My Interests

I work on computational and mathematical linguistics as well as logic. This web page might give you a more specific idea of what I have been doing so far. It is not meant to state exclusive areas of interests.

Computational and Mathematical Linguistics

I consider myself a theoretical computational linguist. Just like an ordinary theoretical computer scientist I rarely use the computer to get work done. Rather, my interest is in understanding how the computer can be used to achieve something. This means that I ask the following questions:
  1. How can the problem be adequately formalized?
  2. Can a problem in principle be solved and if so, how?
  3. What is the most efficient way to solve a problem?
Much work in computational linguistics is or practical nature, using certain methods or experimenting with them. There is however as always the need to understand exactly what these methods yield and what we use them for in order to properly assess the merits of the practical research that is otherwise being done. Also, a lot of theoretical work in linguistics is not in a form that allows immediate implementation. Thus, a rigorous formalization is needed before one can proceed to the implementation. This is the work that I do (just like Ed Stabler). Some of the topics mentioned below as general linguistics are also treated mathematically, but I will mention here only things that are immediately related to these questions.

Formalisation of Generative Grammar

Originally, I have been interested in formal syntax, in particular formal properties of Government and Binding Theory, though I have concerned myself also with the foundations of GPSG and HPSG. In a series of three papers I dealt with the main innovative tools of transformational grammar: the use of command relations, the use of adjunction, and the use of chains. Apart from theoretical interests (for example, showing that multidominance structures are practically equivalent to the more common trace chain structures that you see in the textbooks), there is another motive behind the research. Generative grammar is a complicated theory, and it is often very difficult to gauge the consequences of particular assumptions. Ultimately, we want to be assured that everything works as promised. Such an assurance may come in the form of a theorem prover where you can enter a principle or grammatical fact and you are told whether or not it holds. Such a prover has been written in Prolog by Ed Stabler. The unique trait about my own approach is that I use dynamic logic rather than predicate logic or monadic second order logic. I have recently establish that the dynamic logic of the multidominance structures is decidable (see here). This is not all that we need, but I am slowly getting there.

Formal Languages

The hierarchy or hierarchies of languages have played an essential role in proofs of adequacy of formal models (witness the results by Peters and Ritchie or the history of GPSG). My own interest is in knowing where to place the limits either way. One result was the proof that GB implies strong context freeness if head movement is bounded (see Syntactic Codes and Grammar Refinement). This result was independently established by James Rogers. I also got interested in Linear Context Free Rewrite Systems. It is known (thanks to Jens Michaelis and Henk Harkema) that the Minimalist Program is weakly equivalent to these grammars. Together with Jens Michaelis I had earlier established that languages with stacked cases are not semilinear, and thus cannot be generated by these grammars (and therefore the Minimalist Program). For the paper see here. A summary of case marking and semilinearity can be found here. Semilinearity has been a hobby since then. I managed to establish a new proof of a theorem by Spanier and Ginsburg (see A New Proof of a Result by Ginsburg and Spanier). I get a lot of requests for it but it is already included in "The Mathematics of Language". Another nice result is the proof that Ogden's Lemma (also known as the Pumping Lemma) is vastly insufficient to characterise context free languages; not even a strengthening suggested by Alexis Manaster-Ramer is enough (see here).


Teaching programming involves writing stuff myself. I am doing OCaML. I have a tool for manipulating finite state automata. I expect to do more when I teach it again. Also, I have implemented the ideas on referent systems exposed below. This will eventually allow to test the theory and run it on large scale dictionaries. This work is still experimental. If you want to obtain the source code, please email me.

Logic and Grammar

There is a research area called model theoretic syntax into which these interests partly fit. One half of the enterprise has been explained above, namely the analysis of theories (HPSG, Generative Grammar) with the help of a logical language (in my case dynamic logic). Part of it has to do with obtaining decidability proofs. There are also deeper connections to be uncovered. In Inessential Features I have introduced the notion of an inessential feature. This is a feature whose distribution is controlled by the other features. It is, in logical terms, implicitly defined. If one can obtain an explicit definition (not always guaranteed!) then the feature can be eliminated from the grammar. This was meant to explain the divergence between HPSG, where every detail is in the representation, and generative grammar where "inessential" stuff is omitted. Needless to say that linguists will not necessarily see things the way I have explained, but if one is interested in seeing the common features behind the grammar formalisms, an analysis of this sort is called for.

General Linguistics

Agreement Morphology and Case

Recently, I have become (again) more interested in semantics. For the most part I studied the interaction between morphology and semantics. In the monograph Agreement Morphology, Argument Structure and Syntax, I have spelled out in detail a theory that maps morphological strings onto semantical structures and back. This work has shown to be just the beginning of an investigation that leads to a reassessment of the role of morphological elements, most notably cases. A sideline of this research is the particular structure of local cases. One particular conclusion is that local cases consist of several layers (see Against the Feature Bundle Theorey of Case). Ordinary case selection is selection of two heads, not one; the case where just one head is selected has not hitherto been identified, but does occur quite frequently (see Directionality Selection).


Another issue that has concerned me recently is the notion of compositionality. My main intuition about compositionality is that present theories do not contain a notion of part of an expression. I have published on this in Strict Compositionality and Literal Movement Grammars. I have made the notion of compositionality a central concern of the book "Mathematics of Language". While in that book I studied mainstream theories, such as Montague Grammar, in depth, I have recently moved to investigate different approaches to the question. Suppose, for example, that meanings are not typed λ-terms but rather truth conditions. Suppose also that while meanings contain to indication of syntax and syntactic representations nothing of value for semantics. Then the thesis of compositionality becomes highly nontrivial; the conversion to Chomsky Normal Form does not any longer come for free. Now we can actually show that nontrivial syntactic facts arise out of these assumptions (that Dutch is not strongly context free, for example). If you are interested, read more in here). Meanwhile, I have written a book on this subject; to see it and read surrounding material, go to "The Compositionality Saga".

The Language of Space

The language of space is a small area compared to the language of time. Tense and aspect is a popular topic in linguistics, while space is often only considered in its function as a metaphor for something else (see the work of Langacker). I got interested in space both because of my interest in Uralic languages (which have lots of local cases) and because of the fact that space as a formal object is well studied and so we can actually say with enough precision what certain words mean. Phrases such as "into the city", "200 m from the wall" and so on have reasonably precise meaning. I have been looking at a wide range of languages (Indo-European, Uralic, Oceanic, Native American) and looked at the morphological and syntactic patterns for expressions of space. A reader is on the website, but it is in a very rough state right now.

The Lattices of Modal Logics

I have been concerned mostly with the theory of modal propositional logic, but almost every issue that falls under that heading. My initial research was centered around the lattice of modal logics, in particular splittings of that lattice. Later I have worked on the relationship between monomodal and polymodal logic. This work was done in close collaboration with Frank Wolter. The basic result is that the lattice of n-modal logics is for any given number n isomorphic to some interval in the lattice of monomodal logics, and this isomophism is faithful to almost any property of modal logics. This is a very effective tool in producing counterexamples. My book "Tools and Techniques in Modal Logic" (Elsevier, 1999) contains a pretty good picture of my research and research interests in modal logic.

Reduction Functions

Consider the problem whether a formula φ follows from a set of premises Γ in a logic M containing L. It is then known that there is a finite set H, that depends on both A and Γ, such that φ follows from Γ;Δ in L. Suppose now that there is an algorithm to compute Δ from φ and Γ. Then if L is decidable, so is M. Moreover, explicit complexity bounds can be given. If L has the finite model property, then M does, too. Finally, there is a criterium (easy to verify) which allows to deduce that M has interpolation if only L does. I have introduced the method in "Tools and Techniques" and later refined it. It has great potential (the best known space bounds can be derived with its help). Moreover, it is finitistic and so it does away with all the infinite models that make introductory textbooks so heavy.

Modal Predicate Logic

Modal predicate logic is of interest to linguists and philosophers and they have contributed the most to the subject, both in the form of puzzles (the Pierre Puzzle) and in the form of proposals for semantics. The most popular brands are Kripke-semantics and counterpart semantics. Both are known to be highly incomplete. That means in practice that even for systems whose propositional part is Kripke-complete one may not be able to provide semantics of the mentioned kind. Valentin Shehtman and Dimiter Skvortsov have given the first generally complete semantics. Their semantics is based ob hyperdoctrines and is quite uninituitive. Together with Oliver Kutz I have worked on general semantics, first for counterpart logics and then for modal predicate logics (first and second order). We have proved a general completeness theorem and showed how to systematically inperpret this semantics in other semantics.
Marcus Kracht
Last Modified: Wed Feb 14 15:00:00 2005