Calculus of constructions
In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. It can serve as both a typed programming language and as constructive foundation for mathematics. For this second reason, the CoC and its variants have been the basis for Coq and other proof assistants.
Some of its variants include the calculus of inductive constructions (which adds inductive types), the calculus of (co)inductive constructions (which adds coinduction), and the predicative calculus of inductive constructions (which removes some impredicativity).
The CoC is a higher-order typed lambda calculus, initially developed by Thierry Coquand. It is well known for being at the top of Barendregt's lambda cube. It is possible within CoC to define functions from integers to integers, as well as integers to types, types to types, and types to integers.
The CoC is strongly normalizing, although, it is impossible to prove this property within the CoC since it implies consistency, which by Gödel's incompleteness theorem is impossible to prove from within the system itself.
Variants of the CoC are used in other proof assistants, such as Matita.
The basics of the calculus of constructions
The Calculus of Constructions can be considered an extension of the Curry–Howard isomorphism. The Curry–Howard isomorphism associates a term in the simply typed lambda calculus with each natural-deduction proof in intuitionistic propositional logic. The Calculus of Constructions extends this isomorphism to proofs in the full intuitionistic predicate calculus, which includes proofs of quantified statements (which we will also call "propositions").
A term in the calculus of constructions is constructed using the following rules:
- T is a term (also called Type);
- P is a term (also called Prop, the type of all propositions);
- Variables (x, y, ...) are terms;
- If A and B are terms, then so is (A B);
- If A and B are terms and x is a variable, then the following are also terms:
- (λx:A . B),
- (∀x:A . B).
In other words, the term syntax, in BNF, is then:
The calculus of constructions has five kinds of objects:
- proofs, which are terms whose types are propositions;
- propositions, which are also known as small types;
- predicates, which are functions that return propositions;
- large types, which are the types of predicates (P is an example of a large type);
- T itself, which is the type of large types.
The calculus of constructions allows proving typing judgments:
Which can be read as the implication
- If variables have types , then term has type .
The valid judgments for the calculus of constructions are derivable from a set of inference rules. In the following, we use to mean a sequence of type assignments , and we use K to mean either P or T. We shall write to mean the result of substituting the term for the free variable in the term .
An inference rule is written in the form
- If is a valid judgment, then so is
Inference rules for the calculus of constructions
Defining logical operators
The calculus of constructions has very few basic operators: the only logical operator for forming propositions is . However, this one operator is sufficient to define all the other logical operators:
Defining data types
The basic data types used in computer science can be defined within the Calculus of Constructions:
- Disjoint union
Note that Booleans and Naturals are defined in the same way as in Church encoding. However additional problems arise from propositional extensionality and proof irrelevance .
- Coquand, Thierry; Huet, Gérard (1988). "The Calculus of Constructions" (PDF). Information and Computation. 76 (2–3): 95–120. doi:10.1016/0890-5401(88)90005-3.
- Also available freely accessible online: Coquand, Thierry; Huet, Gérard (1986). "The calculus of constructions". Technical Report 530. INRIA, Centre de Rocquencourt. Note terminology is rather different. For instance, () is written [x : A] B.
- Bunder, M. W.; Seldin, Jonathan P. (2004). "Variants of the Basic Calculus of Constructions". CiteSeerX 10.1.1.88.9497. Cite journal requires
- Frade, Maria João (2009). "Calculus of Inductive Constructions" (talk).
- Gérard Huet (1988). K. Fuchi and M. Nivat (ed.). Induction Principles Formalized in the Calculus of Constructions (PDF). North-Holland. pp. 205–216. — An application of the CoC