Initial Algebra Research Articles

Initial algebras and terminal coalgebras in many-sorted sets

We prove that the iterative construction of initial algebras converges for endofunctors F of many-sorted sets whenever F has an initial algebra. In the case of one-sorted sets, the convergence takes n steps where n is either an infinite regular cardinal or is at most 3. Dually, the existence of a many-sorted terminal coalgebra implies that the iterative construction of a terminal coalgebra converges. Moreover, every endofunctor with a fixed-point pair larger than the number of sorts is proved to have a terminal coalgebra. As demonstrated by James Worell, the number of steps here need not be a cardinal even in the case of a single sort: it is ω + ω for the finite power-set functor. The above results do not hold for related categories, such as graphs: we present non-constructive initial algebras and terminal coalgebras.

Inversive meadows and divisive meadows

Inversive meadows are commutative rings with a multiplicative identity element and a total multiplicative inverse operation satisfying 0 − 1 = 0 . Divisive meadows are inversive meadows with the multiplicative inverse operation replaced by a division operation. We give finite equational specifications of the class of all inversive meadows and the class of all divisive meadows. It depends on the angle from which they are viewed whether inversive meadows or divisive meadows must be considered more basic. We show that inversive and divisive meadows of rational numbers can be obtained as initial algebras of finite equational specifications. In the spirit of Peacockʼs arithmetical algebra, we study variants of inversive and divisive meadows without an additive identity element and/or an additive inverse operation. We propose simple constructions of variants of inversive and divisive meadows with a partial multiplicative inverse or division operation from inversive and divisive meadows. Divisive meadows are more basic if these variants are considered as well. We give a simple account of how mathematicians deal with 1/0, in which meadows and a customary convention among mathematicians play prominent parts, and we make plausible that a convincing account, starting from the popular computer science viewpoint that 1/0 is undefined, by means of some logic of partial functions is not attainable.

The Recursion Scheme from the Cofree Recursive Comonad

We instantiate the general comonad-based construction of recursion schemes for the initial algebra of a functor F to the cofree recursive comonad on F. Differently from the scheme based on the cofree comonad on F in a similar fashion, this scheme allows not only recursive calls on elements structurally smaller than the given argument, but also subsidiary recursions. We develop a Mendler formulation of the scheme via a generalized Yoneda lemma for initial algebras involving strong dinaturality and hint a relation to circular proofs à la Cockett, Santocanale.

An improved algorithm for determinization of weighted and fuzzy automata

For a given weighted finite automaton over a strong bimonoid we construct its reduced Nerode automaton, which is crisp-deterministic and equivalent to the original weighted automaton with respect to the initial algebra semantics. We show that the reduced Nerode automaton is even smaller than the Nerode automaton, which was previously used in determinization related to this semantics. We determine necessary and sufficient conditions under which the reduced Nerode automaton is finite and provide an efficient algorithm which computes the reduced Nerode automaton whenever it is finite. In determinization of weighted finite automata over semirings and fuzzy finite automata over lattice-ordered monoids this algorithm gives smaller crisp-deterministic automata than any other known determinization algorithm.

The determinantal ideals of extended Hankel matrices

In this paper, we use the tools of Gröbner bases and combinatorial secant varieties to study the determinantal ideals I t of the extended Hankel matrices. Denote by c -chain a sequence a 1 , … , a k with a i + c < a i + 1 for all i = 1 , … , k − 1 . Using the results of c -chain, we solve the membership problem for the symbolic powers I t ( s ) and we compute the primary decomposition of the product I t 1 ⋯ I t k of the determinantal ideals. Passing through the initial ideals and algebras we prove that the product I t 1 ⋯ I t k has a linear resolution and the multi-homogeneous Rees algebra Rees ( I t 1 , … , I t k ) is defined by a Gröbner basis of quadrics.

Initial Algebra Semantics for Cyclic Sharing Tree Structures

Terms are a concise representation of tree structures. Since they can be naturally defined by an inductive type, they offer data structures in functional programming and mechanised reasoning with useful principles such as structural induction and structural recursion. However, for graphs or "tree-like" structures - trees involving cycles and sharing - it remains unclear what kind of inductive structures exists and how we can faithfully assign a term representation of them. In this paper we propose a simple term syntax for cyclic sharing structures that admits structural induction and recursion principles. We show that the obtained syntax is directly usable in the functional language Haskell and the proof assistant Agda, as well as ordinary data structures such as lists and trees. To achieve this goal, we use a categorical approach to initial algebra semantics in a presheaf category. That approach follows the line of Fiore, Plotkin and Turi's models of abstract syntax with variable binding.

The initial meadows

AbstractAmeadowis a commutative ring with an inverse operator satisfying 0−1= 0. We determine the initial algebra of the meadows of characteristic 0 and prove a normal form theorem for it. As an immediate consequence we obtain the decidability of the closed term problem for meadows and the computability of their initial object.

On Coalgebras over Algebras

We extend Barr's well-known characterization of the final coalgebra of a Set-endofunctor as the completion of its initial algebra to the Eilenberg-Moore category of algebras for a Set-monad M for functors arising as liftings. As an application we introduce the notion of commuting pair of endofunctors with respect to the monad M and show that under reasonable assumptions, the final coalgebra of one of the endofunctors involved can be obtained as the free algebra generated by the initial algebra of the other endofunctor.

Higher-order Algebras and Coalgebras from Parameterized Endofunctors

The study of algebras and coalgebras involve parametric description of a family of endofunctors. Such descriptions can often be packaged as parameterized endofunctors. A parameterized endofunctor generates a higher-order endofunctor on a functor category. We characterize initial algebras and final coalgebras for these higher-order endofunctors, generalizing several results in the literature.

Determinization of weighted finite automata over strong bimonoids

We consider weighted finite automata over strong bimonoids, where these weight structures can be considered as semirings which might lack distributivity. Then, in general, the well-known run semantics, initial algebra semantics, and transition semantics of an automaton are different. We prove an algebraic characterization for the initial algebra semantics in terms of stable finitely generated submonoids. Moreover, for a given weighted finite automaton we construct the Nerode automaton and Myhill automaton, both being crisp-deterministic, which are equivalent to the original automaton with respect to the initial algebra semantics, respectively, the transition semantics. We prove necessary and sufficient conditions under which the Nerode automaton and the Myhill automaton are finite, and we provide efficient algorithms for their construction. Also, for a given weighted finite automaton, we show sufficient conditions under which a given weighted finite automaton can be determinized preserving its run semantics.

Two extensions of system F with (co)iteration and primitive (co)recursion principles

This paper presents two extensions of the second order polymorphic lambda calculus, system F, with monotone (co)inductive types supporting (co)iteration, primitive (co)recursion and inversion principles as primitives. One extension is inspired by the usual categorical approach to programming by means of initial algebras and final coalgebras; whereas the other models dialgebras, and can be seen as an extension of Hagino's categorical lambda calculus within the framework of parametric polymorphism. The systems are presented in Curry-style, and are proven to be terminating and type-preserving. Moreover their expressiveness is shown by means of several programming examples, going from usual data types to lazy codata types such as streams or infinite trees.

Some Remarks on Type Systems for Course-of-value Recursion

Course-of-value recursion is a scheme which allows us to define the value of a function in some argument of an inductive structure by using not only the immediate, but arbitrary previously computed values. In the categorical approach to typed (total) functional programming, where datatypes (codatatypes) are initial algebras (final coalgebras), one models this principle by a construction called histomorphism. On the other hand, it is known that other categorical principles such as catamorphisms and anamorphisms representing definitions by iteration and coiteration have been successfully used to implement safe type systems extending the second-order polymorphic lambda calculus, system F. Hence it is natural to pursuit the definition and implementation of fold operators corresponding to course-of-value recursion as well. This paper proposes some new such extensions and states some important remarks emerged while verifying the correctness and safety properties of their operational semantics, relying not only on the categorical, but also on the logical approach based on fixed-point operators. Our observations should be considered as a starting point for a deeper study of the interrelation between these two approaches.

An induction principle for nested datatypes in intensional type theory

AbstractNested datatypes are families of datatypes that are indexed over all types such that the constructors may relate different family members (unlike the homogeneous lists). Moreover, the argument types of the constructors refer to indices given by expressions in which the family name may occur. Especially in this case of true nesting, termination of functions that traverse these data structures is far from being obvious. A joint paper with A. Abel and T. Uustalu (Theor. Comput. Sci., 333 (1–2), 2005, pp. 3–66) proposed iteration schemes that guarantee termination not by structural requirements but just by polymorphic typing. They are generic in the sense that no specific syntactic form of the underlying datatype “functor” is required. However, there was no induction principle for the verification of the programs thus obtained, although they are well known in the usual model of initial algebras on endofunctor categories. The new contribution is a representation of nested datatypes in intensional type theory (more specifically, in the calculus of inductive constructions) that is still generic and covers true nesting, guarantees termination of all expressible programs, and has an induction principle that allows to prove functoriality of monotonicity witnesses (maps for nested datatypes) and naturality properties of iteratively defined polymorphic functions.

A principled approach to programming with nested types in Haskell

Initial 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.

Finally tagless, partially evaluated: Tagless staged interpreters for simpler typed languages

AbstractWe have built the first family of tagless interpretations for a higher-order typed object language in a typed metalanguage (Haskell or ML) that require no dependent types, generalized algebraic data types, or postprocessing to eliminate tags. The statically type-preserving interpretations include an evaluator, a compiler (or staged evaluator), a partial evaluator, and call-by-name and call-by-value continuation-passing style (CPS) transformers. Our principal technique is to encode de Bruijn or higher-order abstract syntax using combinator functions rather than data constructors. In other words, we represent object terms not in an initial algebra but using the coalgebraic structure of the λ-calculus. Our representation also simulates inductive maps from types to types, which are required for typed partial evaluation and CPS transformations. Our encoding of an object term abstracts uniformly over the family of ways to interpret it, yet statically assures that the interpreters never get stuck. This family of interpreters thus demonstrates again that it is useful to abstract over higher-kinded types.

Bootstrapping Inductive and Coinductive Types in HasCASL

We discuss the treatment of initial datatypes and final process types in the wide-spectrum language HasCASL. In particular, we present specifications that illustrate how datatypes and process types arise as bootstrapped concepts using HasCASL's type class mechanism, and we describe constructions of types of finite and infinite trees that establish the conservativity of datatype and process type declarations adhering to certain reasonable formats. The latter amounts to modifying known constructions from HOL to avoid unique choice; in categorical terminology, this means that we establish that quasitoposes with an internal natural numbers object support initial algebras and final coalgebras for a range of polynomial functors, thereby partially generalising corresponding results from topos theory. Moreover, we present similar constructions in categories of internal complete partial orders in quasitoposes.

A Categorical Model of the Fusion Calculus

We provide a categorical presentation of the Fusion calculus. First we give Working in a suitable category of presheaves, we describe the syntax as initial algebra of a signature endofunctor, and the semantics as coalgebras of a “behaviour” endofunctor. To this end, we first give a a new, congruence-free presentation of the Fusion calculus; then, the behaviour endofunctor is constructed by adding in a systematic way a notion of “state” to the intuitive endofunctor induced by the LTS. Coalgebras can be given a concrete presentation as “stateful indexed labelled transition systems”; the bisimilarity over these systems is a congruence, and corresponds to hyperequivalence. Then, we model the labelled transition system of Fusion by abstract categorical rules. As a consequence, we get a semantics for the Fusion calculus which is both compositional and fully abstract: two processes have the same semantics iff they are bisimilar, that is, hyperequivalent.

On initial algebras of multiplicative invariants

Consider the ring of multiplicative invariants, k [ A ] G , of the group algebra k [ A ] of a faithful G -lattice A over base field k . Among other things, we will determine the cardinality of the set of all initial algebras, in ≽ ( k [ A ] G ) , of the ring of multiplicative invariants over all possible admissible orders ≽ on A .

An irregular filter model

In this paper we introduce a new filter model, which is of a kind that has escaped investigation up to now: it is induced by an intersection type theory generated in a non-standard way, by a preorder which puts into relation an atom with an arrow type, without equating them. We study the domain-theoretic implications of this choice, that are not trivial: in order to describe this filter model a new category is introduced and a special purpose functor defined. The filter model is then characterized as the initial algebra of the functor.

Foundations for structured programming with GADTs

GADTs are at the cutting edge of functional programming and becomemore 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 derivean 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 bederived.