The IO and OI hierarchies revisited

We study languages of λ-terms generated by IO and OI unsafe grammars. These languages can be used to model meaning representations in the formal semantics of natural languages following the tradition of Montague [19]. Using techniques pertaining to the denotational semantics of the simply typed λ-calculus, we show that the emptiness and membership problems for both types of grammars are decidable. In the course of the proof of the decidability results for OI, we identify a decidable variant of the λ-definability problem, and prove a stronger form of Statman's finite completeness Theorem [28].

[1]  Giulio Manzonetto,et al.  Loader and Urzyczyn Are Logically Related , 2012, ICALP.

[2]  Christopher H. Broadbent.  The Limits of Decidability for First Order Logic on CPDA Graphs , 2012, STACS.

[3]  Axel Haddad.  IO vs OI in Higher-Order Recursion Schemes , 2012, FICS.

[4]  Sylvain Salvati.  On the Membership Problem for Non-Linear Abstract Categorial Grammars , 2010, J. Log. Lang. Inf..

[5]  Igor Walukiewicz,et al.  Recursive Schemes, Krivine Machines, and Collapsible Pushdown Automata , 2012, RP.

[6]  Sylvain Salvati,et al.  A Datalog Recognizer for Almost Affine λ-CFGs , 2011, MOL.

[7]  Joost Engelfriet,et al.  Iterated Stack Automata and Complexity Classes , 1991, Inf. Comput..

[8]  Kazushige Terui.  Semantic Evaluation, Intersection Types and Complexity of Simply Typed Lambda Calculus , 2012, RTA.

[9]  Los Angeles,et al.  Generating Copies: An investigation into structural identity in language and grammar , 2006 .

[10]  Richard Statman,et al.  Completeness, invariance and λ-definability , 1982, Journal of Symbolic Logic.

[11]  C.-H. Luke Ong,et al.  On Model-Checking Trees Generated by Higher-Order Recursion Schemes , 2006, 21st Annual IEEE Symposium on Logic in Computer Science (LICS'06).

[12]  Philippe de Groote,et al.  Towards Abstract Categorial Grammars , 2001, ACL.

[13]  Reinhard Muskens,et al.  Lambda grammars and the syntax-semantics interface , 2001 .

[14]  Yiannis N. Moschovakis,et al.  Sense and denotation as algorithm and value , 1993 .

[15]  Makoto Kanazawa,et al.  Parsing and Generation as Datalog Queries , 2007, ACL.

[16]  Pawel Urzyczyn,et al.  Higher-Order Pushdown Trees Are Easy , 2002, FoSSaCS.

[17]  Kyle Johnson,et al.  A step-by-step guide to ellipsis resolution , 2001 .

[18]  Gregory M. Kobele,et al.  Generating copies : an investigation into structural identity in language and grammar , 2006 .

[19]  Pawel Parys.  On the Significance of the Collapse Operation , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

[20]  Michael J. Fischer,et al.  Grammars with Macro-Like Productions , 1968, SWAT.

[21]  Klaus Aehlig,et al.  Safety Is not a Restriction at Level 2 for String Languages , 2005, FoSSaCS.

[22]  Sylvain Salvati.  Recognizability in the Simply Typed Lambda-Calculus , 2009, WoLLIC.

[23]  Ralph Loader.  The Undecidability of λ-Definability , 2001 .

[24]  Richard Statman,et al.  On Statman's Finite Completeness Theorem , 1992 .

[25]  J. Roger Hindley,et al.  Lambda-Calculus and Combinators: An Introduction , 2008 .

[26]  Marcus Kracht,et al.  Semilinearity as a Syntactic Invariant , 1996, LACL.

[27]  Roberto M. Amadio,et al.  Domains and lambda-calculi , 1998, Cambridge tracts in theoretical computer science.

[28]  Gregory M. Kobele,et al.  CoTAGs and ACGs , 2012, LACL.

[29]  Ekaterina Lebedeva,et al.  Expressing Discourse Dynamics Through Continuations , 2012 .

[30]  Philippe de Groote,et al.  Towards a Montagovian Account of Dynamics , 2006 .

[31]  Werner Damm,et al.  The IO- and OI-Hierarchies , 1982, Theor. Comput. Sci..

[32]  Philippe de Groote,et al.  Presupposition Accommodation as Exception Handling , 2010, SIGDIAL Conference.

[33]  Alfred V. Aho,et al.  Indexed Grammars—An Extension of Context-Free Grammars , 1967, SWAT.

[34]  Igor Walukiewicz,et al.  An alternate proof of Statman's finite completeness theorem , 2012, Inf. Process. Lett..

[35]  Roberto M. Amadio,et al.  Domains and Lambda-Calculi (Cambridge Tracts in Theoretical Computer Science) , 2008 .

[36]  William Blum,et al.  The Safe Lambda Calculus , 2007, TLCA.

[37]  R. Montague.  Formal philosophy; selected papers of Richard Montague , 1974 .

[38]  Iris van Rooij,et al.  The Tractable Cognition Thesis , 2008, Cogn. Sci..