A Theory of Templating Languages

Joe Gregorio

Working on the URI Templating specification has made me realize that there is a pretty substantial hole in computer science theory: a lack of a theory of templating languages. For example, the current version of URI Templates is not Turing-complete, which excludes a whole bunch of possible attacks. In the specification I state:

On the balance, the template processing is not Turing complete, thus avoiding a number of security issues, ala the billion-laughs attack of XML DTDs.

I was rightly called out on this on the W3C URI mailing list:

This reads a little odd, as not being Turing-complete is not sufficient
to avoid the attack.  (And DTDs are not Turing-complete either.)

The criticism is correct. The problem is that I don't know of any finer grained levels of classifications of templating languages than Turing/non-Turing, and not only for security reasons, but for general capabilities.

For example, if there were classes of templating languages, I could say that URI Templates fell into 'class X', and if that class had a known set of limitations and capabilities then I could say that URI Templates thus had those limitations and capabilities. The weakness to the billion laughs attack comes from two facets of DTD usage, the first being that templates can be defined in terms of other templates, and the second is that the depth of template definition, in terms of other templates, isn't limited. But the converse isn't true, that is, I don't have a general theory of templating to lean on that says since URI Template expansions are never defined in terms of other expansions then URI Templates are immune to such resource exhaustion attacks.

I did find one paper that makes a start at such work, "Enforcing Strict Model-View Separation in Template Engines" (pdf), but the theory is a little weak and it focuses on the nebulous idea of separation of model and view, as opposed to a classification of capabilities and limitations. I'd love to be proven wrong and find that there's a rich theory that I'm over-looking, if so let me know in the comments.

hello, i always wanted to ask this: please, how do i know if some language is turing-complete or not?

Posted by ju on 2007-11-28

the billions laugh attack relies on the fact that non-recursive (i.e finite) DTD expansion still has more than one stage, so it's possible to express an n^m output with an n x m input. so I think categorically, it would be helpful to separate recursive vs non-recursive and n vs 1.

Posted by Assaf on 2007-11-28

Not providing a proof here, but I think URI templates are regular.

Posted by Lenny on 2007-11-28

Consider an implementation that parses an IRI template by looking the template up in a dictionary that maps from strings to parse trees. Such an implementation would return O(2^20n) of space to work correctly. This illustrates the impracticality (impossibility) of specifying URI templates in a way that prevents inefficient implementations.

Instead, you can prove that there exists an efficient, O(n), method of parsing URI templates and an efficient, O(n), method of variable substitution. Both of these are trivial: for the former, simply prove that the grammar is regular, LL(k), LR(k), or any other class known to require only constant space and O(n) time to parse. The latter can be proven by showing that the number of substitutions required at the start is O(n), and then show that the number of pending substitutions decreases by one for each substitution done.

Couple this with an example that illustrates that an implementation that expands templates at more than one level is incorrect.

By the way, the grammar you currently have is ambiguous, I believe. In particular, how should http://foo/{join|}|x} be parsed? Note that "}" is in "unreserved", and so it is a valid "arg".

URI templates as currently specified suffer from the fact that braces are in "unreserved", and so "http://foo.com/{bar}" is a valid URI, even when not expanded. (If there are already braces that need to be included in the template, they should be percent-encoded, right?) If you had chosen to use "%{" and %}" then an unexpanded URI template would never be a valid URI, and then you would also not have the problem mentioned in the preceding paragraph.

What does this template mechanism have to do with URIs except that it requires some substituted values to be percent-encoded? Really, it isn't much different the generic StringTemplate mechanism that is described in Terence Parr's paper or other general string substitution mechanisms.

I also consider the over-conservative quoting to be a unfortunate. I think you mentioned this in an earlier blog post: you cannot have a mailto:{email} URI that expands to "mailto:bill@example.org" because the URI template mechanism requires the "@" to be percent-encoded. I think a mailto example should be included in the spec to illustrate this. Another problematic example is any URI that allows another URI in its query string: e.g. "http://example.org/foo?atom:id=http://somewhere/this", gets expanded into something nasty to read. This will mean that URI templates cannot be used in applications that require character-for-character equality tests for URIs (atom:id for example). This is a problem not just with URI templates, though; IRI-URI round-tripping has similar issues. And, allowing customization of the percent-encoding requirements would add complexity.

Posted by Brian on 2007-11-28

I take it back, "}" is not in unreserved, it is in the set that is neither reserved nor unreserved. I keep forgetting that set.

Posted by Brian on 2007-11-28


From the draft:

To provide feedback on this Internet-Draft, join the W3C URI mailing list (http://lists.w3.org/Archives/Public/uri/).

Posted by Joe on 2007-11-28

I'm rather rusty on this stuff and could be way off, but I believe URI Templates fall into Type 3 of the Chomsky hierarchy. Given a template and a set of inputs, there is a finite set of very predictable results. Further, given the fact that there is no recursion, attacks similar to the million laughs attack are not possible.

Posted by James Snell on 2007-11-29

I think Brian's designation of a complexity class makes sense, but I think the class has to be at least O(TD) where T and D are the size of the template and data.

Consider the template {a}{a}..{a}{a} with data a=zz..zz - the output is of length O(TD), so assuming you want a regular string as output it's going to take that long.

This criterion distinguishes URI templating from DTDs in complexity. A DTD parser (that outputs text nodes as strings) can't be implemented with better than O(2^n) complexity, as the output size of billion laughs demonstrates.

An example efficient implementation is an another way of proving the deliberately low power of the language.

Posted by Sam McCall on 2007-11-29

I think there needs to be a bright line drawn between the complexity of parsing the template versus the possible runtime of actually expanding such templates. That latter is what I was talking about, and the two are not related. For example, we can have a trivial template language that is easily parsed by a regular expression:


If we define template substitution to be variables of one letter, where variables are substituted with their values recursively until no more matches are found, then this simple template


with the definition

a := aa

will expand forever.

In addition the examples given so far only cover completely self-contained templates like URI Template expansions, there's much more complex constructs, like Cheetah's #for, #end for construct. I'm interested in the the total complexity of the output based on the template and the variables supplied, not only for putting a bound on the time and space complexity of the expansion, but also in the types of constructs that can be expressed. That is, given a template language and allowable input data structures can I - fill in the options for a combo box? fill out a row-oriented table? fill out a column-oriented table? etc.

From the responses so far I guess there isn't a rich theory of templating languages that I just happened to overlook.

Posted by Joe on 2007-11-29

Ju, You show some language/process is Turning complete by implementing a Turing Machine in that language or process. Or, equivalently, implement something which is itself Turning Complete (e.g the lambda calculus or the Rule 110 1d cellular automaton).

Posted by matthew on 2007-11-29

From the responses so far I guess there isn't a rich theory of templating languages that I just happened to overlook.
Formal grammars are likely the closest you're going to get. To the original issue that brought up the discussion, however, as has been pointed out, the main reasons that URI Templates are not susceptible to attacks like billion laughs are the fact that they're not recursive and the fact that non-unreserved chars have to be escaped. The input parameters, the amount of time it takes to process the template and the set of possible outcomes are finite and predictable.

Posted by James Snell on 2007-11-29

OK I'm ignorant. I doubt I'm alone. Request please Joe. Pointer to some reading on templating languages (are there others?) or a 100 word intro to what they're about/useful for etc? E.g. why they peaked your interest?

Posted by Dave Pawson on 2007-11-30

I saw the headline for this in my reader and the first thing I did was hit Google for Terrence Parr's pdf. Turns out this post is already in the front page!

I think in terms of theory, you could look at the work done around OWL and description logics in general, if you're prepared to treat the evaluator for a URI template as a theorem prover. A lot of work has been put into making sure DLs will return a result in a predictable amount of time.

Fwiw on the human factors side - and I don't why this is - django templates for me have an almost perfect balance between expressive power and functionality.

Posted by Bill de hOra on 2007-11-30


Dave, when I say templating languages, I mean the input to a template processor, such as Cheetah (Python), Smarty (PHP), or Velocity (Java). Most of the the templating languages listed on that Wikipedia page are Turing complete and so can produce any sort of output, but what I'm interested in are more restricted templating languages, for example URI Templates.

The situations with URI Templates is a little different in that you have a situation where an untrusted party can hand you a template which you fill in using your own data structure. This is different from a normal templating situation where you would control both the template and the data structure input. I'd like to have a general way of classifying the expressiveness of a templating language along with the worst case complexity of the expansion process both in time and space. Obviously a Turing complete templating language is O(\infinity) in both time and space.

Posted by Joe on 2007-11-30

matthew, thanks!

Posted by ju on 2007-12-02

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