208,682 research outputs found
Relational Parametricity and Separation Logic
Separation logic is a recent extension of Hoare logic for reasoning about
programs with references to shared mutable data structures. In this paper, we
provide a new interpretation of the logic for a programming language with
higher types. Our interpretation is based on Reynolds's relational
parametricity, and it provides a formal connection between separation logic and
data abstraction
Applications of Finite Model Theory: Optimisation Problems, Hybrid Modal Logics and Games.
There exists an interesting relationships between two seemingly distinct fields: logic from the field of Model Theory, which deals with the truth of statements about discrete structures; and Computational Complexity, which deals with the classification of problems by how much of a particular computer resource is required in order to compute a solution. This relationship is known as Descriptive Complexity and it is the primary application of the tools from Model Theory when they are restricted to the finite; this restriction is commonly called Finite Model Theory.
In this thesis, we investigate the extension of the results of Descriptive Complexity from classes of decision problems to classes of optimisation problems. When dealing with decision problems the natural mapping from true and false in logic to yes and no instances of a problem is used but when dealing with optimisation problems, other features of a logic need to be used. We investigate what these features are and provide results in the form of logical frameworks that can be used for describing optimisation problems in particular classes, building on the existing research into this area.
Another application of Finite Model Theory that this thesis investigates is the relative expressiveness of various fragments of an extension of modal logic called hybrid modal logic. This is achieved through taking the Ehrenfeucht-Fraïssé game from Model Theory and modifying it so that it can be applied to hybrid modal logic. Then, by developing winning strategies for the players in the game, results are obtained that show strict hierarchies of expressiveness for fragments of hybrid modal logic that are generated by varying the quantifier depth and the number of proposition and nominal symbols available
From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions
Permissive-Nominal Logic (PNL) extends first-order predicate logic with
term-formers that can bind names in their arguments. It takes a semantics in
(permissive-)nominal sets. In PNL, the forall-quantifier or lambda-binder are
just term-formers satisfying axioms, and their denotation is functions on
nominal atoms-abstraction.
Then we have higher-order logic (HOL) and its models in ordinary (i.e.
Zermelo-Fraenkel) sets; the denotation of forall or lambda is functions on full
or partial function spaces.
This raises the following question: how are these two models of binding
connected? What translation is possible between PNL and HOL, and between
nominal sets and functions?
We exhibit a translation of PNL into HOL, and from models of PNL to certain
models of HOL. It is natural, but also partial: we translate a restricted
subsystem of full PNL to HOL. The extra part which does not translate is the
symmetry properties of nominal sets with respect to permutations. To use a
little nominal jargon: we can translate names and binding, but not their
nominal equivariance properties. This seems reasonable since HOL---and ordinary
sets---are not equivariant.
Thus viewed through this translation, PNL and HOL and their models do
different things, but they enjoy non-trivial and rich subsystems which are
isomorphic
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
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