8,432 research outputs found
From IF to BI: a tale of dependence and separation
We take a fresh look at the logics of informational dependence and
independence of Hintikka and Sandu and Vaananen, and their compositional
semantics due to Hodges. We show how Hodges' semantics can be seen as a special
case of a general construction, which provides a context for a useful
completeness theorem with respect to a wider class of models. We shed some new
light on each aspect of the logic. We show that the natural propositional logic
carried by the semantics is the logic of Bunched Implications due to Pym and
O'Hearn, which combines intuitionistic and multiplicative connectives. This
introduces several new connectives not previously considered in logics of
informational dependence, but which we show play a very natural role, most
notably intuitionistic implication. As regards the quantifiers, we show that
their interpretation in the Hodges semantics is forced, in that they are the
image under the general construction of the usual Tarski semantics; this
implies that they are adjoints to substitution, and hence uniquely determined.
As for the dependence predicate, we show that this is definable from a simpler
predicate, of constancy or dependence on nothing. This makes essential use of
the intuitionistic implication. The Armstrong axioms for functional dependence
are then recovered as a standard set of axioms for intuitionistic implication.
We also prove a full abstraction result in the style of Hodges, in which the
intuitionistic implication plays a very natural r\^ole.Comment: 28 pages, journal versio
An Effective Fixpoint Semantics for Linear Logic Programs
In this paper we investigate the theoretical foundation of a new bottom-up
semantics for linear logic programs, and more precisely for the fragment of
LinLog that consists of the language LO enriched with the constant 1. We use
constraints to symbolically and finitely represent possibly infinite
collections of provable goals. We define a fixpoint semantics based on a new
operator in the style of Tp working over constraints. An application of the
fixpoint operator can be computed algorithmically. As sufficient conditions for
termination, we show that the fixpoint computation is guaranteed to converge
for propositional LO. To our knowledge, this is the first attempt to define an
effective fixpoint semantics for linear logic programs. As an application of
our framework, we also present a formal investigation of the relations between
LO and Disjunctive Logic Programming. Using an approach based on abstract
interpretation, we show that DLP fixpoint semantics can be viewed as an
abstraction of our semantics for LO. We prove that the resulting abstraction is
correct and complete for an interesting class of LO programs encoding Petri
Nets.Comment: 39 pages, 5 figures. To appear in Theory and Practice of Logic
Programmin
A Labelled Analytic Theorem Proving Environment for Categorial Grammar
We present a system for the investigation of computational properties of
categorial grammar parsing based on a labelled analytic tableaux theorem
prover. This proof method allows us to take a modular approach, in which the
basic grammar can be kept constant, while a range of categorial calculi can be
captured by assigning different properties to the labelling algebra. The
theorem proving strategy is particularly well suited to the treatment of
categorial grammar, because it allows us to distribute the computational cost
between the algorithm which deals with the grammatical types and the algebraic
checker which constrains the derivation.Comment: 11 pages, LaTeX2e, uses examples.sty and a4wide.st
Proceedings of International Workshop "Global Computing: Programming Environments, Languages, Security and Analysis of Systems"
According to the IST/ FET proactive initiative on GLOBAL COMPUTING, the goal is to obtain techniques (models, frameworks, methods, algorithms) for constructing systems that are flexible, dependable, secure, robust and efficient.
The dominant concerns are not those of representing and manipulating data efficiently but rather those of handling the co-ordination and interaction, security, reliability, robustness, failure modes, and control of risk of the entities in the system and the overall design, description and performance of the system itself.
Completely different paradigms of computer science may have to be developed to tackle these issues effectively. The research should concentrate on systems having the following characteristics: • The systems are composed of autonomous computational entities where activity is not centrally controlled, either because global control is impossible or impractical, or because the entities are created or controlled by different owners.
• The computational entities are mobile, due to the movement of the physical platforms or by movement of the entity from one platform to another.
• The configuration varies over time. For instance, the system is open to the introduction of new computational entities and likewise their deletion.
The behaviour of the entities may vary over time.
• The systems operate with incomplete information about the environment.
For instance, information becomes rapidly out of date and mobility requires information about the environment to be discovered.
The ultimate goal of the research action is to provide a solid scientific foundation for the design of such systems, and to lay the groundwork for achieving effective principles for building and analysing such systems.
This workshop covers the aspects related to languages and programming environments as well as analysis of systems and resources involving 9 projects (AGILE , DART, DEGAS , MIKADO, MRG, MYTHS, PEPITO, PROFUNDIS, SECURE) out of the 13 founded under the initiative. After an year from the start of the projects, the goal of the workshop is to fix the state of the art on the topics covered by the two clusters related to programming environments and analysis of systems as well as to devise strategies and new ideas to profitably continue the research effort towards the overall objective of the initiative.
We acknowledge the Dipartimento di Informatica and Tlc of the University of Trento, the Comune di Rovereto, the project DEGAS for partially funding the event and the Events and Meetings Office of the University of Trento for the valuable collaboration
Constructive Provability Logic
We present constructive provability logic, an intuitionstic modal logic that
validates the L\"ob rule of G\"odel and L\"ob's provability logic by permitting
logical reflection over provability. Two distinct variants of this logic, CPL
and CPL*, are presented in natural deduction and sequent calculus forms which
are then shown to be equivalent. In addition, we discuss the use of
constructive provability logic to justify stratified negation in logic
programming within an intuitionstic and structural proof theory.Comment: Extended version of IMLA 2011 submission of the same titl
From coinductive proofs to exact real arithmetic: theory and applications
Based on a new coinductive characterization of continuous functions we
extract certified programs for exact real number computation from constructive
proofs. The extracted programs construct and combine exact real number
algorithms with respect to the binary signed digit representation of real
numbers. The data type corresponding to the coinductive definition of
continuous functions consists of finitely branching non-wellfounded trees
describing when the algorithm writes and reads digits. We discuss several
examples including the extraction of programs for polynomials up to degree two
and the definite integral of continuous maps
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Proof-theoretic investigations into integrated logical and functional programming
This thesis is a proof-theoretic investigation of logic programming based on hereditary Harrop logic (as in lambdaProlog). After studying various proof systems for the first-order hereditary Harrop logic, we define the proof-theoretic semantics of a logic LFPL, intended as the basis of logic programming with functions, which extends higher-order hereditary Harrop logic by providing definition mechanisms for functions in such a way that the logical specification of the function rather than the function may be used in proof search. In Chap. 3, we define, for the first-order hereditary Harrop fragment of LJ, the class of uniform linear focused (ULF) proofs (suitable for goal-directed search with backchaining and unification) and show that the ULF-proofs are in 1-1 correspondence with the expanded normal deductions, in Prawitz's sense. We give a system of proof-term annotations for LJ-proofs (where proof-terms uniquely represent proofs). We define a rewriting system on proof-terms (where rules represent a subset of Kleene's permutations in LJ) and show that: its irreducible proof- terms are those representing ULF-proofs; it is weakly normalising. We also show that the composition of Prawitz's mappings between LJ and NJ, restricted to ULF-proofs, is the identity. We take the view of logic programming where: a program P is a set of formulae; a goal G is a formula; and the different means of achieving G w.r.t. P correspond to the expanded normal deductions of G from the assumptions in P (rather than the traditional view, whereby the different means of goal-achievement correspond to the different answer substitutions). LFPL is defined in Chap. 4, by means of a sequent calculus. As in LeFun, it extends logic programming with functions and provides mechanisms for defining names for functions, maintaining proof search as the computation mechanism (contrary to languages such as ALF, Babel, Curry and Escher, based on equational logic, where the computation mechanism is some form of rewriting). LFPL also allows definitions for declaring logical properties of functions, called definitions of dependent type. Such definitions are of the form: (f,x) =def(A, w) : EX:RF, where f is a name for A and x is a name for w, a proof-term witnessing that the formula [A/x]F holds (i.e. A meets the specification Ex:rF). When searching for proofs, it may suffice to use the formula [A/x]F rather than A itself. We present an interpretation of LFPL into NNlambdanorm, a natural deduction system for hereditary Harrop logic with lambda-terms. The means of goal-achievement in LFPL are interpreted in NNlambdanorm essentially by cut-elimination, followed by an interpretation of cut-free sequent calculus proofs as normal deductions. We show that the use of definitions of dependent type may speed up proof search because the equivalent proofs using no such definitions may be much longer and because normalisation may be done lazily, since not all parts of the proof need to be exhibited. We sketch two methods for implementing LFPL, based on goal-directed proof search, differing in the mechanism for selecting definitions of dependent type on which to backchain. We discuss techniques for handling the redundancy arising from the equivalence of each proof using such a definition to one using no such definitions
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
PSPACE Bounds for Rank-1 Modal Logics
For lack of general algorithmic methods that apply to wide classes of logics,
establishing a complexity bound for a given modal logic is often a laborious
task. The present work is a step towards a general theory of the complexity of
modal logics. Our main result is that all rank-1 logics enjoy a shallow model
property and thus are, under mild assumptions on the format of their
axiomatisation, in PSPACE. This leads to a unified derivation of tight
PSPACE-bounds for a number of logics including K, KD, coalition logic, graded
modal logic, majority logic, and probabilistic modal logic. Our generic
algorithm moreover finds tableau proofs that witness pleasant proof-theoretic
properties including a weak subformula property. This generality is made
possible by a coalgebraic semantics, which conveniently abstracts from the
details of a given model class and thus allows covering a broad range of logics
in a uniform way
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