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A formal grammar specifies how to generate a formal language by repeated applications of production rules. Write $G=(N,\Sigma ,P,S)$, where $N$ is a finite set of nonterminal symbols, $\Sigma$ is a disjoint finite alphabet of terminal symbols, $S\in N$ is the start symbol, and $P$ is a finite set of production rules $\alpha \to \beta$. In an unrestricted formal grammar, the word $\alpha$ must contain at least one nonterminal symbol; $\beta$ may be any word over $N\cup \Sigma$.

A grammar derivation replaces an occurrence of $\alpha$ inside a word by $\beta$. The generated formal language isIntermediate words may still contain nonterminal symbols; these are sentential forms. For example, the production rules $S\to aSb$ and $S\to \epsilon$ give $S\Rightarrow aSb\Rightarrow aaSbb\Rightarrow aabb$ and generate exactly $\{a^n{b}^n:n\ge 0\}$. Restricting each left side to one nonterminal symbol gives a context-free grammar.^[ref]

Fix an effective numbering ${\phi }_e$ of the partial computable functions with a universal partial computable function and the parameter compilation supplied by the S-m-n theorem. For every total computable function $f:\mathbb{N}\to \mathbb{N}$, there is a program index $e$ such thatHere $\simeq$ means equality as partial functions, including the inputs on which computation diverges. The conclusion concerns a program's behavior, not equality of program indices: $f(n)=n+1$ has no numerical fixed point, but still has this semantic fixed point.^[ref]

For a proof, the S-m-n theorem gives a total computable function $d$ compiling $a$ into a program that, on input $x$, first computes ${\phi }_a(a)$ and then runs the program with that resulting index on $x$. Thuswhere both sides diverge if the inner computation does. Choose an index $b$ for the total computable function $a\mapsto f(d(a))$. Setting $e=d(b)$ givesThe theorem makes rigorous a program's ability to use a description of itself. For example, let $f(a)$ compile a program that ignores its input and prints $a$. Its semantic fixed point is a program that prints its own index. This does not require a program to read its source file.