Certified Context-Free Parsing: A formalisation of Valiant's Algorithm in Agda

Valiant (1975) has developed an algorithm for recognition of context free languages. As of today, it remains the algorithm with the best asymptotic complexity for this purpose. In this paper, we present an algebraic specification, implementation, and proof of correctness of a generalisation of Valiant's algorithm. The generalisation can be used for recognition, parsing or generic calculation of the transitive closure of upper triangular matrices. The proof is certified by the Agda proof assistant. The certification is representative of state-of-the-art methods for specification and proofs in proof assistants based on type-theory. As such, this paper can be read as a tutorial for the Agda system

Feat: Functional Enumeration of Algebraic Types

In mathematics, an enumeration of a set S is a bijective function from (an initial segment of) the natural numbers to S. We define "functional enumerations" as efficiently computable such bijections. This paper describes a theory of functional enumeration and provides an algebra of enumerations closed under sums, products, guarded recursion and bijections. We partition each enumerated set into numbered, finite subsets. We provide a generic enumeration such that the number of each part corresponds to the size of its values (measured in the number of constructors). We implement our ideas in a Haskell library called testing-feat, and make the source code freely available. Feat provides efficient "random access" to enumerated values. The primary application is property-based testing, where it is used to define both random sampling (for example QuickCheck generators) and exhaustive enumeration (in the style of SmallCheck). We claim that functional enumeration is the best option for automatically generating test cases from large groups of mutually recursive syntax tree types. As a case study we use Feat to test the pretty-printer of the Template Haskell library (uncovering several bugs)

Level-p-complexity of Boolean functions using Thinning, Memoization, and Polynomials

This paper describes a purely functional library for computing level-$p$-complexity of Boolean functions, and applies it to two-level iterated majority. Boolean functions are simply functions from $n$ bits to one bit, and they can describe digital circuits, voting systems, etc. An example of a Boolean function is majority, which returns the value that has majority among the $n$ input bits for odd $n$. The complexity of a Boolean function $f$ measures the cost of evaluating it: how many bits of the input are needed to be certain about the result of $f$. There are many competing complexity measures but we focus on level-$p$-complexity -- a function of the probability $p$ that a bit is 1. The level-$p$-complexity $D_p(f)$ is the minimum expected cost when the input bits are independent and identically distributed with Bernoulli($p$) distribution. We specify the problem as choosing the minimum expected cost of all possible decision trees -- which directly translates to a clearly correct, but very inefficient implementation. The library uses thinning and memoization for efficiency and type classes for separation of concerns. The complexity is represented using polynomials, and the order relation used for thinning is implemented using polynomial factorisation and root-counting. Finally we compute the complexity for two-level iterated majority and improve on an earlier result by J.~Jansson.Comment: 20 pages, 10 figure

Trust as subject content: Advancing students’ reasoning on democracy through displacement

Purpose: The article explores how the tension between embracing and scrutinising democracy can be productively overcome through social science teaching about democracy that focuses on trust as a subject content. Design/methodology/approach: Empirical materials were collected through focus group interviews before and after an inquiry-based teaching segment on trust, and the materials were analysed qualitatively through three grounded themes. Findings: It is argued that working with the displacement of subject content in inquiry-based teaching about democracy enhances the possibilities for students to deepen their knowledge about democracy, while enabling them to scrutinise the democratic system critically. Research limitations/implications: The article reports from a small-scale study of four classes in two upper secondary schools in Sweden, and the study provides tentative observations and conclusions that should be investigated further in future research. Practical implications: The article shows how trust as a subject content can contribute to problematising students’ understandings of democracy, and how the displacement of content can be important in formulating compelling questions and in designing inquiries on democracy

The impact of uncertainty on optimal emission policies

We apply a computational framework for specifying and solving sequential decision problems to study the impact of three kinds of uncertainties on optimal emission policies in a stylized sequential emission problem.We find that uncertainties about the implementability of decisions on emission reductions (or increases) have a greater impact on optimal policies than uncertainties about the availability of effective emission reduction technologies and uncertainties about the implications of trespassing critical cumulated emission thresholds. The results show that uncertainties about the implementability of decisions on emission reductions (or increases) call for more precautionary policies. In other words, delaying emission reductions to the point in time when effective technologies will become available is suboptimal when these uncertainties are accounted for rigorously. By contrast, uncertainties about the implications of exceeding critical cumulated emission thresholds tend to make early emission reductions less rewarding

Naisdivinaattorien pyydyksissä? : Katsaus Hesekielin kirjan jakeiden 13:17–23 taustalla mahdollisesti vaikuttaviin ilmiöihin

Tarkastelen tässä pro gradu -tutkielmassa Hesekielin kirjan jakeiden 13:17–23 sisältämään kuvaukseen mahdollisesti vaikuttaneita uskonnollisia ilmiöitä. Nämä jakeet sisältävät pelkästään naisista koostuvalle ryhmälle suunnatun tuomion, jossa kiintoisasti viitataan sekä profetiaan että erikoisilla välineillä tapahtuvaan sielujen pyydystämiseen. Tästä syystä jakeita koskeva tutkimus on jakaantunut useampaan eri suuntaan. Osa tutkijoista on katsonut, että jakeiden takaa on mahdollista löytää ryhmä, joka olisi harjoittanut esimerkiksi noituutta, parantamista tai kuolleiden henkien kautta tapahtuvaa divinaatiota eli nekromantiaa. Toisaalta osa tutkijoista on tuonut esille, että jakeet peittävät poleemisella kuvauksellaan ryhmän todellista toimintaa, mikä estää tuomion kohteena olleen ryhmän löytämisen. Tutkielma pyrkii selvittämään, mikä olisi todennäköisin ja parhaiten perusteltavissa ratkaisu kuvauksen erikoisilta vaikuttaville piirteille. Metodologisesti pro gradu -tutkielmaa voidaan pitää uskonnonhistoriallisena tutkimuksena, jossa kiinnitetään huomiota vertailevaan menetelmään ja tekstien mahdolliseen ideologisuuteen. Olennaista tutkielmassa on erilaisten Vanhan testamentin ja muinaisen Lähi-idän uskonnollisten ilmiöiden tarkastelu. Tämän lisäksi tarkastelen Hesekielin kirjaa yleisesti tutkimuskysymyksen kannalta keskeisistä näkökulmista. Tutkielman kolmannessa luvussa käsittelen jakeiden 13:17–23 rakennetta ja sanastoa. Jakeet ovat analyysin perusteella hyvin monitulkintaiset. Toisaalta niissä käytetään naisryhmästä profetiaan mahdollisesti viittaavaa verbiä mutta myös pyydystämiseen liittyvää kuvastoa. Pyydystämisessä käytetyt välineet jäävät jakeiden perusteella moniselitteisiksi. Tulkinnanvaraista on myös se, pyydystävätkö naiset vertauskuvallisesti ihmisiä vai heidän sielujaan. Moniselitteisyys luonnollisesti johtaa erilaisiin painotuksiin, mikä näkyy lukuisissa jakeita käsittelevissä tutkimuksissa. Analyysi ja taustoitus toimivat pohjana eri tutkimuksissa tehtyjen ratkaisujen arvioinnille. Taustoituksen ja analyysin valossa tehty eri tutkimuksien arviointi mielestäni osoittaa, että jakeiden tuomion kieli selittyy parhaiten sillä, että kirjoittaja on hyödyntänyt käsityksiä vaarallisista naisista tai noidista. Sen sijaan jakeiden kuvaus ei suoraan vaikuta vastaavan kuvaa, joka esimerkiksi parantamisesta tai nekromantiasta on eri lähteiden perusteella mahdollista saada. Näin ollen naisista voidaan kohtalaisella varmuudella sanoa ainoastaan, että he olivat naispuolisia divinaation harjoittajia, jotka eivät sopineet kirjoittajan tavoittelemaan uskonnolliseen järjestykseen

Extensional equality preservation and verified generic programming

In verified generic programming, one cannot exploit the structure of concrete data types but has to rely on well chosen sets of specifications or abstract data types (ADTs). Functors and monads are at the core of many applications of functional programming. This raises the question of what useful ADTs for verified functors and monads could look like. The functorial map of many important monads preserves extensional equality. For instance, if $f, g : A \rightarrow B$ are extensionally equal, that is, $\forall x \in A, \ f \ x = g \ x$, then $map \ f : List \ A \rightarrow List \ B$ and $map \ g$ are also extensionally equal. This suggests that preservation of extensional equality could be a useful principle in verified generic programming. We explore this possibility with a minimalist approach: we deal with (the lack of) extensional equality in Martin-L\"of's intensional type theories without extending the theories or using full-fledged setoids. Perhaps surprisingly, this minimal approach turns out to be extremely useful. It allows one to derive simple generic proofs of monadic laws but also verified, generic results in dynamical systems and control theory. In turn, these results avoid tedious code duplication and ad-hoc proofs. Thus, our work is a contribution towards pragmatic, verified generic programming.Comment: Manuscript ID: JFP-2020-003

Generic programming with C++ concepts and Haskell type classes—a comparison

Earlier studies have introduced a list of high-level evaluation criteria to assess how well a language supports generic programming. Languages that meet all criteria include Haskell, because of its type classes, and C++ with the concept feature. We refine these criteria into a taxonomy that captures commonalities and differences between type classes in Haskell and concepts in C++, and discuss which differences are incidental and which ones are due to other language features. The taxonomy allows for an improved understanding of language support for generic programming, and the comparison is useful for the ongoing discussions among language designers and users of both languages

Types, equations, dimensions and the Pi theorem

The languages of mathematical physics and modelling are endowed with a rich "grammar of dimensions" that common abstractions of programming languages fail to represent. We propose a dependently typed domain-specific language (embedded in Idris) that captures this grammar. We apply it to explain basic notions of dimensional analysis and Buckingham's Pi theorem. We hope that the language makes mathematical physics more accessible to computer scientists and functional programming more palatable to modelers and physicists.Comment: Submitted for publication in the "Journal of Functional Programming" in August 202