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Towards extracting explicit proofs from totality checking in twelf 论从总合检验中提取明确的证明
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2013-07-05 DOI: 10.1145/2503887.2503893
Yuting Wang, G. Nadathur
{"title":"Towards extracting explicit proofs from totality checking in twelf","authors":"Yuting Wang, G. Nadathur","doi":"10.1145/2503887.2503893","DOIUrl":"https://doi.org/10.1145/2503887.2503893","url":null,"abstract":"The Edinburgh Logical Framework (LF) is a dependently type lambda calculus that can be used to encode formal systems. The versatility of LF allows specifications to be constructed also about the encoded systems. The Twelf system exploits the correspondence between formulas and types to give specifications in LF a logic programming interpretation. By interpreting particular arguments as input and others as output, specifications can be seen as describing non-deterministic functions. We can then prove meta-theorems about the encoded systems by showing particular such functions to be total. Twelf provides tools for establishing totality. However, the resulting proofs of meta-theorems are implicit in that they do not yield a certificate that can be given to a proof checker. We begin the process here of making these proofs explicit. We treat the restricted situation in Twelf where context definitions (regular worlds), mutually recursive definitions and lemmas are not used. In this setting we describe and prove correct a translation of the steps in totality checking into an actual proof in the companion logic M2.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134604600","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}
引用次数: 8
Multi-level Contextual Type Theory 多层次语境类型理论
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2011-10-31 DOI: 10.4204/EPTCS.71.3
Mathieu Boespflug, B. Pientka
{"title":"Multi-level Contextual Type Theory","authors":"Mathieu Boespflug, B. Pientka","doi":"10.4204/EPTCS.71.3","DOIUrl":"https://doi.org/10.4204/EPTCS.71.3","url":null,"abstract":"Contextual type theory distinguishes between bound variables and meta-variables to write potentially incomplete terms in the presence of binders. It has found good use as a framework for concise explanations of higher-order unification, characterize holes in proofs, and in developing a foundation for programming with higher-order abstract syntax, as embodied by the programming and reasoning environment Beluga. However, to reason about these applications, we need to introduce meta^2-variables to characterize the dependency on meta-variables and bound variables. In other words, we must go beyond a two-level system granting only bound variables and meta-variables. \u0000In this paper we generalize contextual type theory to n levels for arbitrary n, so as to obtain a formal system offering bound variables, meta-variables and so on all the way to meta^n-variables. We obtain a uniform account by collapsing all these different kinds of variables into a single notion of variabe indexed by some level k. We give a decidable bi-directional type system which characterizes beta-eta-normal forms together with a generalized substitution operation.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114286780","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}
引用次数: 4
An Improved Implementation and Abstract Interface for Hybrid 一种改进的混合实现和抽象接口
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2011-10-31 DOI: 10.4204/EPTCS.71.6
Alan J. Martin, A. Felty
{"title":"An Improved Implementation and Abstract Interface for Hybrid","authors":"Alan J. Martin, A. Felty","doi":"10.4204/EPTCS.71.6","DOIUrl":"https://doi.org/10.4204/EPTCS.71.6","url":null,"abstract":"Hybrid is a formal theory implemented in Isabelle/HOL that provides an interface for representing and reasoning about object languages using higher-order abstract syntax (HOAS). This interface is built around an HOAS variable-binding operator that is constructed definitionally from a de Bruijn index representation. In this paper we make a variety of improvements to Hybrid, culminating in an abstract interface that on one hand makes Hybrid a more mathematically satisfactory theory, and on the other hand has important practical benefits. We start with a modification of Hybrid’s type of terms that better hides its implementation in terms of de Bruijn indices, by excluding at the type level terms with dangling indices. We present an improved set of definitions, and a series of new lemmas that provide a complete characterization of Hybrid’s primitives in terms of properties stated at the HOAS level. Benefits of this new package include a new proof of adequacy and improvements to reasoning about object logics. Such proofs are carried out at the higher level with no involvement of the lower level de Bruijn syntax.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125268528","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}
引用次数: 1
A Lambda Term Representation Inspired by Linear Ordered Logic 基于线性有序逻辑的Lambda项表示
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2011-10-31 DOI: 10.4204/EPTCS.71.1
Andreas Abel, Nicolai Kraus
{"title":"A Lambda Term Representation Inspired by Linear Ordered Logic","authors":"Andreas Abel, Nicolai Kraus","doi":"10.4204/EPTCS.71.1","DOIUrl":"https://doi.org/10.4204/EPTCS.71.1","url":null,"abstract":"We introduce a new nameless representation of lambda terms inspired by ordered logic. At a lambda abstraction, number and relative position of all occurrences of the bound variable are stored, and application carries the additional information where to cut the variable context into function and argument part. This way, complete information about free variable occurrence is available at each subterm without requiring a traversal, and environments can be kept exact such that they only assign values to variables that actually occur in the associated term. Our approach avoids space leaks in interpreters that build function closures. In this article, we prove correctness of the new representation and present an experimental evaluation of its performance in a proof checker for the Edinburgh Logical Framework.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117136793","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}
引用次数: 4
Nominal Logic with Equations Only 只有方程式的名义逻辑
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2011-10-31 DOI: 10.4204/EPTCS.71.4
Ranald Clouston
{"title":"Nominal Logic with Equations Only","authors":"Ranald Clouston","doi":"10.4204/EPTCS.71.4","DOIUrl":"https://doi.org/10.4204/EPTCS.71.4","url":null,"abstract":"Many formal systems, particularly in computer science, may be captured by equations modulated by side conditions asserting the “freshness of names”; these can be reasoned about with Nominal Equational Logic (NEL). Like most logics of this sort NEL employs this notion of freshness as a first class logical connective. However, this can become inc onvenient when attempting to translate results from standard equational logic to the nominal setti ng. This paper presents proof rules for a logic whose only connectives are equations, which we call Nominal Equation-only Logic (NEoL). We prove that NEoL is just as expressive as NEL. We then give a simple description of equality in the empty NEoL-theory, then extend that result to describe freshness in the empty NEL-theory.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127756890","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}
引用次数: 3
Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets 标称亨金语义:标称集合中的简单类型lambda-微积分模型
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2011-10-31 DOI: 10.4204/EPTCS.71.5
M. Gabbay, D. Mulligan
{"title":"Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets","authors":"M. Gabbay, D. Mulligan","doi":"10.4204/EPTCS.71.5","DOIUrl":"https://doi.org/10.4204/EPTCS.71.5","url":null,"abstract":"We investigate a class of nominal algebraic Henkin-style models for the simply typed lambda-calculus in which variables map to names in the denotation and lambda-abstraction maps to a (non-functional) name-abstraction operation. The resulting denotations are smaller and better-behaved, in ways we make precise, than functional valuation-based models. \u0000Using these new models, we then develop a generalisation of lambda-term syntax enriching them with existential meta-variables, thus yielding a theory of incomplete functions. This incompleteness is orthogonal to the usual notion of incompleteness given by function abstraction and application, and corresponds to holes and incomplete objects.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122713634","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}
引用次数: 13
A Bigraph Relational Model 图关系模型
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2011-10-01 DOI: 10.4204/EPTCS.71.2
Maxime Beauquier, C. Schürmann
{"title":"A Bigraph Relational Model","authors":"Maxime Beauquier, C. Schürmann","doi":"10.4204/EPTCS.71.2","DOIUrl":"https://doi.org/10.4204/EPTCS.71.2","url":null,"abstract":"In this paper, we present a model based on relations for bigraphical reactive systems [Milner09]. Its defining characteristics are that validity and reaction relations are captured as traces in a multi-set rewriting system. The relational model is derived from Milner's graphical definition and directly amenable to implementation.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"119 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134271035","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}
引用次数: 1
Closed nominal rewriting and efficiently computable nominal algebra equality 闭标称改写与高效可计算的标称代数等式
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2010-09-14 DOI: 10.4204/EPTCS.34.5
M. Fernández, M. Gabbay
{"title":"Closed nominal rewriting and efficiently computable nominal algebra equality","authors":"M. Fernández, M. Gabbay","doi":"10.4204/EPTCS.34.5","DOIUrl":"https://doi.org/10.4204/EPTCS.34.5","url":null,"abstract":"We analyse the relationship between nominal algebra and nominal rewriting, giving a new and concise presentation of equational deduction in nominal theories. With some new results, we characterise a subclass of equational theories for which nominal rewriting provides a complete procedure to check nominal algebra equality. This subclass includes specifications of the lambda-calculus and first-order logic.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129529196","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}
引用次数: 23
A Monadic Formalization of ML5 ML5的一元形式化
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2010-09-14 DOI: 10.4204/EPTCS.34.7
Daniel R. Licata, R. Harper
{"title":"A Monadic Formalization of ML5","authors":"Daniel R. Licata, R. Harper","doi":"10.4204/EPTCS.34.7","DOIUrl":"https://doi.org/10.4204/EPTCS.34.7","url":null,"abstract":"ML5 is a programming language for spatially distributed computing, based on a Curry-Howard correspondence with the modal logic S5. Despite being designed by a correspondence with S5 modal logic, the ML5 programming language differs from the logic in several ways. In this paper, we explain these discrepancies between ML5 and S5 by translating ML5 into a slightly different logic: intuitionistic S5 extended with a lax modality that encapsulates effectful computations in a monad. This translation both explains the existing ML5 design and suggests some simplifications and generalizations. We have formalized our translation within the Agda proof assistant. Rather than formalizing lax S5 as a proof theory, we emph{embed} it as a universe within the the dependently typed host language, with the universe elimination given by implementing the modal logic's Kripke semantics. This representation technique saves us the work of defining a proof theory for the logic and proving it correct, and additionally allows us to inherit the equational theory of the meta-language, which can be exploited in proving that the semantics validates the operational semantics of ML5.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129199061","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}
引用次数: 7
Generating Bijections between HOAS and the Natural Numbers 在HOAS和自然数之间产生双射
International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice Pub Date : 2010-09-14 DOI: 10.4204/EPTCS.34.4
J. Boyland
{"title":"Generating Bijections between HOAS and the Natural Numbers","authors":"J. Boyland","doi":"10.4204/EPTCS.34.4","DOIUrl":"https://doi.org/10.4204/EPTCS.34.4","url":null,"abstract":"A provably correct bijection between higher-order abstract syntax (HOAS) and the natural numbers enables one to define a “not equals” relationship between ter ms and also to have an adequate encoding of sets of terms, and maps from one term family to another. Sets and maps are useful in many situations and are preferably provided in a library of some sort. I have released a map and set library for use with Twelf which can be used with any type for which a bijection to the natural numbers exists. Since creating such bijections is tedious and error-prone, I have created a “bijection generator” that generates such bijections automatically together wit h proofs of correctness, all in the context of Twelf.","PeriodicalId":262518,"journal":{"name":"International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133750833","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}
引用次数: 2
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