Type Theory

Bridge Category theory x Functional programming - functors as type constructors

Fields: Mathematics, Computer_Science, Type_Theory, Logic

The Curry-Howard-Lambek correspondence establishes a three-way isomorphism between typed lambda calculus, intuitionistic logic, and Cartesian closed categories; monads in Haskell are exactly monads in...

Bridge Category theory (Eilenberg & Mac Lane 1945) is the semantic foundation of functional programming: types are objects, functions are morphisms, functors are type constructors, monads are monoids in the category of endofunctors, and the Curry-Howard correspondence makes propositions = types and proofs = programs.

Fields: Mathematics, Computer Science, Type Theory, Functional Programming

Category theory — the abstract mathematics of structure-preserving maps — is not merely analogous to functional programming; it is the precise mathematical semantics of statically-typed functional lan...

Bridge The Cook-Levin theorem (1971) establishes SAT as NP-complete; Gödel's incompleteness theorems and Turing's halting problem both derive from diagonalization; the Curry-Howard correspondence identifies programs with proofs and types with propositions; interactive proof systems (IP=PSPACE) reveal that probabilistic verification is exponentially more powerful than deterministic checking — mathematics and computer science study the same logical limits from different directions.

Fields: Mathematics, Logic, Computer Science, Complexity Theory, Proof Theory, Type Theory

The Cook-Levin theorem (Cook 1971, Levin 1973): SAT is NP-complete — every problem in NP polynomially reduces to Boolean satisfiability. P vs NP (Clay Millennium Problem): does every efficiently verif...

Bridge The Curry-Howard correspondence proves that propositions in intuitionistic logic are identical to types in typed lambda calculus, and proofs of those propositions are identical to programs of those types — mathematics and computation are the same formal system viewed from two perspectives.

Fields: Mathematical Logic, Type Theory, Computer Science, Proof Theory, Programming Language Theory

The Curry-Howard isomorphism (independently discovered by Haskell Curry in 1934 for combinatory logic and William Howard in 1969 for natural deduction) establishes an exact correspondence between the ...

Bridge The Curry-Howard correspondence identifies types in programming languages with propositions in logic and programs with proofs — making proof assistants (Coq, Lean) and systems languages (Rust borrow checker) instances of applied type theory.

Fields: Mathematics, Computer Science, Logic, Type Theory, Programming Languages

The Curry-Howard isomorphism (Curry 1934 combinatory logic; Howard 1969 natural deduction) establishes: types ↔ propositions; programs ↔ proofs; program execution ↔ proof normalization; function types...