|logic||category theory||type theory|
|true||terminal object/(-2)-truncated object||h-level 0-type/unit type|
|false||initial object||empty type|
|proposition||(-1)-truncated object||h-proposition, mere proposition|
|cut rule||composition of classifying morphisms / pullback of display maps||substitution|
|cut elimination for implication||counit for hom-tensor adjunction||beta reduction|
|introduction rule for implication||unit for hom-tensor adjunction||eta conversion|
|disjunction||coproduct ((-1)-truncation of)||sum type (bracket type of)|
|implication||internal hom||function type|
|negation||internal hom into initial object||function type into empty type|
|universal quantification||dependent product||dependent product type|
|existential quantification||dependent sum ((-1)-truncation of)||dependent sum type (bracket type of)|
|equivalence||path space object||identity type|
|equivalence class||quotient||quotient type|
|induction||colimit||inductive type, W-type, M-type|
|higher induction||higher colimit||higher inductive type|
|completely presented set||discrete object/0-truncated object||h-level 2-type/preset/h-set|
|set||internal 0-groupoid||Bishop set/setoid|
|universe||object classifier||type of types|
|modality||closure operator, (idemponent) monad||modal type theory, monad (in computer science)|
|linear logic||(symmetric, closed) monoidal category||linear type theory/quantum computation|
|proof net||string diagram||quantum circuit|
|(absence of) contraction rule||(absence of) diagonal||no-cloning theorem|
|synthetic mathematics||domain specific embedded programming language|
A certification of a computer program is a formalized guarantee – a proof – that the program has given specified properties. For instance, it could be guaranteed to compute a given output based on a given input, or to always terminate, or to not include a certain kind of security hole.
Certifications often take the form of a proof that a program, regarded as a term of some sort (under programs as proofs), has a specified type. Thus, programming languages based on highly expressive type theories (including dependent types) are a natural place to do certified programming “natively”. Examples are Coq and Agda. In this case, the program is written at the same time as a proof of its certification. One often then wants to “extract” the executable code or “ignore” the proof part of the terms when actually running the code, for performance reasons; Coq and Agda include mechanisms designed for this.
It is also possible to write a program in some less strongly typed language and provide an “external” certification for it, rather than one built into the program itself. Computer proof assistants like Coq and Agda are also used for this, using a formal representation of some other programming language. There are also other program analysis tools which can produce automated proofs of certain aspects of a computer program, such as safety and termination (although of course a complete solution to termination-checking is impossible, being the halting problem).
So far, fully certified programming in the type-theoretic sense is largely an academic endeavor, see for instance (SpittersKrebbersvdWeegen); the tools available at present usually require too much time and effort to be worth the payoff in industry. As automation progresses, this may change.
Discussion of the need for certified programming in scientific computation is in