What programming languages are context-free?

Depending on how you understand the question, the answer changes. But IMNSHO, the proper answer is that all modern programming languages are in fact context sensitive. For example there is no context free grammar that accepts only syntactically correct C programs. People who point to yacc/bison context free grammars for C are missing the point.

What programming languages are context-free? [...]

My gut tells me that functional languages might be context-free [...]

The short version: There are hardly any real-world programming languages that are context-free in any meaning of the word. Whether a language is context-free or not has nothing to do with it being functional. It is simply a matter of how complex the syntax is.

Here's a CFG for the imperative language Brainfuck:

Program → Instr Program | ε
Instr → '+' | '-' | '>' | '<' | ',' | '.' | '[' Program ']'

And here's a CFG for the functional SKI combinator calculus:

Program → E
E → 'S' E E E
E → 'K' E E
E → 'I'
E → '(' E ')'

These CFGs recognize all valid programs of the two languages because they're so simple.

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The longer version: Usually, context-free grammars (CFGs) are only used to roughly specify the syntax of a language. One must distinguish between syntactically correct programs and programs that compile/evaluate correctly. Most commonly, compilers split language analysis into syntax analysis that builds and verifies the general structure of a piece of code, and semantic analysis that verifies the meaning of the program.

If by "context-free language" you mean "... for which all programs compile", then the answer is: hardly any. Languages that fit this bill hardly have any rules or complicated features, like the existence of variables, whitespace-sensitivity, a type system, or any other context: Information defined in one place and relied upon in another.

If, on the other hand, "context-free language" only means "... for which all programs pass syntax analysis", the answer is a matter of how complex the syntax alone is. There are many syntactic features that are hard or impossible to describe with a CFG alone. Some of these are overcome by adding additional state to parsers for keeping track of counters, lookup tables, and so on.

Examples of syntactic features that are not possible to express with a CFG:

• Indentation- and whitespace-sensitive languages like Python and Haskell. Keeping track of arbitrarily nested indentation levels is essentially context-sensitive and requires separate counters for the indentation level; both how many spaces that are used for each level and how many levels there are.

  Allowing only a fixed level of indentation using a fixed amount of spaces would work by duplicating the grammar for each level of indentation, but in practice this is inconvenient.

• The C Typedef Parsing Problem says that C programs are ambiguous during lexical analysis because it cannot know from the grammar alone if something is a regular identifier or a typedef alias for an existing type.

  The example is:

    typedef int my_int;
    my_int x;
  

  At the semicolon, the type environment needs to be updated with an entry for my_int. But if the lexer has already looked ahead to my_int, it will have lexed it as an identifier rather than a type name.

  In context-free grammar terms, the X → ... rule that would trigger on my_int is ambiguous: It could be either one that produces an identifier, or one that produces a typedef'ed type; knowing which one relies on a lookup table (context) beyond the grammar itself.

• Macro- and template-based languages like Lisp, C++, Template Haskell, Nim, and so on. Since the syntax changes as it is being parsed, one solution is to make the parser into a self-modifying program. See also Is C++ context-free or context-sensitive?

• Often, operator precedence and associativity are not expressed directly in CFGs even though it is possible. For example, a CFG for a small expression grammar where ^ binds tighter than ×, and × binds tighter than +, might look like this:

    E → E ^ E
    E → E × E
    E → E + E
    E → (E)
    E → num
  

  This CFG is ambiguous, however, and is often accompanied by a precedence / associativity table saying e.g. that ^ binds tightest, × binds tighter than +, that ^ is right-associative, and that × and + are left-associative.

  Precedence and associativity can be encoded into a CFG in a mechanical way such that it is unambiguous and only produces syntax trees where the operators behave correctly. An example of this for the grammar above:

    E₀ → EA E₁
    EA → E₁ + EA
    EA → ε
    E₁ → EM E₂
    EM → E₂ × EM
    EM → ε
    E₂ → E₃ EP
    EP → ^ E₃ EP
    E₃ → num
    E₃ → (E₀)
  

  But ambiguous CFGs + precedence / associativity tables are common because they're more readable and because various types of LR parser generator libraries can produce more efficient parsers by eliminating shift/reduce conflicts instead of dealing with an unambiguous, transformed grammar of a larger size.

In theory, all finite sets of strings are regular languages, and so all legal programs of bounded size are regular. Since regular languages are a subset of context-free languages, all programs of bounded size are context-free. The argument continues,

While it can be argued that it would be an acceptable limitation for a language to allow only programs of less than a million lines, it is not practical to describe a programming language as a regular language: The description would be far too large.
     — Torben Morgensen's Basics of Compiler Design, ch. 2.10.2

The same goes for CFGs. To address your sub-question a little differently,

Which programming languages are defined by a context-free grammar?

Most real-world programming languages are defined by their implementations, and most parsers for real-world programming languages are either hand-written or uses a parser generator that extends context-free parsing. It is unfortunately not that common to find an exact CFG for your favourite language. When you do, it's usually in Backus-Naur form (BNF), or a parser specification that most likely isn't purely context-free.

Examples of grammar specifications from the wild:

• BNF for Standard ML
• BNF-like for Haskell
• BNF for SQL
• Yacc grammar for PHP

The set of programs that are syntactically correct is context-free for almost all languages.

The set of programs that compile is not context-free for almost all languages. For example, if the set of all compiling C programs were context free, then by intersecting with a regular language (also known as a regex), the set of all compiling C programs that match

^int main\(void\) { int a+; a+ = a+; return 0; }$

would be context-free, but this is clearly isomorphic to the language a^kba^kba^k, which is well-known not to be context-free.

To go for the most dramatic example of a non-context-free grammar, Perl's grammar is, as I understand it, turing-complete.