4 research outputs found
Expressing Ecumenical Systems in the ??-Calculus Modulo Theory
Systems in which classical and intuitionistic logics coexist are called ecumenical. Such a system allows for interoperability and hybridization between classical and constructive propositions and proofs. We study Ecumenical STT, a theory expressed in the logical framework of the ??-calculus modulo theory. We prove soudness and conservativity of four subtheories of Ecumenical STT with respect to constructive and classical predicate logic and simple type theory. We also prove the weak normalization of well-typed terms and thus the consistency of Ecumenical STT
Some axioms for type theories
The -calculus modulo theory is a logical framework in which many
type systems can be expressed as theories. We present such a theory, the theory
, where proofs of several logical systems can be expressed.
Moreover, we identify a sub-theory of corresponding to each of
these systems, and prove that, when a proof in uses only symbols
of a sub-theory, then it is a proof in that sub-theory
Learning Definable Hypotheses on Trees
We study the problem of learning properties of nodes in tree structures.
Those properties are specified by logical formulas, such as formulas from
first-order or monadic second-order logic. We think of the tree as a database
encoding a large dataset and therefore aim for learning algorithms which depend
at most sublinearly on the size of the tree. We present a learning algorithm
for quantifier-free formulas where the running time only depends polynomially
on the number of training examples, but not on the size of the background
structure. By a previous result on strings we know that for general first-order
or monadic second-order (MSO) formulas a sublinear running time cannot be
achieved. However, we show that by building an index on the tree in a linear
time preprocessing phase, we can achieve a learning algorithm for MSO formulas
with a logarithmic learning phase.Comment: Full version of ICDT 2019 pape
A modular construction of type theories
The lambda-Pi-calculus modulo theory is a logical framework in which manytype systems can be expressed as theories. We present such a theory, the theoryU, where proofs of several logical systems can be expressed. Moreover, weidentify a sub-theory of U corresponding to each of these systems, and provethat, when a proof in U uses only symbols of a sub-theory, then it is a proofin that sub-theory