At its simplest, a Twelf program is a collection of declarations of type families and constants that inhabit those type families. For example, the following is the standard definition of the natural numbers, with z standing for zero and s the successor operator. nat : type. z : nat. s : nat -> nat.
Here nat is a type, and z and s are constant terms. As a dependently typed system, types can be indexed by terms, which allows the definition of more interesting type families. Here is a definition of addition: plus : nat -> nat -> nat -> type. plus_zero : plus M z M. plus_succ : plus M <- plus M N P.
The type familyplus is read as a relation between three natural numbers M, N and P, such that M + N = P. We then give the constants that define the relation: plus_zero indicates that any natural numberM plus zero is still M. The quantifier can be read as "for all M of type nat". The constant plus_succ defines the case for when the second argument is the successor of some other number N. The result is the successor of P, where P is the sum of M and N. This recursive call is made via the subgoal plus M N P, introduced with <-. The arrow can be understood operationally as Prolog's :-, or as logical implication =, or most faithfully to the type theory, as the type of the constant plus_succ . Twelf features type reconstruction and supports implicit parameters, so in practice one usually does not need to explicitly write above. These simple examples do not display LF's higher-order features, nor any of its theorem checking capabilities. See the Twelf distribution for its included examples.
Uses
Twelf is used in several different ways.
Logic programming
Twelf signatures can be executed via a search procedure, so Twelf can be used as a logic programming language. Its core is more sophisticated than Prolog, since it is higher-order and dependently typed, but it is restricted to pure operators: there is no cut or other extralogical operators as are often found in Prolog implementations, which may make it less well-suited for practical logic programming applications. Some of the use of cut rule as used in Prolog is obtained through the ability to declare that certain operators belong to deterministic type families, which avoids recalculation. Also, like λProlog, Twelf generalizes the Horn clauses underlying Prolog to hereditary Harrop formulas, which allow for logically well-founded operational notions of fresh-name generation and scoped extension of the clause database.
Formalizing mathematics
Twelf's main use today is as a system for formalizing mathematics. Used this way it is closely related to Coq and Isabelle/HOL/HOL Light. However, unlike those systems, Twelf proofs are typically developed by hand. Despite this, for the problem domains at which it excels, Twelf proofs are often shorter and easier to develop than in the automated, general-purpose systems. Twelf is particularly well suited to the encoding of programming languages and logics, because it has a built-in notion of binding and substitution. Most logics and programming languages of interest make use of binding and substitution. When implemented in Twelf, binders can often be directly encoded using the technique of higher-order abstract syntax, in which the meta-language binders are used to represent the object-level binders. As a consequence, standard theorems such as type-preserving substitution and alpha conversion come "for free". Twelf has been used to formalize many different logics and programming languages. Among the larger projects are a proof of safety for the Standard ML programming language, a foundational typed assembly language system from CMU, and a foundational proof carrying code system from Princeton.
