Hans Joerg Tiede

Lambek Calculus Proofs and Tree Automata

We investigate natural deduction proofs of the Lambek calculus from the point of view of tree automata. The main result is that the set of proofs of the Lambek calculus cannot be accepted by a finite tree automaton. The proof is extended to cover the proofs used by grammars based on the Lambek calculus, which typically use only a subset of the set of all proofs. While Lambek grammars can assign regular tree languages as structural descriptions, there exist Lambek grammars that assign non-regular structural descriptions, both when considering normal and non-normal proof trees. Combining the results of Pentus (1993) and Thatcher (1967), we can conclude that Lambek grammars, although generating only context-free languages, can extend the strong generative capacity of context-free grammars. Furthermore, we show that structural descriptions that disregard the use of introduction rules cannot be used for a compositional semantics following the Curry-Howard isomorphism.

Inessential Features, Ineliminable Features, and Modal Logics for Model Theoretic Syntax

While monadic second-order logic (MSO) has played a prominent role in model theoretic syntax, modal logics have been used in this context since its inception. When comparing propositional dynamic logic (PDL) to MSO over trees, Kracht (1997) noted that there are tree languages that can be defined in MSO that can only be defined in PDL by adding new features whose distribution is predictable. He named such features “inessential features”. We show that Kracht’s observation can be extended to other modal logics of trees in two ways. First, we demonstrate that for each stronger logic, there exists a tree language that can only be defined in a weaker logic with inessential features. Second, we show that any tree language that can be defined in a stronger logic, but not in some weaker logic, can be defined with inessential features. Additionally, we consider Kracht’s definition of inessential features more closely. It turns out that there are features whose distribution can be predicted, but who fail to be inessential in Kracht’s sense. We will look at ways to modify his definition.

Proof Theory and Formal Grammars: Applications of Normalization

We survey results about the proof theoretic approach to formal grammars, concentrating on those that can be obtained from the normalization theorem. In particular, we survey results that relate proof theoretic grammars to context-free grammars and results on the strong generative capacity of proof theoretic grammars. The paper concludes with a list of open problems concerning different issues of complexity of proof theoretical grammars.