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Sun, 08 Jul 2018

Undershoot: Parsing theory in 1965

The difference between theory and practice is that in theory there is no difference between theory and practice, but in practice, there is.[1]

Once it was taken seriously that humans might have the power to, for example, "read" a chessboard in a way that computers could not beat. This kind of "computational mysticism" has taken a beating. But it survives in one last stronghold -- parsing theory.

In a previous post, I asked "Why is parsing considered solved?" If the state of the art of computer parsing is taken as anything close to its ultimate solution, then it is a case of "human exceptionalism" -- the human brain has some power that makes it much better at parsing than computers can be. It is very unlikely resorting to human exceptionalism as an explanation would be accepted for any other problem in computer science. Why is it accepted for parsing theory?[2]

The question really requires two separate answers:

In one sense, the answer to both questions is the same -- because of the consensus created by Knuth's 1965 paper "On the translation of languages from left to right". In a previous post, I looked at Knuth 1965 and I answered the practitioner question in detail. But, for the sake of brevity, I answered the question about the theoreticians in outline. This post expands on that outline.

The Practitioners

To summarize, in 1965, practitioners accepted the parsing problem as solved for the following reasons.

What about the theorists?

The practitioners of 1965, then, were quite reasonable in feeling that LR-parsing was as good as anything they were likely to be able to implement any time soon. And they were being told by the theorists that, in fact, it never would get any better -- there were theoretical limits on parsers that faster hardware could not overcome.

We now know that the theorists were wrong -- there are non-LR parsers which are better than the LR parsers are at LR grammars. What made the theorists go astray?

How theorists work

As the epigraph for this post reminds us, theorists who hope to guide practitioners have to confront a big problem -- theory is practice only in theory. Theoreticians (or at least the better ones, like Knuth) know this, but they try to make theory as reliable a guide to practice as possible.

One of the most important examples of the theoretician's successes is asymptotic notation, which we owe to Knuth[4]. Asymptotic notation is more commonly referred to as big-O notation. The term "asymptotic notation" emphasizes its most dangerous aspect from a practical point of view: Asymptotic notation assumes that the behavior of most interest is the behavior for arbitrarily large inputs.

Practical inputs can be very large but, by definition, they are never arbitrarily large. Results in asymptotic terms might be what is called "galactic" -- they might have relevance only in situations which cannot possibly occur in practice.

Fortunately for computer science, asymptotic results usually are not "galactic". Most often asymptotic results are not only relevant to practice -- they are extremely relevant. Wikipedia pages for algorithms put the asymptotic complexities in special displays, and these displays are one of the first things that some practitioners look at.


Since coming up with a theoretical model that is equivalent to "practical" is impossible, theoreticians often work like artillerists. Artillerists often deliberately overshoot and undershoot, before they "fire for effect". "Bracketing" their target in this way has disadvantages -- it reduces the element of surprise, and can even allow the enemy to get their counter-fire in first. But, nasty as these consequences could be, the advantage in accuracy is usually held to outweigh them.

The practice of theoretical computer science is less risky, which makes "bracketing" a very attractive approach to tricky problems. Theoreticians often try to "bracket" practice between an "undershoot" and an "overshoot". The undershoots are models simple and efficient enough to be practical, but too weak to capture all the needs of practice. The overshoots are models which capture everything a practitioner needs, but which are too complicated and/or too resource-intensive for practice.

The P vs. NP problem is an active example of a bracketing technique. You will sometimes read that the P/NP boundary is expected to be that between practical and impractical, but this is an extreme simplification. P includes complexities like O(n^1000000), where the complexity for even n == 2 is a nunber which, in decimal form, fills many pages. Modulo bold advances in quantum computing, I cannot imagine that O(n^1000000) will ever be practical. And you can make the complexities much harder than O(n^1000000) without ever reaching P-hard.

So P-hard is beyond any reasonable definition of "practical" -- it is an "overshoot". But the P vs. NP question is almost certainly very relevant to what is "practical". Resolving the P vs. NP question is likely to be an important or even necessary step. It is a mystery that such a seemingly obvious question has resisted the best efforts of the theoreticians for so long, and the solution of P vs. NP is likely to bring new insights into asymptotic complexity.

Bracketing practical parsing

When Knuth published his 1965, "practical parsing" was already bracketed. On the overshoot side, Irons had already published a parser for context-free grammars. Worst case, this ran in exponential time, and it was, and remains, expected that general context-free parsing was not going to be practical.[5]

On the undershoot side, there were regular expressions and recursive descent. Regular expressions are fast and very practical, but parse a very limited set of grammars. Recursive descent is also fast and, since it parses a larger set of grammars, was the closest undershoot.

Mistake 1: The misdefinition of "language"

To curry respect from the behaviourists, American linguistics for many years banned any reference to meaning. Behaviorists looked down on hypothesized mental states as not worthy of "science", and it's hard to have a theory of meaning without conjectures about mental states. Without mental states, language was just a set of utterances. So in 1926 the linguist Leonard Bloomfield dutifully defined a "language" as a set of "utterances" (for our purposes, "strings"), and through the 30s and 40s most American linguists followed him.

After a brief nod to this tradition, Noam Chomsky restored sanity to linguistics. But it was too late for computer science. Automata theory adopted the semantics-free definition. In 1965, Knuth inherited a lot of prior work, almost all of which ignored, not just meaning or semantics, but even syntax and structure.[6]

Language extension versus language intension

Knuth, of course, wanted to make contact with prior art. The definition he had inherited seemed to work well enough and Knuth's 1965 defines a language as a set of strings. Most subsequent work has refused to breach this tradition.

In most people's idea of what a language is, the utterances/strings mean something -- you cannot take just any set of meaningless strings and call it a language. So the parsing theorists and everybody else had two different definitions of language.

But parsing theory also hoped to produce results relevant to practice, and few people are interested in recognizing meaningless strings -- almost everybody who parses is interested in (at a minimum) finding some kind of structure in what they parse, in order to do something with the result of the parse. Parsing theorists ended up using the word "language" in one sense, but implying that results they found worked for the word "language" in the usual sense.

At this point both senses of the word "language" have gotten entrenched in parsing theory. Instead of making up a new terminology for this blog post, I will borrow a distinction from linguistics and speak of the extension of a language and the intension of a language. The extension of a language is the Bloomfieldian defintion -- the set of utterances/strings in the language. The intension of a language, for our purposes here, can be regarded as its BNF grammar. Each language intension will have (if it is well-defined) exactly one extension. But multiple language intensions can have the same extension.

Red Herring 1: The stack machine model as a natural boundary

The temptation to use language extensions as a proxy for LR-grammars must have been overwhelming. It turns out that the language extension of deterministic stack machines is exactly that of the LR grammars. Further, the language extension of the context-free grammars is exactly that of the non-deterministic stack machines. (Non-deterministic stack machines are stack machines which can "fork" new instances of themselves on the fly.)

If you take language extensions as the proxy for grammars, things fall into place very neatly: the LR-parsers are the deterministic subset of the context-free parsers. And "deterministic" seemed like a very good approximation of practical. Certainly non-deterministic parsing is probably not practical. And the best practical parsers in 1965 were deterministic stack parsers.

Viewed this way, LR-parsing looked like the equivalent of practical parsing. It was a "direct hit", or as close to a exact equivalent of practical parsing as theory was going to get.

As we shall see, with this red herring, the reasoning went astray. But disaster was not inevitable. The whole point of bracketing, after all, is that it allows you to correct errors. Another red herring, however, resulted in parsing theory going on a decades-long wrong turn.

Red Herring 2: LR parsers are not good at LR grammars

The second red herring led to the mis-bracketing of practical parsing. Having seemingly established that LR-parsing is a natural boundary in the hierarchy of languages, Knuth discovered that general LR-parsers were very far from practical. LR parsing goes out to LR(k) for arbitrary k, but even LR(1) parsing was impractical in 1965 -- in fact, it is rare in practical use today. As the k in LR(k) grows, the size of the tables grows exponentially, while the value of the additional lookahead rapidly diminishes. It is not likely that LR(2) parsing will ever see much practical use, never mind LR(k) for any k greater than 2.

From this it was concluded that LR-parsing is an overshoot. In reality, as Joop Leo was to show, it is an undershoot, and in practical terms a very large one. If you mistake an undershoot for an overshoot, bracketing no longer works, and you are not likely to hit your target.

The Wrong Turn

Summing up, parsing theorists concluded, based on the results of Knuth 1965, that

Signs of trouble ignored

There were, in hindsight, clear signs that LR language extensions were not a good proxy for LR grammars. LR grammars form a hierarchy -- for every k≥0, there is an LR grammar which is LR(k+1), but which is not LR(k).

But if you look at extensions instead of grammars, the hierarchy immediately collapses -- every LR(k) language extension is also an LR(1) language extension, as long as k≥1. Only LR(0) remains distinct.

It gets worse. In most practical applications, you can add an end-of-input marker to a grammar. If you do this the LR extension hierarchy collapses totally -- every LR(k) language extension is also an LR(0) language extension.

In short, it seems that, as a proxy for LR grammars, LR language extensions are likely to be completely worthless.

Why didn't Knuth see the problem?

Why didn't Knuth see the problem? Knuth certainly noted the strange behavior of the LR hierarchy in extensional terms -- he discovered it, and devoted several dense pages of his 1965 to laying out the complicated mathematics involved.

So why did Knuth expect to get away with punning intension and extension, even in the face of some very unsettling results? Here, the answer is very simple -- "punning" had always worked before.

Regular expressions are easily turned into parsers[7], so the language extension of a regular grammar is an adequate approximation to its intension. Context-free recognition has the same complexity, and in practice uses the same algorithms, as context-free parsing, so here again, language extension is a good approximation of language intension.

And the LL language extensions follow a strict hierarchy -- for every k≥0, LL(k+1) is a proper superset of LL(k). This fact forces LL grammars to follow the same hierarchy[8]. So, when studying complexity, LL language extensions are an excellent proxy for LL grammars.

Based on past experience, Knuth had reason to believe he could use language extensions as a proxy for grammars, and that the result would be a theory that was a reliable guide to practice.


In my timeline of parsing, I describe what happened next. Briefly, theory focused on finding a useful subset of LR(1). One, LALR, became the favorite and the basis of the yacc and bison tools.

Research into parsing of supersets of LR became rare. The theorists were convinced the LR parsing was the solution. These were so convinced that when, in 1991, Joop Leo discovered a practical way to parse an LR superset, the result went unimplemented for decades.

In 1965, the theoreticians gave a lot of weight to the evidence from the world of practice, but probably not undue weight. Going forward, it was a different story.

Leo had, in essence, disproved the implied conjecture of Knuth 1965. But the question is not an explicit mathematical question, like that of P vs. NP. It is a slipprier one -- capturing practice. Practitioners left it to the theoreticians to keep up with the literature. But theoreticians, as long as LR-superset methods did not come into use in the world of practice, felt no need to revisit their conclusions.

Comments, etc.

I encourage those who want to know more about the story of Parsing Theory to look at my Parsing: a timeline 3.0. To learn about Marpa, my Earley/Leo-based parsing project, there is the semi-official web site, maintained by Ron Savage. The official, but more limited, Marpa website is my personal one. Comments on this post can be made in Marpa's Google group, or on our IRC channel: #marpa at freenode.net.


1. Attributed to Jan L. A. van de Snepscheut and Yogi Berra. See https://en.wikiquote.org/wiki/Jan_L._A._van_de_Snepscheut, accessed 1 July 2018. I quote my preferred form of this -- the one it takes in Doug Rosenberg and Matt Stephens, Use Case Driven Object Modeling with UML: Theory and Practice, 2007, p. xxvii. Rosenberg and Stephens is also the accepted authority for its attribution.

2. As an aside, I am open to the idea that the human mind has abilities that Turing machines cannot improve on or even duplicate. When it comes to survival heuristics tied to the needs of human bodies, for example, it seems very reasonable to at least entertain the conjecture that the human mind might be near-optimal, particularly in big-O terms. But when it comes to ability to solve problems which can be formalized as "puzzles" -- and syntactic analysis is one of these -- I think that resort to human exceptionalism is a sign of desperation.

3. Oettinger, Anthony. "Automatic Syntactic Analysis and the Pushdown Store", Proceedings of Symposia in Applied Mathematics, Volume 12, American Mathematical Society, 1961. Oettinger describes "push" and "pop" stack operations in "Iversion notation" -- what later became APL. See the discussion of Oettinger in my "Why is parsing considered solved?" post.

4. Knuth did not invent asymptotic notation -- it comes from calculus -- but he introduced it to computer science and motivated its use.

5. The best lower bound for context-free parsing is still O(n). So it is even possible that there is a practical linear-time general context-free parser. But its discovery would be a big surprise.

6. In another blog post, I talk about the use of the word "language" in parsing theory in much more detail.

7. For example, regular expressions can be extended with "captures". Captures cannot handle recursion, but neither can regular expressions, so captures are usually sufficient to provide all the structure an application wants.

8. The discussion of the LL(k) hierarchy is in a sense anachronistic -- the LL(k) hierachy was not studied until after 1965. But Knuth certainly was aware of recursive descent, and it seems reasonable to suppose that, even in 1965, he had a sense of what the LL hierarchy would look like.

posted at: 21:02 | direct link to this entry

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