744 research outputs found
On the mathematical synthesis of equational logics
We provide a mathematical theory and methodology for synthesising equational
logics from algebraic metatheories. We illustrate our methodology by means of
two applications: a rational reconstruction of Birkhoff's Equational Logic and
a new equational logic for reasoning about algebraic structure with
name-binding operators.Comment: Final version for publication in Logical Methods in Computer Scienc
Relational Parametricity and Control
We study the equational theory of Parigot's second-order
λμ-calculus in connection with a call-by-name continuation-passing
style (CPS) translation into a fragment of the second-order λ-calculus.
It is observed that the relational parametricity on the target calculus induces
a natural notion of equivalence on the λμ-terms. On the other hand,
the unconstrained relational parametricity on the λμ-calculus turns
out to be inconsistent with this CPS semantics. Following these facts, we
propose to formulate the relational parametricity on the λμ-calculus
in a constrained way, which might be called ``focal parametricity''.Comment: 22 pages, for Logical Methods in Computer Scienc
Dual-Context Calculi for Modal Logic
We present natural deduction systems and associated modal lambda calculi for
the necessity fragments of the normal modal logics K, T, K4, GL and S4. These
systems are in the dual-context style: they feature two distinct zones of
assumptions, one of which can be thought as modal, and the other as
intuitionistic. We show that these calculi have their roots in in sequent
calculi. We then investigate their metatheory, equip them with a confluent and
strongly normalizing notion of reduction, and show that they coincide with the
usual Hilbert systems up to provability. Finally, we investigate a categorical
semantics which interprets the modality as a product-preserving functor.Comment: Full version of article previously presented at LICS 2017 (see
arXiv:1602.04860v4 or doi: 10.1109/LICS.2017.8005089
Partial Horn logic and cartesian categories
A logic is developed in which function symbols are allowed to represent partial functions. It has the usual rules of logic (in the form of a sequent calculus) except that the substitution rule has to be modified. It is developed here in its minimal form, with equality and conjunction, as “partial Horn logic”.
Various kinds of logical theory are equivalent: partial Horn theories, “quasi-equational” theories (partial Horn theories without predicate symbols), cartesian theories and essentially algebraic theories.
The logic is sound and complete with respect to models in , and sound with respect to models in any cartesian (finite limit) category.
The simplicity of the quasi-equational form allows an easy predicative constructive proof of the free partial model theorem for cartesian theories: that if a theory morphism is given from one cartesian theory to another, then the forgetful (reduct) functor from one model category to the other has a left adjoint.
Various examples of quasi-equational theory are studied, including those of cartesian categories and of other classes of categories. For each quasi-equational theory another, , is constructed, whose models are cartesian categories equipped with models of . Its initial model, the “classifying category” for , has properties similar to those of the syntactic category, but more precise with respect to strict cartesian functors
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