":0.0,"publicationDate":"2018-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88194875","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}
We present attributed hierarchical port graphs (AHP) as an extension of port graphs that aims at facilitating the design of modular port graph models for complex systems. AHP consist of a number of interconnected layers, where each layer defines a port graph whose nodes may link to layers further down the hierarchy; attributes are used to store user-defined data as well as visualisation and run-time system parameters. We also generalise the notion of strategic port graph rewriting (a particular kind of graph transformation system, where port graph rewriting rules are controlled by user-defined strategies) to deal with AHP following the Single Push-out approach. We outline examples of application in two areas: functional programming and financial modelling.
{"title":"Attributed Hierarchical Port Graphs and Applications","authors":"N. Ene, M. Fernández, Bruno Pinaud","doi":"10.4204/EPTCS.265.2","DOIUrl":"https://doi.org/10.4204/EPTCS.265.2","url":null,"abstract":"We present attributed hierarchical port graphs (AHP) as an extension of port graphs that aims at facilitating the design of modular port graph models for complex systems. AHP consist of a number of interconnected layers, where each layer defines a port graph whose nodes may link to layers further down the hierarchy; attributes are used to store user-defined data as well as visualisation and run-time system parameters. We also generalise the notion of strategic port graph rewriting (a particular kind of graph transformation system, where port graph rewriting rules are controlled by user-defined strategies) to deal with AHP following the Single Push-out approach. We outline examples of application in two areas: functional programming and financial modelling.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"19 1","pages":"2-19"},"PeriodicalIF":0.0,"publicationDate":"2018-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74864578","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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