474 research outputs found
Fibrational induction meets effects
No full textThis paper provides several induction rules that can be used to prove properties of effectful data types. Our results are semantic in nature and build upon Hermida and Jacobs’ fibrational formulation of induction for polynomial data types and its extension to all inductive data types by Ghani, Johann, and Fumex. An effectful data type μ(TF) is built from a functor F that describes data, and a monad T that computes effects. Our main contribution is to derive induction rules that are generic over all functors F and monads T such that μ(TF) exists. Along the way, we also derive a principle of definition by structural recursion for effectful data types that is similarly generic. Our induction rule is also generic over the kinds of properties to be proved: like the work on which we build, we work in a general fibrational setting and so can accommodate very general notions of properties, rather than just those of particular syntactic forms. We give examples exploiting the generality of our results, and show how our results specialize to those in the literature, particularly those of Filinski and Støvring
Variations on inductive-recursive definitions
No full textDybjer and Setzer introduced the definitional principle of inductive-recursively defined families — i.e. of families ( U : Set , T : U → D ) such that the inductive definition of U may depend on the recursively defined T — by defining a type DS D E of codes. Each c : DS D E defines a functor J c K : Fam D → Fam E , and ( U , T ) = μ J c K : Fam D is exhibited as the initial algebra of J c K . This paper considers the composition of DS -definable functors: Given F : Fam C → Fam D and G : Fam D → Fam E , is G ◦ F : Fam C → Fam E DS -definable, if F and G are? We show that this is the case if and only if powers of families are DS -definable, which seems unlikely. To construct composition, we present two new systems UF and PN of codes for inductive-recursive definitions, with UF ↪ → DS ↪ → PN . Both UF and PN are closed under composition. Since PN defines a potentially larger class of functors, we show that there is a model where initial algebras of PN -functors exist by adapting Dybjer-Setzer’s proof for DS
Corrigendum to Practices of Law Number 6292 and Evaluation of Lands Taken Out of Forest Boundary: The Case of Finike District Year 2020, Volume 22, Issue 1, 222 - 231, https://doi.org/10.24011/barofd.659281
No full textThe authors regret there was an error in the author line related to a forgotten author and the corrected author Line is provided below. Mohammad CHEHREH GHANI 1,*, Nimet VELİOĞLU 2 1İstanbul Üniversitesi, Orman Fakültesi, Orman Mühendisliği Bölümü, , İSTANBUL 2 *Sorumlu Yazar (Corresponding Author): İstanbul Üniversitesi, Orman Fakültesi, Orman Mühendisliği Bölümü, , İSTANBUL These changes would not affect the results and conclusion of the whole manuscript. The authors would like to apologize for any inconvenience caused.*The authors regret there was an error in the author line related to a forgotten author and the corrected author Line is provided below. Mohammad CHEHREH GHANI 1,*, Nimet VELİOĞLU 2 1İstanbul Üniversitesi, Orman Fakültesi, Orman Mühendisliği Bölümü, , İSTANBUL 2 *Sorumlu Yazar (Corresponding Author): İstanbul Üniversitesi, Orman Fakültesi, Orman Mühendisliği Bölümü, , İSTANBUL These changes would not affect the results and conclusion of the whole manuscript. The authors would like to apologize for any inconvenience caused
Foundations for structured programming with GADTs
No full textGADTs are at the cutting edge of functional programming and become more widely used every day. Nevertheless, the semantic foundations underlying GADTs are not well understood. In this paper we solve this problem by showing that the standard theory of data types as carriers of initial algebras of functors can be extended from algebraic and nested data types to GADTs. We then use this observation to derive an initial algebra semantics for GADTs, thus ensuring that all of the accumulated knowledge about initial algebras can be brought to bear on them. Next, we use our initial algebra semantics for GADTs to derive expressive and principled tools — analogous to the well-known and widely-used ones for algebraic and nested data types — for reasoning about, programming with, and improving the performance of programs involving, GADTs; we christen such a collection of tools for a GADT an initial algebra package. Along the way, we give a constructive demonstration that every GADT can be reduced to one which uses only the equality GADT and existential quantification. Although other such reductions exist in the literature, ours is entirely local, is independent of any particular syntactic presentation of GADTs, and can be implemented in the host language, rather than existing solely as a metatheoretical artifact. The main technical ideas underlying our approach are (i) to modify the notion of a higher-order functor so that GADTs can be seen as carriers of initial algebras of higher-order functors, and (ii) to use left Kan extensions to trade arbitrary GADTs for simpler-but-equivalent ones for which initial algebra semantics can be derive
Coproducts of ideal monads
No full textThe question of how to combine monads arises naturally in many areas
with much recent interest focusing on the coproduct of two monads.
In general, the coproduct of arbitrary monads does not always exist.
Although a rather general construction was given by
Kelly [Bull. Austral. Math. Soc.
22 (1980) 1–83], its generality is reflected in its
complexity which limits the applicability of this construction.
Following our own research [C. Lüth and N. Ghani,
Lect. Notes Artif. Intell. 2309 (2002) 18–32], and that of
Hyland, Plotkin and Power [IFIP Conf. Proc.
223 (2002) 474–484], we are looking
for specific situations when simpler constructions are available.
This paper uses fixed points to give a simple construction of the
coproduct of two ideal monads
A principled approach to programming with nested types in Haskell
No full textInitial algebra semantics is one of the cornerstones of the theory of modern functional programming languages. For each inductive data type, it provides a Church encoding for that type, a build combinator which constructs data of that type, a fold combinator which encapsulates structured recursion over data of that type, and a fold/build rule which optimises modular programs by eliminating from them data constructed using the buildcombinator, and immediately consumed using the foldcombinator, for that type. It has long been thought that initial algebra semantics is not expressive enough to provide a similar foundation for programming with nested types in Haskell. Specifically, the standard folds derived from initial algebra semantics have been considered too weak to capture commonly occurring patterns of recursion over data of nested types in Haskell, and no build combinators or fold/build rules have until now been defined for nested types. This paper shows that standard folds are, in fact, sufficiently expressive for programming with nested types in Haskell. It also defines buildcombinators and fold/build fusion rules for nested types. It thus shows how initial algebra semantics provides a principled, expressive, and elegant foundation for programming with nested types in Haskell
KARISMA K.H. MUHAMMAD ZAINI ABDUL GHANI DAN PERAN SOSIALNYA (1942-2005)
No full textIn this exposure, the author proposed about KH Muhammad Zaini Abdul Ghani, the study on charisma and social role and Kharisma of KH Muhammad Zaini Abd. Ghani also supported by the ability to master the classical Islamic sciences. He was able to explain the concepts and to construct elaborate religious in simple language. Therefore, when listening to the description of his talk was easy to understand, understood and grown the love (awe) to him. Feelings of love and admiration were also growing in the hearts of thousands of his disciples. Therefore, a lot of people who liked to tell his karamah - Karamah, although teachers themselves said, Karamah not very important for one's piety, but the consistency in doing good, more better than a thousand Karamah. Once passed away into Rahmatullah, the tomb of K.H. Muhammad Zaini Abd. Ghani always visited by the public until now. Hundreds of people come every day from various parts of this city. This shows that his charisma is not lost when the person dies
غنی خان کی شاعری میں پسے ہوئے طبقے کی ترجمانی: REPRESENTATION OF THE OPPRESED CLASSES IN THE POETRY OF GHANI KHAN
No full textA great poet of Pakhto language author, scholar, painter, painter, sculptor, engineer, politician and philosopher, Ghani khan was such a multi-dimensional personality of the 20th century who is, deservedly, the focus of Pakhtun nation’s pride۔ The person of Ghani khan is a veritable whole of numerous talents۔ His interests and work encompasses quite varied and broad subjects but his prominence is primarily based on his stature as a great poet۔ Although, his poetry consist of many equally important aspects, this research paper aims to bring into relief only those portion of his poetry which portrays the plight and fain of the poor, oppressed and down-trodden section of the society۔
Fibrational induction rules for initial algebras
No full textThis paper provides an induction rule that can be used to prove properties of data structures whose types are inductive, i.e., are carriers of initial algebras of functors. Our results are semantic in nature and are inspired by Hermida and Jacobs’ elegant algebraic formulation of induction for polynomial data types. Our contribution is to derive, under slightly different assumptions, an induction rule that is generic over all inductive types, polynomial or not. Our induction rule is generic over the kinds of properties to be proved as well: like Hermida and Jacobs, we work in a general fibrational setting and so can accommodate very general notions of properties on inductive types rather than just those of particular syntactic forms. We establish the correctness of our generic induction rule by reducing induction to iteration. We show how our rule can be instantiated to give induction rules for the data types of rose trees, finite hereditary sets, and hyperfunctions. The former lies outside the scope of Hermida and Jacobs’ work because it is not polynomial; as far as we are aware, no induction rules have been known to exist for the latter two in a general fibrational framework. Our instantiation for hyperfunctions underscores the value of working in the general fibrational setting since this data type cannot be interpreted as a set
Indexed induction and coinduction, fibrationally.
No full textThis paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions. First, we present a sound coinduction rule for any data type arising as the final coalgebra of a functor, thus relaxing Hermida and Jacobs’ restriction to polynomial data types. For this we introduce the notion of a quotient category with equality (QCE), which both abstracts the standard notion of a fibration of relations constructed from a given fibration, and plays a role in the theory of coinduction dual to that of a comprehension category with unit (CCU) in the theory of induction. Second, we show that indexed inductive and coinductive types also admit sound induction and coinduction rules. Indexed data types often arise as initial algebras and final coalgebras of functors on slice categories, so our key technical results give sufficent conditions under which we can construct, from a CCU (QCE) U : E -> B, a fibration with base B/I that models indexing by I and is also a CCU (QCE)
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