The ternary relation B(x,y,z) of betweenness states that an element y is between the elements x and z, in some sense depending on the considered structure. In a partially ordered set (N,≤), B(x,y,z):⇔x<y<z∨z<y<x, and the corresponding betweenness structure is (N,B). The class of betweenness structures of linear orders is first-order definable. That of partial orders is monadic second-order definable. An order-theoretic tree is a partial order such that the set of elements larger that any element is linearly ordered and any two elements have an upper-bound. Finite or infinite rooted trees ordered by the ancestor relation are order-theoretic trees. In an order-theoretic tree, B(x,y,z) means that x<y<z or z<y<x or x<y≤x⊔z or z<y≤x⊔z, where x⊔z is the least upper-bound of incomparable elements x and z. In a previous article, we established that the corresponding class of betweenness structures is monadic second-order definable.We prove here that the induced substructures of the betweenness structures of the countable order-theoretic trees form a monadic second-order definable class, denoted by IBO. The proof uses a variant of cographs, the partitioned probe cographs, and their known six finite minimal excluded induced subgraphs called the bounds of the class. This proof links two apparently unrelated topics: cographs and order-theoretic trees.However, the class IBO has finitely many bounds, i.e., minimal excluded finite induced substructures. Hence it is first-order definable. The proof of finiteness uses well-quasi-orders and does not provide the finite list of bounds. Hence, the associated first-order defining sentence is not known.
We consider nondeterministic higher-order recursion schemes as recognizers of languages of finite words or finite trees. We propose a type system that allows to solve the simultaneous-unboundedness problem (SUP) for schemes, which asks, given a set of letters A and a scheme G, whether it is the case that for every number n the scheme accepts a word (a tree) in which every letter from A appears at least n times. Using this type system we prove that SUP is (m-1)-EXPTIME-complete for word-recognizing schemes of order m, and m-EXPTIME-complete for tree-recognizing schemes of order m. Moreover, we establish the reflection property for SUP: out of an input scheme G one can create its enhanced version that recognizes the same language but is aware of the answer to SUP.
In this paper, we facilitate the reasoning about impure programming languages, by annotating terms with “decorations”that describe what computational (side) effect evaluation of a term may involve. In a point-free categorical language,called the “decorated logic”, we formalize the mutable state and the exception effects first separately, exploiting anice duality between them, and then combined. The combined decorated logic is used as the target language forthe denotational semantics of the IMP+Exc imperative programming language, and allows us to prove equivalencesbetween programs written in IMP+Exc. The combined logic is encoded in Coq, and this encoding is used to certifysome program equivalence proofs.
Reseña del libro:Grupo de Intervención y Responsabilidad Social (2014). Intervención social y el debate sobre lo público: reflexiones conceptuales y casos locales. Colección "El Sur es Cielo Roto", No. 9. Cali, Colombia: Universidad Icesi, Facultad de Derecho y Ciencias Sociales. 258 pp.
In arXiv:1207.0332 [cs.LO] was proposed a graphic lambda calculus formalism, which has sectors corresponding to untyped lambda calculus and emergent algebras. Here we explore the sector covering knot diagrams, which are constructed as macros over the graphic lambda calculus.
One of the central open questions in bounded arithmetic is whether Buss'
hierarchy of theories of bounded arithmetic collapses or not. In this paper, we
reformulate Buss' theories using free logic and conjecture that such theories
are easier to handle. To show this, we first prove that Buss' theories prove
consistencies of induction-free fragments of our theories whose formulae have
bounded complexity. Next, we prove that although our theories are based on an
apparently weaker logic, we can interpret theories in Buss' hierarchy by our
theories using a simple translation. Finally, we investigate finitistic G\"odel
sentences in our systems in the hope of proving that a theory in a lower level
of Buss' hierarchy cannot prove consistency of induction-free fragments of our
theories whose formulae have higher complexity.
Logics for security protocol analysis require the formalization of an
adversary model that specifies the capabilities of adversaries. A common model
is the Dolev-Yao model, which considers only adversaries that can compose and
replay messages, and decipher them with known keys. The Dolev-Yao model is a
useful abstraction, but it suffers from some drawbacks: it cannot handle the
adversary knowing protocol-specific information, and it cannot handle
probabilistic notions, such as the adversary attempting to guess the keys. We
show how we can analyze security protocols under different adversary models by
using a logic with a notion of algorithmic knowledge. Roughly speaking,
adversaries are assumed to use algorithms to compute their knowledge; adversary
capabilities are captured by suitable restrictions on the algorithms used. We
show how we can model the standard Dolev-Yao adversary in this setting, and how
we can capture more general capabilities including protocol-specific knowledge
and guesses.
Relational data exchange is the problem of translating relational data from a
source schema into a target schema, according to a specification of the
relationship between the source data and the target data. One of the basic
issues is how to answer queries that are posed against target data. While
consensus has been reached on the definitive semantics for monotonic queries,
this issue turned out to be considerably more difficult for non-monotonic
queries. Several semantics for non-monotonic queries have been proposed in the
past few years. This article proposes a new semantics for non-monotonic
queries, called the GCWA*-semantics. It is inspired by semantics from the area
of deductive databases. We show that the GCWA*-semantics coincides with the
standard open world semantics on monotonic queries, and we further explore the
(data) complexity of evaluating non-monotonic queries under the
GCWA*-semantics. In particular, we introduce a class of schema mappings for
which universal queries can be evaluated under the GCWA*-semantics in
polynomial time (data complexity) on the core of the universal solutions.
Analytic proof calculi are introduced for box and diamond fragments of basic
modal fuzzy logics that combine the Kripke semantics of modal logic K with the
many-valued semantics of G\"odel logic. The calculi are used to establish
completeness and complexity results for these fragments.
The Description Logic EL has recently drawn considerable attention since, on
the one hand, important inference problems such as the subsumption problem are
polynomial. On the other hand, EL is used to define large biomedical
ontologies. Unification in Description Logics has been proposed as a novel
inference service that can, for example, be used to detect redundancies in
ontologies. The main result of this paper is that unification in EL is
decidable. More precisely, EL-unification is NP-complete, and thus has the same
complexity as EL-matching. We also show that, w.r.t. the unification type, EL
is less well-behaved: it is of type zero, which in particular implies that
there are unification problems that have no finite complete set of unifiers.
We study tree languages that can be defined in \Delta_2 . These are tree
languages definable by a first-order formula whose quantifier prefix is forall
exists, and simultaneously by a first-order formula whose quantifier prefix is
. For the quantifier free part we consider two signatures, either the
descendant relation alone or together with the lexicographical order relation
on nodes. We provide an effective characterization of tree and forest languages
definable in \Delta_2 . This characterization is in terms of algebraic
equations. Over words, the class of word languages definable in \Delta_2 forms
a robust class, which was given an effective algebraic characterization by Pin
and Weil.
Refinement types sharpen systems of simple and dependent types by offering
expressive means to more precisely classify well-typed terms. We present a
system of refinement types for LF in the style of recent formulations where
only canonical forms are well-typed. Both the usual LF rules and the rules for
type refinements are bidirectional, leading to a straightforward proof of
decidability of typechecking even in the presence of intersection types.
Because we insist on canonical forms, structural rules for subtyping can now be
derived rather than being assumed as primitive. We illustrate the expressive
power of our system with examples and validate its design by demonstrating a
precise correspondence with traditional presentations of subtyping. Proof
irrelevance provides a mechanism for selectively hiding the identities of terms
in type theories. We show that LF refinement types can be interpreted as
predicates using proof irrelevance, establishing a uniform relationship between
two previously studied concepts in type theory. The interpretation and its
correctness proof are surprisingly complex, lending support to the claim that
refinement types are a fundamental construct rather than just a convenient
surface syntax for certain uses of proof irrelevance.
We propose a new algorithm for minimal unsatisfiable core extraction, based on a deeper exploration of resolution-refutation properties. We provide experimental results on formal verification benchmarks confirming that our algorithm finds smaller cores than suboptimal algorithms; and that it runs faster than those algorithms that guarantee minimality of the core. (A more complete version of this paper may be found at arXiv.org/pdf/cs.LO/0605085.)
In [M. Pedicini and F. Quaglia. A parallel implementation for optimal lambda-calculus reduction PPDP '00: Proceedings of the 2nd ACM SIGPLAN international conference on Principles and practice of declarative programming, pages 3-14, ACM, 2000, M. Pedicini and F. Quaglia. PELCR: Parallel environment for optimal lambda-calculus reduction. CoRR, cs.LO/0407055, accepted for publication on TOCL, ACM, 2005], PELCR has been introduced as an implementation derived from the Geometry of Interaction in order to perform virtual reduction on parallel/distributed computing systems. In this paper we provide an extension of PELCR with computational effects based on directed virtual reduction [V. Danos, M. Pedicini, and L. Regnier. Directed virtual reductions. In M. Bezem D. van Dalen, editor, LNCS 1258, pages 76-88. EACSL, Springer Verlag, 1997], namely a restriction of virtual reduction [V. Danos and L. Regnier. Local and asynchronous beta-reduction (an analysis of Girard's EX-formula). LICS, pages 296-306. IEEE Computer Society Press, 1993], which is a particular way to compute the Geometry of Interaction [J.-Y. Girard. Geometry of interaction 1: Interpretation of system F. In R. Ferro, et al. editors Logic Colloquium '88, pages 221-260. North-Holland, 1989] in analogy with Lamping's optimal reduction [J. Lamping. An algorithm for optimal lambda calculus reduction. In Proc. of 17th Annual ACM Symposium on Principles of Programming Languages. ACM, San Francisco, California, pages 16-30, 1990]. Moreover, the proposed solution preserves scalability of the parallelism arising from local and asynchronous reduction as studied in [M. Pedicini and F. Quaglia. PELCR: Parallel environment for optimal lambda-calculus reduction. CoRR, cs.LO/0407055, accepted for publication on TOCL, ACM, 2005].