Linguistics

Contains reports on three research projects.U. S. Air Force (Electronics Systems Division) under Contract AF 19(628)-2487Joint Services Electronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DA 36-039-AMC-03200(E)National Science Foundation (Grant GK-835)National Institutes of Health (Grant 2 P01 MH-04737-06)National Aeronautics and Space Administration (Grant NsG-496

Recent Advances in Σ-definability over Continuous Data Types

The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Sigma-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the reals by the structure of hereditarily finite sets. In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals. We prove Engeler's Lemma for Sigma-definability over the reals without the equality test which relates Sigma-definability with definability in the constructive infinitary language L_{omega_1 omega}. Thus, a relation over the real numbers is Sigma-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas. This result reveals computational aspects of Sigma-definability and also gives topological characterisation of Sigma-definable relations over the reals without the equality test. We also illustrate how computability over the real numbers can be expressed in the language of Sigma-formulas

A Chomsky-Schützenberger-Stanley type characterization of the class of slender context-free languages

Slender context-free languages have a complete algebraic characterization by L. Ilie in [13]. In this paper we give another characterization of this class of languages. In particular, using linear Dyck languages instead of unrestricted ones, we obtain a Chomsky-Schützenberger-Stanley type characterization of slender context-free languages

Fixed Points on Abstract Structures without the Equality Test

In this paper we present a study of definability properties of fixed points of effective operators on abstract structures without the equality test. In particular we prove that Gandy theorem holds for abstract structures. This provides a useful tool for dealing with recursive definitions using Sigma-formulas. One of the applications of Gandy theorem in the case of the reals without the equality test is that it allows us to define universal Sigma-predicates. It leads to a topological characterisation of Sigma-relations on |R

Analyzing program analyses

We want to prove that a static analysis of a given program is complete, namely, no imprecision arises when asking some query on the program behavior in the concrete (i.e., for its concrete semantics) or in the abstract (i.e., for its abstract interpretation). Completeness proofs are therefore useful to assign confidence to alarms raised by static analyses. We introduce the completeness class of an abstraction as the set of all programs for which the abstraction is complete. Our first result shows that for any nontrivial abstraction, its completeness class is not recursively enumerable. We then introduce a stratified deductive system a2A to prove the completeness of program analyses over an abstract domain A. We prove the soundness of the deductive system. We observe that the only sources of incompleteness are assignments and Boolean tests \u2014 unlikely a common belief in static analysis, joins do not induce incompleteness. The first layer of this proof system is generic, abstraction-agnostic, and it deals with the standard constructs for program composition, that is, sequential composition, branching and guarded iteration. The second layer is instead abstraction-specific: the designer of an abstract domain A provides conditions for completeness in A of assignments and Boolean tests which have to be checked by a suitable static analysis or assumed in the completeness proof as hypotheses. We instantiate the second layer of this proof system first with a generic nonrelational abstraction in order to provide a sound rule for the completeness of assignments. Orthogonally, we instantiate it to the numerical abstract domains of Intervals and Octagons, providing necessary and sufficient conditions for the completeness of their Boolean tests and of assignments for Octagons