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Questions tagged [lambda-calculus]

λ-calculus is a formal system for function definition, function application and recursion which forms the mathematical basis of functional programming.

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600 views

Can a calculus have incremental copying and closed scopes?

A few days ago, I proposed the Abstract Calculus, a minimal untyped language that is very similar to the Lambda Calculus, except for the main difference that substitutions are ...
308 views

Advantages of algorithm W over algorithm J for type inference in Hindley-Milner type system

According to A modern eye on ML type inference Furthermore, for some unknown reason, W appears to have become more popular than J, even though the latter is viewed—with reason!—by Milner as ...
132 views

When can you “invert” an equation in the lambda calculus

Suppose that $M$ is a full model of the simply typed lambda calculus. Suppose each base type is infinite. Now suppose that $f$ and $g$ are two functions in $M$ (not necessarily in the same domain) ...
118 views

Using naturality to prove $f: \forall\alpha. \alpha\times\alpha\to\alpha$ must be a projection

Suppose we have a System F term $f : \forall \alpha. \alpha\times\alpha\to\alpha$, interpreted in a parametric model which is a bicartesian closed category. I was wondering if in such context it is ...
31 views

Unbounded-time programs in lambda calculus?

The Turing machine model has been extended to “infinitary turing machines”, which are Turing machines that can perform a countably and uncountably infinite amount of computations in finite time. Is ...
70 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, ...
683 views

Barendregt's Variable Convention: what does it mean?

Barendregt's Variable Convention: If $M_1,...,M_n$ occur in a certain mathematical context (e.g. definition, proof), then in these terms all bound variables are chosen to be different from the free ...
287 views

Capture Avoiding Substitution of multiple variables at once

In articles you often find the terminus "capture avoiding substitution" that saves the author(s) from the tedious process to re-define a recursive function -including alpha-conversion and the ...
58 views

Is it possible to reduce functional equations to SAT?

The problem of finding a solution for functional equations can be defined as: Let A0, A1, A2... An, B0, B1, B2... Bn, X be terms of the lambda calculus, all terms known, except for X, unknown. ...
93 views

Bounded existential polymorphism

In his "Types and Programming Languages", Pierce, at the very end, presents the most powerful system in the book: $F^{\omega}_{<:}$. He, however, does not explain how bounded existential ...
64 views

Difference between “functional programming languages” and “lambda calculus based languages”?

In "Can programming be liberated from the Von Neumann Style?", John Backus states: The main reason FP systems are considerably simpler than either conventional languages or lambda-calculus-based ...
81 views

How could one write typing rules with variables defined at call-site?

I'm trying to write typing rules for a simple language, which is basically a lambda calculus with SSA-like $\phi$-nodes, which basically exchange formal parameters for actual parameters. For ...
133 views

36 views

Are isomorphic (untyped) lambda expressions semantically equivalent?

"Isomorphic" is defined as having the same shape of syntax trees and the same bindings of variables. However, the variable names might be completely different. In other words, it is to say that we ...
30 views

Construct a lambda term from a Böhm tree

Given an acyclic graph, how can I build a lambda calculus term such that this graph is the term's Böhm tree? If the Böhm tree is a finite tree (so the result is a strongly normalizing term). If the ...
87 views

Elegant algorithm to semi-decide if two lambda calculus terms are equivalent

Given two lambda terms $t_1$ and $t_2$, it is semi-decidable if they are equivalent (i.e. can be rewritten as each other using alpha, beta, and eta conversions). An algorithm to do this is to try ...
52 views

Are there any interesting terms in pure LF or $\lambda\Pi$?

In my searching, I've seen that if Church numerals are encoded in a dependently typed Lambda calculus, that we can't derive induction or that $0 \neq 1$. I know that LF and the dependently typed ...
35 views

Understanding First conversion rules from church's lambda bible

While going through Church's text on Lambda calculus , I cam across the first set of conversion rules . Before writing out my query I would like to put the notation that church has used for ...
61 views

Term for weak head normal forms that cannot be reduced in any environment

In my understanding, a lambda expression is a normal form (NF) when it has no redexes. For instance, $\lambda x.x$ is a NF, but $(\lambda x.x)y$ is not. A lambda expression is a weak head normal form (...
52 views

Automatic learning/discovery of logics

Are there efforts to automatically discover new logics? Logics are simple structures - they have formal language, deduction rules, semantics and certain properties that are proved or discarded for ...
66 views

Lambda calculus closure expansion

The set of lambda calculus expressions $Expr$ is generated by the grammar $$Expr \ni e ::= x \mid \lambda x\ldotp e \mid e_1 e_2$$ We can define an interpreter without explicit substitution by ...
I came across the paper: "Efficient Self-Interpretation in Lambda Calculus" by Torben Mogensen, 1994: http://repository.readscheme.org/ftp/papers/topps/D-128.pdf It talks about the intepreter $E$ ...