343 research outputs found
On maximal intermediate predicate constructive logics
We extend to the predicate frame a previous characterization of the
maximal intermediate propositional constructive logics. This provides
a technique to get maximal intermediate predicate constructive logics
starting from suitable sets of classically valid predicate formulae we
call maximal nonstandard predicate constructive logics. As an exam-
ple of this technique, we exhibit two maximal intermediate predicate
constructive logics, yet leaving open the problem of stating whether
the two logics are distinct. Further properties of these logics will be
also investigated
Syntactic Interpolation for Tense Logics and Bi-Intuitionistic Logic via Nested Sequents
We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting to embeddings, semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants
Synthesizing nested relational queries from implicit specifications: via model theory and via proof theory
Derived datasets can be defined implicitly or explicitly. An implicit
definition (of dataset O in terms of datasets I) is a logical specification
involving the source data I and the interface data O. It is a valid definition
of O in terms of I, if any two models of the specification agreeing on I agree
on O. In contrast, an explicit definition is a query that produces O from I.
Variants of Beth's theorem state that one can convert implicit definitions to
explicit ones. Further, this conversion can be done effectively given a proof
witnessing implicit definability in a suitable proof system.
We prove the analogous implicit-to-explicit result for nested relations:
implicit definitions, given in the natural logic for nested relations, can be
converted to explicit definitions in the nested relational calculus (NRC) We
first provide a model-theoretic argument for this result, which makes some
additional connections that may be of independent interest. between NRC
queries, interpretations, a standard mechanisms for defining
structure-to-structure translation in logic, and between interpretations and
implicit to definability "up to unique isomorphism". The latter connection
makes use of a variation of a result of Gaifman concerning "relatively
categorical" theories.
We also provide a proof-theoretic result that provides an effective argument:
from a proof witnessing implicit definability, we can efficiently produce an
NRC definition. This will involve introducing the appropriate proof system for
reasoning with nested sets, along with some auxiliary Beth-type results for
this system. As a consequence, we can effectively extract rewritings of NRC
queries in terms of NRC views, given a proof witnessing that the query is
determined by the views.Comment: arXiv admin note: substantial text overlap with arXiv:2209.08299,
arXiv:2005.0650
Synthesizing nested relational queries from implicit specifications: via model theory and via proof theory
Derived datasets can be defined implicitly or explicitly. An implicit definition (of dataset O in terms of datasets ) is a logical specification involving two distinguished sets of relational symbols. One set of relations is for the âsource dataâ , and the other is for the âinterface dataâ O. Such a specification is a valid definition of O in terms of , if any two models of the specification agreeing on agree on O. In contrast, an explicit definition is a transformation (or âqueryâ below) that produces O from . Variants of Bethâs theorem [Bet53] state that one can convert implicit definitions to explicit ones. Further, this conversion can be done effectively given a proof witnessing implicit definability in a suitable proof system. We prove the analogous implicit-to-explicit result for nested relations: implicit definitions, given in the natural logic for nested relations, can be converted to explicit definitions in the nested relational calculus (NRC). We first provide a model-theoretic argument for this result, which makes some additional connections that may be of independent interest, between NRC queries, interpretations, a standard mechanism for defining structure-to-structure translation in logic, and between interpretations and implicit to definability âup to unique isomorphismâ. The latter connection uses a variation of a result of Gaifman concerning ârelatively categoricalâ theories. We also provide a proof-theoretic result that provides an effective argument: from a proof witnessing implicit definability, we can efficiently produce an NRC definition. This will involve introducing the appropriate proof system for reasoning with nested sets, along with some auxiliary Beth-type results for this system. As a consequence, we can effectively extract rewritings of NRC queries in terms of NRC views, given a proof witnessing that the query is determined by the views
Lewis meets Brouwer: constructive strict implication
C. I. Lewis invented modern modal logic as a theory of "strict implication".
Over the classical propositional calculus one can as well work with the unary
box connective. Intuitionistically, however, the strict implication has greater
expressive power than the box and allows to make distinctions invisible in the
ordinary syntax. In particular, the logic determined by the most popular
semantics of intuitionistic K becomes a proper extension of the minimal normal
logic of the binary connective. Even an extension of this minimal logic with
the "strength" axiom, classically near-trivial, preserves the distinction
between the binary and the unary setting. In fact, this distinction and the
strong constructive strict implication itself has been also discovered by the
functional programming community in their study of "arrows" as contrasted with
"idioms". Our particular focus is on arithmetical interpretations of the
intuitionistic strict implication in terms of preservativity in extensions of
Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years
later
Logics with rigidly guarded data tests
The notion of orbit finite data monoid was recently introduced by Bojanczyk
as an algebraic object for defining recognizable languages of data words.
Following Buchi's approach, we introduce a variant of monadic second-order
logic with data equality tests that captures precisely the data languages
recognizable by orbit finite data monoids. We also establish, following this
time the approach of Schutzenberger, McNaughton and Papert, that the
first-order fragment of this logic defines exactly the data languages
recognizable by aperiodic orbit finite data monoids. Finally, we consider
another variant of the logic that can be interpreted over generic structures
with data. The data languages defined in this variant are also recognized by
unambiguous finite memory automata
On the complexity of the disjunction property in intuitionistic and modal logics
In this paper we study the complexity of disjunction property for Intuitionistic Logic, the modal logics S3, S4.1, Grzegorczyk Logic, Godel-Lob Logic and the intuitionistic counterpart of the modal logic K. For S4 we even prove the feasible interpolation theorem and we provide a lower bound for the length of proofs. The techniques we use do not require to prove structural properties of the calculi in hand, such as the Cut-elimination Theorem or the Normalization Theorem. This is a key-point of our approach, since it allows us to treat logics for which only Hilbert-style characterizations are known
Datalog and Constraint Satisfaction with Infinite Templates
On finite structures, there is a well-known connection between the expressive
power of Datalog, finite variable logics, the existential pebble game, and
bounded hypertree duality. We study this connection for infinite structures.
This has applications for constraint satisfaction with infinite templates. If
the template Gamma is omega-categorical, we present various equivalent
characterizations of those Gamma such that the constraint satisfaction problem
(CSP) for Gamma can be solved by a Datalog program. We also show that
CSP(Gamma) can be solved in polynomial time for arbitrary omega-categorical
structures Gamma if the input is restricted to instances of bounded treewidth.
Finally, we characterize those omega-categorical templates whose CSP has
Datalog width 1, and those whose CSP has strict Datalog width k.Comment: 28 pages. This is an extended long version of a conference paper that
appeared at STACS'06. In the third version in the arxiv we have revised the
presentation again and added a section that relates our results to
formalizations of CSPs using relation algebra
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