Polynomial Time and Dependent Types

We combine dependent types with linear type systems that soundly and completely capture polynomial time computation. We explore two systems for capturing polynomial time: one system that disallows construction of iterable data, and one, based on the LFPL system of Martin Hofmann, that controls construction via a payment method. Both of these are extended to full dependent types via Quantitative Type Theory, allowing for arbitrary computation in types alongside guaranteed polynomial time computation in terms. We prove the soundness of the systems using a realisability technique due to Dal Lago and Hofmann. Our long-term goal is to combine the extensional reasoning of type theory with intensional reasoning about the resources intrinsically consumed by programs. This paper is a step along this path, which we hope will lead both to practical systems for reasoning about programs' resource usage, and to theoretical use as a form of synthetic computational complexity theory

Reasoning about the garden of forking paths

Lazy evaluation is a powerful tool for functional programmers. It enables the concise expression of on-demand computation and a form of compositionality not available under other evaluation strategies. However, the stateful nature of lazy evaluation makes it hard to analyze a program's computational cost, either informally or formally. In this work, we present a novel and simple framework for formally reasoning about lazy computation costs based on a recent model of lazy evaluation: clairvoyant call-by-value. The key feature of our framework is its simplicity, as expressed by our definition of the clairvoyance monad. This monad is both simple to define (around 20 lines of Coq) and simple to reason about. We show that this monad can be effectively used to mechanically reason about the computational cost of lazy functional programs written in Coq.Comment: 28 pages, accepted by ICFP'2

The syntax and semantics of quantitative type theory

We present Quantitative Type Theory, a Type Theory that records usage information for each variable in a judgement, based on a previous system by McBride. The usage information is used to give a realizability semantics using a variant of Linear Combinatory Algebras, refining the usual realizability semantics of Type Theory by accurately tracking resource behaviour. We define the semantics in terms of Quantitative Categories with Families, a novel extension of Categories with Families for modelling resource sensitive type theories

The complexity of theorem proving in autoepistemic logic

Autoepistemic logic is one of the most successful formalisms for nonmonotonic reasoning. In this paper we provide a proof-theoretic analysis of sequent calculi for credulous and sceptical reasoning in propositional autoepistemic logic, introduced by Bonatti and Olivetti [5]. We show that the calculus for credulous reasoning obeys almost the same bounds on the proof size as Gentzen's system LK. Hence proving lower bounds for credulous reasoning will be as hard as proving lower bounds for LK. This contrasts with the situation in sceptical autoepistemic reasoning where we obtain an exponential lower bound to the proof length in Bonatti and Olivetti's calculus

Polynomial time and dependent types

We combine dependent types with linear type systems that soundly and completely capture polynomial time computation. We explore two systems for capturing polynomial time: one system that disallows construction of iterable data, and one, based on the LFPL system of Martin Hofmann, that controls construction via a payment method. Both of these are extended to full dependent types via Quantitative Type Theory, allowing for arbitrary computation in types alongside guaranteed polynomial time computation in terms. We prove the soundness of the systems using a realisability technique due to Dal Lago and Hofmann. Our long-term goal is to combine the extensional reasoning of type theory with intensional reasoning about the resources intrinsically consumed by programs. This paper is a step along this path, which we hope will lead both to practical systems for reasoning about programs’ resource usage, and to theoretical use as a form of synthetic computational complexity theory

Polynomial Path Orders

This paper is concerned with the complexity analysis of constructor term rewrite systems and its ramification in implicit computational complexity. We introduce a path order with multiset status, the polynomial path order POP*, that is applicable in two related, but distinct contexts. On the one hand POP* induces polynomial innermost runtime complexity and hence may serve as a syntactic, and fully automatable, method to analyse the innermost runtime complexity of term rewrite systems. On the other hand POP* provides an order-theoretic characterisation of the polytime computable functions: the polytime computable functions are exactly the functions computable by an orthogonal constructor TRS compatible with POP*.Comment: LMCS version. This article supersedes arXiv:1209.379

Open Government Data Licensing Framework: An Informal Ontology for Supporting Mashup

Objectives of the thesis are –1) to identify the legal problems coming from mashups of Open Government Data (OGD) and 2) to purpose an informal ontology to help technical reusers of Public Sector Information to utilize datasets according to their intended purpose and in compliance with the legal obligations that govern the rights to reuse the data. A survey of national OGD portals found that the majority of OGD are released under inappropriate licenses, not fully complying with the legal rules that apply to the reuse of the data. Open Government Data can be released and covered by multiple licensing regimes, up to 33 in a single country. We have analysed the European Union (EU) legal framework of reuse of Public Sector Information (PSI), the EU Database Directive and copyright framework and other legal sources (e.g., licenses, legal notices, and terms of use) that can apply to open government Datasets. From this deep analysis we have modelled several major concepts in an Informal Ontology of Open Government Data Licenses Framework for a Mash-up Model (iOGDL4M). The iOGDL4M will be used for qualifying datasets in order to improve the accuracy of their legal annotation. The iOGDL4M also aims to connect each applicable legal rule to official legal texts in order to direct legal experts and reusers to primary sources

Open Government Data Licensing Framework: An Informal Ontology for Supporting Mashup

Objectives of the thesis are –1) to identify the legal problems coming from mashups of Open Govern-ment Data (OGD) and 2) to purpose an informal ontology to help technical reusers of Public Sector Informa-tion to utilize datasets according to their intended purpose and in compliance with the legal obligations that govern the rights to reuse the data. A survey of national OGD portals found that the majority of OGD are released under inappropriate li-censes, not fully complying with the legal rules that apply to the reuse of the data. Open Government Data can be released and covered by multiple licensing regimes, up to 33 in a single country. We have analysed the European Union (EU) legal framework of reuse of Public Sector Information (PSI), the EU Database Directive and copyright framework and other legal sources (e.g., licenses, legal notices, and terms of use) that can apply to open government Datasets. From this deep analysis we have modelled several major concepts in an Informal Ontology of Open Government Data Licenses Framework for a Mash-up Model (iOGDL4M). The iOGDL4M will be used for qualifying datasets in order to improve the accuracy of their legal anno-tation. The iOGDL4M also aims to connect each applicable legal rule to official legal texts in order to direct legal experts and reusers to primary sources. This research aims to present 1) a legal analysis of OGD regulation in the European Union and its mem-ber states; 2) the Survey of National Open Government Data Portals and analysis of the most commonly applied licenses and legal notices and their compatibility; and 3) the Informal Ontology of Open Govern-ment Data Licenses Framework for a Mash-up Model. This thesis is comprised of 4 publications. It consists of presentation of the research, the publications, and annexes that support the research