# Questions tagged [functional-programming]

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### Primitive recursion relative to a logical system

In various places I have read that the normally considered non-primitive recursive Ackermann function is primitive recursive in higher-order logic. It's claimed to be due to "Reynolds, 1985",...
79 views

### How does laziness help functional data structure?

Functional data structures, or immutable data structures, are often achieved by copying old data to new data upon operation. Naively, it looks much less efficient than their imperical counterpart. ...
250 views

### Composition with recursion in functions between types

I always understood functions in functional programming to be modeled by morphisms in the category of types, where any powerful function you write in your code is a morphism that is the composition of ...
102 views

### Can we use relational parametricity to simplify the type $\forall a.\,((a\to r)\to a)\to a$ and similar types?

This question is similar to Can we use relational parametricity to simplify the type $\forall a. ( (a \to r) \to r ) \to (a \to r) \to r$? but looks more complicated. It is about using relational ...
138 views

### Can we use relational parametricity to simplify the type $\forall a. ( (a \to r) \to r ) \to (a \to r) \to r$?

This question is about using relational parametricity to resolve practical questions in pure functional programming in System F. Consider the following types of polymorphic functions:  \forall a.\, (...
286 views

### What is a "Covering Function"?

In Idris2, I will sometimes get an error telling me that a function "is not covering", which is apparently distinct from it not being total (and I do understand what a total function is). I ...
171 views

### Intuitive way to handle variable binding

Suppose we have an algebraic datatype parameterised by a type variable name, e.g. ...
131 views

### Encapsulation of OOP and referential transparency of functional programming

I would like to understand more about the 'orthogonality' of OOP and functional programming. What makes me confused is the 'encapsulation' of OOP and 'referential transparency of functional ...
82 views

### Is function composition associative in non-pure programming languages?

We know that function composition is associative in theoretical programming languages such as STλC, and pure functional programming languages such as Haskell. Is the same true for languages where ...
318 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 ...
73 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 ...
142 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 ...
1 vote
81 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 ...
466 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 ...
1 vote
273 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: ...
317 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 ...
232 views

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

I'm trying to implement a proof of the following type: ...
805 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 ...
468 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 ...
1k 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 ...
368 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, ...
453 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 ...
169 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 ...
1 vote
119 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. ...
132 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 ...
138 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 ...
342 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 ...
216 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. ...
517 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: ...
177 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 ...
294 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: ...
1 vote
47 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 ...
113 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 ...
714 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/...
419 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, ...
167 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  supports computations on infinite regular streams. Coq ...
819 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 ...
483 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 ...
3k 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 ...
166 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 ...
1 vote
151 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 ...
507 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 ...
1 vote
77 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 ...
356 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 ...
1 vote
63 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, ...
1 vote
201 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: ...
1 vote
2k views

### How to implement a functional programming language efficiently?

Thanks to Petr and Andrej for their feedback. I'm rephrasing my question and give a bit of a context: Functional programming languages are mostly based on lambda calculus. Implementing a functional ... 904 views

### How do you encode Lamping's abstract algorithm using interaction combinators?

Interaction combinators have been proposed as a compile target for the λ-calculus before. That paper implements the full λ-calculus. It is also known that it is possible to optimize interaction-net ...