Software is now ubiquitous and involved in complex interactions with the human users and the physical world in so-called cyber-physical systems where the management of time is a major issue. Separation of concerns is a key asset in the development of these ever more complex systems. Two different kinds of separation exist: a first one corresponds to the different steps in a development leading from the abstract requirements to the system implementation and is qualified as vertical. It matches the commonly used notion of refinement. A second one corresponds to the various components in the system architecture at a given level of refinement and is called horizontal. Refinement has been studied thoroughly for the data, functional and concurrency concerns while our work focuses on the time modeling concern. This contribution aims at providing a formal construct for the verification of refinement in time models, through the definition of an order between strict partial orders used to relate the different instants in asynchronous systems. This relation allows the designer at the concrete level to distinguish events that are coincident at the abstract level while preserving the properties assessed at the abstract level. This work has been conducted using the proof assistant Agda and is connected to a previous work on the asynchronous language CCSL, which has also been modelled using the same tool.
{"title":"Ordering Strict Partial Orders to Model Behavioral Refinement","authors":"Mathieu Montin, M. Pantel","doi":"10.4204/EPTCS.282.3","DOIUrl":"https://doi.org/10.4204/EPTCS.282.3","url":null,"abstract":"Software is now ubiquitous and involved in complex interactions with the human users and the physical world in so-called cyber-physical systems where the management of time is a major issue. Separation of concerns is a key asset in the development of these ever more complex systems. Two different kinds of separation exist: a first one corresponds to the different steps in a development leading from the abstract requirements to the system implementation and is qualified as vertical. It matches the commonly used notion of refinement. A second one corresponds to the various components in the system architecture at a given level of refinement and is called horizontal. Refinement has been studied thoroughly for the data, functional and concurrency concerns while our work focuses on the time modeling concern. This contribution aims at providing a formal construct for the verification of refinement in time models, through the definition of an order between strict partial orders used to relate the different instants in asynchronous systems. This relation allows the designer at the concrete level to distinguish events that are coincident at the abstract level while preserving the properties assessed at the abstract level. This work has been conducted using the proof assistant Agda and is connected to a previous work on the asynchronous language CCSL, which has also been modelled using the same tool.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"7 1","pages":"23-38"},"PeriodicalIF":0.0,"publicationDate":"2018-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84388054","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}
{"title":"A Theory of Lazy Imperative Timing","authors":"E. Hehner","doi":"10.4204/EPTCS.282.1","DOIUrl":"https://doi.org/10.4204/EPTCS.282.1","url":null,"abstract":"We present a theory of lazy imperative timing.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"33 1","pages":"1-9"},"PeriodicalIF":0.0,"publicationDate":"2018-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79096221","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}
Urban and Bierman introduced a calculus of proof terms for the sequent calculus LK with a strongly normalizing reduction relation. We extend this calculus to simply-typed higher-order logic with inferences for induction and equality, albeit without strong normalization. We implement thiscalculus in GAPT, our library for proof transformations. Evaluating the normalization on both artificial and real-world benchmarks, we show that this algorithm is typically several orders of magnitude faster than the existing Gentzen-like cut-reduction, and an order of magnitude faster than any other cut-elimination procedure implemented in GAPT.
{"title":"Fast Cut-Elimination using Proof Terms: An Empirical Study","authors":"Gabriel Ebner","doi":"10.4204/EPTCS.281.3","DOIUrl":"https://doi.org/10.4204/EPTCS.281.3","url":null,"abstract":"Urban and Bierman introduced a calculus of proof terms for the sequent calculus LK with a strongly normalizing reduction relation. We extend this calculus to simply-typed higher-order logic with inferences for induction and equality, albeit without strong normalization. We implement thiscalculus in GAPT, our library for proof transformations. Evaluating the normalization on both artificial and real-world benchmarks, we show that this algorithm is typically several orders of magnitude faster than the existing Gentzen-like cut-reduction, and an order of magnitude faster than any other cut-elimination procedure implemented in GAPT.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"16 1","pages":"24-38"},"PeriodicalIF":0.0,"publicationDate":"2018-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81943227","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}
Cyclic pre-proofs can be represented as sets of finite tree derivations with back-links. In the frame of the first-order logic with inductive definitions, the nodes of the tree derivations are labelled by sequents and the back-links connect particular terminal nodes, referred to as buds, to other nodes labelled by a same sequent. However, only some back-links can constitute sound pre-proofs. Previously, it has been shown that special ordering and derivability conditions, defined along the minimal cycles of the digraph representing a particular normal form of the cyclic pre-proof, are sufficient for validating the back-links. In that approach, a same constraint could be checked several times when processing different minimal cycles, hence one may require additional recording mechanisms to avoid redundant computation in order to downgrade the time complexity to polynomial. We present a new approach that does not need to process minimal cycles. It based on a normal form that allows to define the validation conditions by taking into account only the root-bud paths from the non-singleton strongly connected components of its digraph.
{"title":"Validating Back-links of FOLID Cyclic Pre-proofs","authors":"Sorin Stratulat","doi":"10.4204/EPTCS.281.4","DOIUrl":"https://doi.org/10.4204/EPTCS.281.4","url":null,"abstract":"Cyclic pre-proofs can be represented as sets of finite tree derivations with back-links. In the frame of the first-order logic with inductive definitions, the nodes of the tree derivations are labelled by sequents and the back-links connect particular terminal nodes, referred to as buds, to other nodes labelled by a same sequent. However, only some back-links can constitute sound pre-proofs. Previously, it has been shown that special ordering and derivability conditions, defined along the minimal cycles of the digraph representing a particular normal form of the cyclic pre-proof, are sufficient for validating the back-links. In that approach, a same constraint could be checked several times when processing different minimal cycles, hence one may require additional recording mechanisms to avoid redundant computation in order to downgrade the time complexity to polynomial. \u0000We present a new approach that does not need to process minimal cycles. It based on a normal form that allows to define the validation conditions by taking into account only the root-bud paths from the non-singleton strongly connected components of its digraph.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"11 1","pages":"39-53"},"PeriodicalIF":0.0,"publicationDate":"2018-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83856240","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 report on a verification of the Fundamental Theorem of Algebra in ACL2(r). The proof consists of four parts. First, continuity for both complex-valued and real-valued functions of complex numbers is defined, and it is shown that continuous functions from the complex to the real numbers achieve a minimum value over a closed square region. An important case of continuous real-valued, complex functions results from taking the traditional complex norm of a continuous complex function. We think of these continuous functions as having only one (complex) argument, but in ACL2(r) they appear as functions of two arguments. The extra argument is a "context", which is uninterpreted. For example, it could be other arguments that are held fixed, as in an exponential function which has a base and an exponent, either of which could be held fixed. Second, it is shown that complex polynomials are continuous, so the norm of a complex polynomial is a continuous real-valued function and it achieves its minimum over an arbitrary square region centered at the origin. This part of the proof benefits from the introduction of the "context" argument, and it illustrates an innovation that simplifies the proofs of classical properties with unbound parameters. Third, we derive lower and upper bounds on the norm of non-constant polynomials for inputs that are sufficiently far away from the origin. This means that a sufficiently large square can be found to guarantee that it contains the global minimum of the norm of the polynomial. Fourth, it is shown that if a given number is not a root of a non-constant polynomial, then it cannot be the global minimum. Finally, these results are combined to show that the global minimum must be a root of the polynomial. This result is part of a larger effort in the formalization of complex polynomials in ACL2(r).
{"title":"The Fundamental Theorem of Algebra in ACL2","authors":"Ruben Gamboa, J. Cowles","doi":"10.4204/EPTCS.280.8","DOIUrl":"https://doi.org/10.4204/EPTCS.280.8","url":null,"abstract":"We report on a verification of the Fundamental Theorem of Algebra in ACL2(r). The proof consists of four parts. First, continuity for both complex-valued and real-valued functions of complex numbers is defined, and it is shown that continuous functions from the complex to the real numbers achieve a minimum value over a closed square region. An important case of continuous real-valued, complex functions results from taking the traditional complex norm of a continuous complex function. We think of these continuous functions as having only one (complex) argument, but in ACL2(r) they appear as functions of two arguments. The extra argument is a \"context\", which is uninterpreted. For example, it could be other arguments that are held fixed, as in an exponential function which has a base and an exponent, either of which could be held fixed. Second, it is shown that complex polynomials are continuous, so the norm of a complex polynomial is a continuous real-valued function and it achieves its minimum over an arbitrary square region centered at the origin. This part of the proof benefits from the introduction of the \"context\" argument, and it illustrates an innovation that simplifies the proofs of classical properties with unbound parameters. Third, we derive lower and upper bounds on the norm of non-constant polynomials for inputs that are sufficiently far away from the origin. This means that a sufficiently large square can be found to guarantee that it contains the global minimum of the norm of the polynomial. Fourth, it is shown that if a given number is not a root of a non-constant polynomial, then it cannot be the global minimum. Finally, these results are combined to show that the global minimum must be a root of the polynomial. This result is part of a larger effort in the formalization of complex polynomials in ACL2(r).","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"46 1","pages":"98-110"},"PeriodicalIF":0.0,"publicationDate":"2018-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77404183","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 a mechanical proof of the Cauchy-Schwarz inequality in ACL2(r) and a formalisation of the necessary mathematics to undertake such a proof. This includes the formalisation of $mathbb{R}^n$ as an inner product space. We also provide an application of Cauchy-Schwarz by formalising $mathbb R^n$ as a metric space and exhibiting continuity for some simple functions $mathbb R^ntomathbb R$. The Cauchy-Schwarz inequality relates the magnitude of a vector to its projection (or inner product) with another: [|langle u,vrangle| leq |u| |v|] with equality iff the vectors are linearly dependent. It finds frequent use in many branches of mathematics including linear algebra, real analysis, functional analysis, probability, etc. Indeed, the inequality is considered to be among "The Hundred Greatest Theorems" and is listed in the "Formalizing 100 Theorems" project. To the best of our knowledge, our formalisation is the first published proof using ACL2(r) or any other first-order theorem prover.
{"title":"Real Vector Spaces and the Cauchy-Schwarz Inequality in ACL2(r)","authors":"Carl Kwan, M. Greenstreet","doi":"10.4204/EPTCS.280.9","DOIUrl":"https://doi.org/10.4204/EPTCS.280.9","url":null,"abstract":"We present a mechanical proof of the Cauchy-Schwarz inequality in ACL2(r) and a formalisation of the necessary mathematics to undertake such a proof. This includes the formalisation of $mathbb{R}^n$ as an inner product space. We also provide an application of Cauchy-Schwarz by formalising $mathbb R^n$ as a metric space and exhibiting continuity for some simple functions $mathbb R^ntomathbb R$. The Cauchy-Schwarz inequality relates the magnitude of a vector to its projection (or inner product) with another: [|langle u,vrangle| leq |u| |v|] with equality iff the vectors are linearly dependent. It finds frequent use in many branches of mathematics including linear algebra, real analysis, functional analysis, probability, etc. Indeed, the inequality is considered to be among \"The Hundred Greatest Theorems\" and is listed in the \"Formalizing 100 Theorems\" project. To the best of our knowledge, our formalisation is the first published proof using ACL2(r) or any other first-order theorem prover.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"6 1","pages":"111-127"},"PeriodicalIF":0.0,"publicationDate":"2018-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85523838","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 a tool that automates termination proofs for recursive definitions by mining existing termination theorems.
我们提出了一个工具，通过挖掘现有的终止定理来自动化递归定义的终止证明。
{"title":"DefunT: A Tool for Automating Termination Proofs by Using the Community Books (Extended Abstract)","authors":"Matt Kaufmann","doi":"10.4204/EPTCS.280.12","DOIUrl":"https://doi.org/10.4204/EPTCS.280.12","url":null,"abstract":"We present a tool that automates termination proofs for recursive definitions by mining existing termination theorems.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"2 1","pages":"161-163"},"PeriodicalIF":0.0,"publicationDate":"2018-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89962532","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 builds upon our prior formalisation of R^n in ACL2(r) by presenting a set of theorems for reasoning about convex functions. This is a demonstration of the higher-dimensional analytical reasoning possible in our metric space formalisation of R^n. Among the introduced theorems is a set of equivalent conditions for convex functions with Lipschitz continuous gradients from Yurii Nesterov's classic text on convex optimisation. To the best of our knowledge a full proof of the theorem has yet to be published in a single piece of literature. We also explore "proof engineering" issues, such as how to state Nesterov's theorem in a manner that is both clear and useful.
{"title":"Convex Functions in ACL2(r)","authors":"Carl Kwan, M. Greenstreet","doi":"10.4204/EPTCS.280.10","DOIUrl":"https://doi.org/10.4204/EPTCS.280.10","url":null,"abstract":"This paper builds upon our prior formalisation of R^n in ACL2(r) by presenting a set of theorems for reasoning about convex functions. This is a demonstration of the higher-dimensional analytical reasoning possible in our metric space formalisation of R^n. Among the introduced theorems is a set of equivalent conditions for convex functions with Lipschitz continuous gradients from Yurii Nesterov's classic text on convex optimisation. To the best of our knowledge a full proof of the theorem has yet to be published in a single piece of literature. We also explore \"proof engineering\" issues, such as how to state Nesterov's theorem in a manner that is both clear and useful.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"2 1","pages":"128-142"},"PeriodicalIF":0.0,"publicationDate":"2018-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81908586","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 this work, we present an approach towards constructing executable specifications of existing filesystems and verifying their functional properties in a theorem proving environment. We detail an application of this approach to the FAT32 filesystem. We also detail the methodology used to build up this type of executable specification through a series of models which incrementally add features of the target filesystem. This methodology has the benefit of allowing the verification effort to start from simple models which encapsulate features common to many filesystems and which are thus suitable for reuse.
{"title":"Formalising Filesystems in the ACL2 Theorem Prover: an Application to FAT32","authors":"M. Mehta","doi":"10.4204/EPTCS.280.2","DOIUrl":"https://doi.org/10.4204/EPTCS.280.2","url":null,"abstract":"In this work, we present an approach towards constructing executable specifications of existing filesystems and verifying their functional properties in a theorem proving environment. We detail an application of this approach to the FAT32 filesystem. \u0000We also detail the methodology used to build up this type of executable specification through a series of models which incrementally add features of the target filesystem. This methodology has the benefit of allowing the verification effort to start from simple models which encapsulate features common to many filesystems and which are thus suitable for reuse.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"21 1","pages":"18-29"},"PeriodicalIF":0.0,"publicationDate":"2018-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80040710","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}
Smtlink is an extension of ACL2 with Satisfiability Modulo Theories (SMT) solvers. We presented an earlier version at ACL2'2015. Smtlink 2.0 makes major improvements over the initial version with respect to soundness, extensibility, ease-of-use, and the range of types and associated theory-solvers supported. Most theorems that one would want to prove using an SMT solver must first be translated to use only the primitive operations supported by the SMT solver -- this translation includes function expansion and type inference. Smtlink 2.0 performs this translation using a sequence of steps performed by verified clause processors and computed hints. These steps are ensured to be sound. The final transliteration from ACL2 to Z3's Python interface requires a trusted clause processor. This is a great improvement in soundness and extensibility over the original Smtlink which was implemented as a single, monolithic, trusted clause processor. Smtlink 2.0 provides support for FTY defprod, deflist, defalist, and defoption types by using Z3's arrays and user-defined data types. We have identified common usage patterns and simplified the configuration and hint information needed to use Smtlink.
{"title":"Smtlink 2.0","authors":"Yan Peng, M. Greenstreet","doi":"10.4204/EPTCS.280.11","DOIUrl":"https://doi.org/10.4204/EPTCS.280.11","url":null,"abstract":"Smtlink is an extension of ACL2 with Satisfiability Modulo Theories (SMT) solvers. We presented an earlier version at ACL2'2015. Smtlink 2.0 makes major improvements over the initial version with respect to soundness, extensibility, ease-of-use, and the range of types and associated theory-solvers supported. Most theorems that one would want to prove using an SMT solver must first be translated to use only the primitive operations supported by the SMT solver -- this translation includes function expansion and type inference. Smtlink 2.0 performs this translation using a sequence of steps performed by verified clause processors and computed hints. These steps are ensured to be sound. The final transliteration from ACL2 to Z3's Python interface requires a trusted clause processor. This is a great improvement in soundness and extensibility over the original Smtlink which was implemented as a single, monolithic, trusted clause processor. Smtlink 2.0 provides support for FTY defprod, deflist, defalist, and defoption types by using Z3's arrays and user-defined data types. We have identified common usage patterns and simplified the configuration and hint information needed to use Smtlink.","PeriodicalId":10720,"journal":{"name":"CoRR","volume":"1 1","pages":"143-160"},"PeriodicalIF":0.0,"publicationDate":"2018-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86649697","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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