The type theory and type checker of

GF (Grammatical Framework) is a Logical Framework enriched with concrete syntax speciications. Ordinary type-theoretical judgements of typing and deenitional equality specify a theory. In addition, a judgement carries a description of how to produce a string in concrete syntax. The intended principal application area of GF is natural languages. It describes formal languages just as well, although less general tools exist that are optimized for them. Natural and formal languages can be combined in an interface to a proof editor of a logical framework by means of GF. Indeed, such an experimental interface has been made for Agda as an application of GF. This presentation focuses on matters familiar from syntax-directed editors of logical frameworks, although GF also has other applications: in semantically precise multilingual natural language documentation, and in compiling the programming language Shines. 1 Overview of GF This paper presents the type-theoretical structure and type checking principles of GF (Grammatical Framework). GF is a variant of Martin-LL of's higher-level type theory with metavariables and rules for concrete syntax, based on type-theoretical grammar (Ranta 1994). We shall use the term type theory for Martin-LL of's higher-level type theory. Section 2 contains the basic rules of type theory that GF uses. Consider the following speciication of a fragment of arithmetic in type theory extended with deenitions by pattern equations, as in ALF. We would usually like to use a less formal concrete syntax instead of the above abstract syntax. Logical frameworks do not help here, although many have a user interface with layout conventions. They might allow a concrete syntax where 0 stands for zero, x' for succ(x), (x+y) for sum(x,y), and x=y for EqNat(x,y). In GF one may replace these layout conventions by logically rigorous concrete syntax deenitions in the framework itself. This is done by extending the above type-theoretical theory into the GF grammar Nat, Prop : cat zero : Nat-"0" succ : (x:Nat)Nat-x "'" sum : (x:Nat)(y:Nat)Nat-"(" x "+" y ")" sum(x,zero) = x : Nat sum(x,succ(y)) = succ(sum(x,y)) : Nat EqNat : (x:Nat)(y:Nat)Prop-x "=" y