# Questions tagged [functional-programming]

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### Explaining monad transformers in categorical terms

Most resource regarding categorical notions in programming describe monads, but I've never seen a categorical description of monad transformers. How could monad transformers be described in the terms ...
89 views

### Kleisli-like category for applicatives?

I am wondering if there is a good way to complete the following analogy: monad : Kleisli category :: applicative functor : ?? That is, a given monad T on a ...
815 views

### Free theorems, where?

I've found this webapp which lets you generate a free theorem for a given type. The generated theorems quantify over types and relations on these types. These theorems (formulas) are theorems of which ...
67 views

### $[[ \text{fn} \ x => [x]^1]^2 [ \text{fn} \ y => [y]^3]^4]^5$ calls the identity function $\text{fn} \ x => x$ on the argument $\text{fn} \ y => y$?

This is question is related to this past question. I am currently studying the textbook Principles of Program Analysis by Flemming Nielson, Hanne R. Nielson, and Chris Hankin. Chapter 1.4 Constraint ...
106 views

### Phonology and lambda calculus

I wonder whether there is any relationship between lambda calculus and phonology (study of phonemes). Specifically, how one would use the concepts of lambda calculus (typed or untyped) in the study of ...
43 views

### What is the time complexity of substitution algorithms(normalization by evaluation, explicit subtitution)?

I'm studying the substitution algorithms of lambda calculus. I think now I understand how they work, but I couldn't find any materials about their time complexity yet. This is what I've thought about ...
6k views

### What are the simplest turing-complete systems? [closed]

Lambda Calculus is very simple. Are there even simpler turing-complete systems? Which is the simplest of them all?
120k views

### What's new in purely functional data structures since Okasaki?

Since Chris Okasaki's 1998 book "Purely functional data structures", I haven't seen too many new exciting purely functional data structures appear; I can name just a few: IntMap (also invented by ...
241 views

### Areas of research and open problems in functional programming [closed]

What are the major areas of functional programming that require more research and development? For example, I know a lot of people are asking for dependent types in Haskell, and someone at my uni is ...
2k views

### Why do functional programming languages require garbage collection?

What's stopping ghc from translating Haskell into a concatenative programming language such as combinatory logic and then simply using stack allocation for everything? According to Wikipedia, the ...
1k views

### Is it possible to generate a collision free hash function from an equality function?

I'm wondering if it's possible to go from an arbitrary equality function: Eq :: (obj, obj) -> bool to an identity/collision-free hash function: ...
198 views

### How do computers check if two functions are the same?

To prove that two given functions are the same involves proving infinitely many statements. I wonder how to implement so that a computer can check such a statement? An easy example is the following: ...
288 views

### Fixed set of type constructors to simulate all intensional inductive families?

I'm wondering, are there small dependent calculi that can simulate a language with inductive families (that is, has a type isomorphic to each inductive family, at least as powerful of induction ...
213 views

### Structural equality of Pi Types with heterogeneous equality?

I'm trying to implement a proof of the following type: ...
333 views

### Can Non-termination be considered an algebraic effect?

Non-termination is sometimes considered an effect. I have been reading about algebraic effect systems (What is algebraic about algebraic effects and handlers?), and I suspect non-termination (like ...
519 views

### What category are Tagless Final Algebras final In?

The Haskell and Scala community have been very enamored recently with what they call tagless final 'pattern' of programming. These are referenced as dual to initial free algebras, so I was wondering ...
859 views

### What logic correponds via Curry-Howard to a Monad?

According to Moggi's 1991 paper "Notions of computation and monads" one can represent monadic equational logic with the well known monad $(T, \eta, \mu)$ with T an functor and the two natural ...
212 views

### Is it possible to check equality of equi-recursive types, or recursive λ-terms?

Can we determine if two λ-terms are equal? Given two lambda terms, let's say they are equal if their (possibly infinite) Bohm trees are. Under this definition, for example, ...
352 views

### When a type is a value?

In functional programming and in the theoretical setting of the $\lambda$-calculus it is standard to consider a lambda abstraction $\lambda x.M$ as a value. In my understanding, the intuitive reason ...
155 views

### Immutable Space Model

I have heard it said that time is more precious than space because we can reuse space but not time. What if we treat space with this much reverence? What is generally known about models of ...
85 views

### A categorized (?) list of functional pearls in JFP and ICFP

Is there a list of (categorized preferred) functional pearls ever published in ICFP and JFP? I could go to the ICFP proceedings and JFP issues and find all of them, but this would be time-consuming. ...
112 views

### Topology/Space of Recursive Algebraic Datatypes

I have a recursive algebraic datatype. I (somewhat arbitrarily) defined one function to compute distance between instances, and am trying to define a function to approximate a "vector" between ...
113 views

### Stream fusion in total functional language

As I understand, stream fusion consists in converting operations on lists to operations on streams (colists), optimize redundant codata to data and back conversions, fuse operations on streams, and ...
308 views

### What makes a language (and its type-system) capable of proving theorems about its own terms?

I've recently attempted to implement Aaron's Cedille-Core, a minimalist programming language capable of proving mathematical theorems about its own terms. I've also proven induction for λ-encoded ...
152 views

### Can all structurally recursive functions be written without explicit recursion using a catamorphism/fold?

In particular, I am thinking of a function which involves conditionals changing the recursive behavior and multiple F-algebras. ...
496 views

### Is it possible to derive induction by extending CoC with recursion?

Suppose we extended the CoC with primitive recursion; that is, we added a term µ x . t such that equality allowed unrolling recursive terms: ...
3k views

### What is a zipper, and how does it relate to a tree-like structure?

I was reading a chapter in LYAH which didn't really make sense to me. I understand that zippers can arbitrarily traverse a tree-like structure, but I need some clarification on it. Also, can zippers ...
159 views

### Type for “ways values can be different”

I am looking for a concept in type theory that I am sure has probably been explored, but do not know the name behind. Let's consider a ML-like language with product and sum types and a Hindley-Milner ...
285 views

### Can $f^{2^N}(x)$ be computed in polynomial time when $f$ is linear?

Linear functions: definition Let's define a linear function as one expressible as an untyped λ-calculus term with the added restriction that no lambda argument can be used twice. Linear functions: ...
98 views

### Can you assign a type to any term of the λEA-calculus?

The untyped language of System-F and similar is the λ-calculus. That language has terms that can't be typed on System-F, λx.(x x) λx.(x x) being the most obvious ...
44 views

### Is it possible to use arbitrary fixpoint values on EAL without losing strong normalization?

From this question, the answerer states EAL-based languages can use arbitrary fixpoint types without losing strong normalization, because their normalization (and complexity) properties comes from ...
573 views

### Wouldn't the calculus of constructions with linear types be a simple functional core that is consistent and expressive?

I have recently asked if there is a simple functional core that is consistent and expressive. In another question, cody pointed out that this is an open problem to have a language that is: Consistent/...
196 views

### Is it possible to unambiguously read back λ terms from interaction nets without node types?

A class of lambda terms can be evaluated using Lamping's abstract algorithm - that is, converting them to interaction nets and applying a set of rules. In order to get the result, you have to read ...
364 views

### Are there simple core languages which are consistent and expressive?

The Calculus of Constructions is a very simple core functional language with dependent types. Per curry-howard isomorphism, it could, potentially, be very useful for writing programs and proofs. It, ...
3k views

### Category theory, computational complexity, and combinatorics connections?

I have been trying to read “Pearls of Functional Algorithm design”, and subsequently “The Algebra of Programming”, and there is an obvious correspondence between recursively (and polynomially) defined ...
110 views

### Programming language supporting infinitary rewriting of regular term graphs?

Do any practical programming languages support term graph rewriting of infinite but regular terms? For example the toy language CoCaml [1] supports computations on infinite regular streams. Coq ...
1k views

### Trampoline that automatically balances heap and stack

While working through Fogus' Functional Javascript, I came across the trampoline function, which can be used to make safe recursive functions that don't blow up the stack. In Fogus' words, "Of course ...
685 views

### Purely(ish) functional data structure with fast append and forward iteration

I find I have need for a data structure with a specific set of requirements: It represents an immutable sequence of values (fixed sized integers if this matters) Appending a new value to the end (and ...
443 views

### (How) Could we discover/analyze NP problems in the absence of the Turing model of computation?

From a purely abstract math/computational reasoning point of view, (how) could one even discover or reason about problems like 3-SAT, Subset Sum, Traveling Salesman etc.,? Would we be even able to ...
122 views

### Is there is an intuitive explanation why call-by-name PCF is less expressive than both call-by-value PCF and lazy PCF?

J.C. Mitchell cites in his "Expressive power of programming languages" the result in Riecke's "Fully abstract translations between functional languages" about the call-by-value, call-by-name and lazy ...
118 views

### Observational Equivalence of open terms in PCF

The notion of observational equivalence is rather intuitive, but formally I'm having some doubts in the particular case of open terms. Lets consider the simple case where the terms ...
437 views

### Can we say that Church encoding is a form of Gödelization?

We see here the following statement about Godelization: Gödel numbering in computer science means more or less "source code" and "data in binary format", so I hope the ...
72 views

### Best Asymptotic Complexity for Persistent Union Find

In this paper https://www.lri.fr/~filliatr/ftp/publis/puf-wml07.pdf, they claim to have a practically fast persistent union-find data structure for most use-cases, but it's still not polylogarithmic ...
2k views

### Bootstrapping a Finger Tree Structure

After working with 2-3 finger trees for quite a bit I have been impressed by their speed in most operations. However, the one issue I have run into is the large overhead associated with the initial ...
2k views

### What persistent data structure for a set of partially ordered elements?

I need to store sets of elements of type a. Type a is partially ordered, so comparing $a_1$ and $a_2$ can return smaller, greater, equal or incomparable. One problem with hashtables is that two equal ...
323 views

### What type system fits the subclass of λ-terms that can be reduced optimally?

There is a subset of λ-calculus terms that can be reduced by Lamping's Abstract Algorithm without using the Oracle. That is an interesting subset, because only for those terms it is proven that ...
167 views

### Is the topsort from “Structuring Depth-First Search Algorithms” guaranteed to be (reverse) stable?

In "Structuring Depth-First Search Algorithms in Haskell", implemented in Data.Graph in the Haskell standard library, an algorithm for topologically sorting graphs is given: ...
44 views

### Efficient compilation of interaction combinators with infinite cell types to usual interaction combinators?

It is known that interaction combinators can implement any interaction net system efficiently. Now, let us define a modification of interaction combinators, which, instead of two types of fan cells, ...