2,374 research outputs found
On Descriptive Complexity, Language Complexity, and GB
We introduce , a monadic second-order language for reasoning about
trees which characterizes the strongly Context-Free Languages in the sense that
a set of finite trees is definable in iff it is (modulo a
projection) a Local Set---the set of derivation trees generated by a CFG. This
provides a flexible approach to establishing language-theoretic complexity
results for formalisms that are based on systems of well-formedness constraints
on trees. We demonstrate this technique by sketching two such results for
Government and Binding Theory. First, we show that {\em free-indexation\/}, the
mechanism assumed to mediate a variety of agreement and binding relationships
in GB, is not definable in and therefore not enforcible by CFGs.
Second, we show how, in spite of this limitation, a reasonably complete GB
account of English can be defined in . Consequently, the language
licensed by that account is strongly context-free. We illustrate some of the
issues involved in establishing this result by looking at the definition, in
, of chains. The limitations of this definition provide some insight
into the types of natural linguistic principles that correspond to higher
levels of language complexity. We close with some speculation on the possible
significance of these results for generative linguistics.Comment: To appear in Specifying Syntactic Structures, papers from the Logic,
Structures, and Syntax workshop, Amsterdam, Sept. 1994. LaTeX source with
nine included postscript figure
Polymonadic Programming
Monads are a popular tool for the working functional programmer to structure
effectful computations. This paper presents polymonads, a generalization of
monads. Polymonads give the familiar monadic bind the more general type forall
a,b. L a -> (a -> M b) -> N b, to compose computations with three different
kinds of effects, rather than just one. Polymonads subsume monads and
parameterized monads, and can express other constructions, including precise
type-and-effect systems and information flow tracking; more generally,
polymonads correspond to Tate's productoid semantic model. We show how to equip
a core language (called lambda-PM) with syntactic support for programming with
polymonads. Type inference and elaboration in lambda-PM allows programmers to
write polymonadic code directly in an ML-like syntax--our algorithms compute
principal types and produce elaborated programs wherein the binds appear
explicitly. Furthermore, we prove that the elaboration is coherent: no matter
which (type-correct) binds are chosen, the elaborated program's semantics will
be the same. Pleasingly, the inferred types are easy to read: the polymonad
laws justify (sometimes dramatic) simplifications, but with no effect on a
type's generality.Comment: In Proceedings MSFP 2014, arXiv:1406.153
Gradual Certified Programming in Coq
Expressive static typing disciplines are a powerful way to achieve
high-quality software. However, the adoption cost of such techniques should not
be under-estimated. Just like gradual typing allows for a smooth transition
from dynamically-typed to statically-typed programs, it seems desirable to
support a gradual path to certified programming. We explore gradual certified
programming in Coq, providing the possibility to postpone the proofs of
selected properties, and to check "at runtime" whether the properties actually
hold. Casts can be integrated with the implicit coercion mechanism of Coq to
support implicit cast insertion a la gradual typing. Additionally, when
extracting Coq functions to mainstream languages, our encoding of casts
supports lifting assumed properties into runtime checks. Much to our surprise,
it is not necessary to extend Coq in any way to support gradual certified
programming. A simple mix of type classes and axioms makes it possible to bring
gradual certified programming to Coq in a straightforward manner.Comment: DLS'15 final version, Proceedings of the ACM Dynamic Languages
Symposium (DLS 2015
Extended Initiality for Typed Abstract Syntax
Initial Semantics aims at interpreting the syntax associated to a signature
as the initial object of some category of 'models', yielding induction and
recursion principles for abstract syntax. Zsid\'o proves an initiality result
for simply-typed syntax: given a signature S, the abstract syntax associated to
S constitutes the initial object in a category of models of S in monads.
However, the iteration principle her theorem provides only accounts for
translations between two languages over a fixed set of object types. We
generalize Zsid\'o's notion of model such that object types may vary, yielding
a larger category, while preserving initiality of the syntax therein. Thus we
obtain an extended initiality theorem for typed abstract syntax, in which
translations between terms over different types can be specified via the
associated category-theoretic iteration operator as an initial morphism. Our
definitions ensure that translations specified via initiality are type-safe,
i.e. compatible with the typing in the source and target language in the
obvious sense. Our main example is given via the propositions-as-types
paradigm: we specify propositions and inference rules of classical and
intuitionistic propositional logics through their respective typed signatures.
Afterwards we use the category--theoretic iteration operator to specify a
double negation translation from the former to the latter. A second example is
given by the signature of PCF. For this particular case, we formalize the
theorem in the proof assistant Coq. Afterwards we specify, via the
category-theoretic iteration operator, translations from PCF to the untyped
lambda calculus
Relational semantics of linear logic and higher-order model-checking
In this article, we develop a new and somewhat unexpected connection between
higher-order model-checking and linear logic. Our starting point is the
observation that once embedded in the relational semantics of linear logic, the
Church encoding of any higher-order recursion scheme (HORS) comes together with
a dual Church encoding of an alternating tree automata (ATA) of the same
signature. Moreover, the interaction between the relational interpretations of
the HORS and of the ATA identifies the set of accepting states of the tree
automaton against the infinite tree generated by the recursion scheme. We show
how to extend this result to alternating parity automata (APT) by introducing a
parametric version of the exponential modality of linear logic, capturing the
formal properties of colors (or priorities) in higher-order model-checking. We
show in particular how to reunderstand in this way the type-theoretic approach
to higher-order model-checking developed by Kobayashi and Ong. We briefly
explain in the end of the paper how his analysis driven by linear logic results
in a new and purely semantic proof of decidability of the formulas of the
monadic second-order logic for higher-order recursion schemes.Comment: 24 pages. Submitte
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