253 research outputs found
A Complete Axiomatisation for Quantifier-Free Separation Logic
We present the first complete axiomatisation for quantifier-free separation
logic. The logic is equipped with the standard concrete heaplet semantics and
the proof system has no external feature such as nominals/labels. It is not
possible to rely completely on proof systems for Boolean BI as the concrete
semantics needs to be taken into account. Therefore, we present the first
internal Hilbert-style axiomatisation for quantifier-free separation logic. The
calculus is divided in three parts: the axiomatisation of core formulae where
Boolean combinations of core formulae capture the expressivity of the whole
logic, axioms and inference rules to simulate a bottom-up elimination of
separating connectives, and finally structural axioms and inference rules from
propositional calculus and Boolean BI with the magic wand
Internal Calculi for Separation Logics
We present a general approach to axiomatise separation logics with heaplet semantics with no external features such as nominals/labels. To start with, we design the first (internal) Hilbert-style axiomatisation for the quantifier-free separation logic SL(?, -*). We instantiate the method by introducing a new separation logic with essential features: it is equipped with the separating conjunction, the predicate ls, and a natural guarded form of first-order quantification. We apply our approach for its axiomatisation. As a by-product of our method, we also establish the exact expressive power of this new logic and we show PSpace-completeness of its satisfiability problem
Disjoint-union partial algebras
Disjoint union is a partial binary operation returning the union of two sets
if they are disjoint and undefined otherwise. A disjoint-union partial algebra
of sets is a collection of sets closed under disjoint unions, whenever they are
defined. We provide a recursive first-order axiomatisation of the class of
partial algebras isomorphic to a disjoint-union partial algebra of sets but
prove that no finite axiomatisation exists. We do the same for other signatures
including one or both of disjoint union and subset complement, another partial
binary operation we define.
Domain-disjoint union is a partial binary operation on partial functions,
returning the union if the arguments have disjoint domains and undefined
otherwise. For each signature including one or both of domain-disjoint union
and subset complement and optionally including composition, we consider the
class of partial algebras isomorphic to a collection of partial functions
closed under the operations. Again the classes prove to be axiomatisable, but
not finitely axiomatisable, in first-order logic.
We define the notion of pairwise combinability. For each of the previously
considered signatures, we examine the class isomorphic to a partial algebra of
sets/partial functions under an isomorphism mapping arbitrary suprema of
pairwise combinable sets to the corresponding disjoint unions. We prove that
for each case the class is not closed under elementary equivalence.
However, when intersection is added to any of the signatures considered, the
isomorphism class of the partial algebras of sets is finitely axiomatisable and
in each case we give such an axiomatisation.Comment: 30 page
Synthetic Undecidability and Incompleteness of First-Order Axiom Systems in Coq
We mechanise the undecidability of various frst-order axiom systems in Coq, employing
the synthetic approach to computability underlying the growing Coq Library of Undecidability Proofs. Concretely, we cover both semantic and deductive entailment in fragments
of Peano arithmetic (PA) as well as ZF and related fnitary set theories, with their undecidability established by many-one reductions from solvability of Diophantine equations, i.e.
Hilbert’s tenth problem (H10), and the Post correspondence problem (PCP), respectively.
In the synthetic setting based on the computability of all functions defnable in a constructive foundation, such as Coq’s type theory, it sufces to defne these reductions as metalevel functions with no need for further encoding in a formalised model of computation.
The concrete cases of PA and the considered set theories are supplemented by a general
synthetic theory of undecidable axiomatisations, focusing on well-known connections to
consistency and incompleteness. Specifcally, our reductions rely on the existence of standard models, necessitating additional assumptions in the case of full ZF, and all axiomatic
extensions still justifed by such standard models are shown incomplete. As a by-product of
the undecidability of set theories formulated using only membership and no equality symbol, we obtain the undecidability of frst-order logic with a single binary relation
Logics of knowledge and action: critical analysis and challenges
International audienceWe overview the most prominent logics of knowledge and action that were proposed and studied in the multiagent systems literature. We classify them according to these two dimensions, knowledge and action, and moreover introduce a distinction between individual knowledge and group knowledge, and between a nonstrategic an a strategic interpretation of action operators. For each of the logics in our classification we highlight problematic properties. They indicate weaknesses in the design of these logics and call into question their suitability to represent knowledge and reason about it. This leads to a list of research challenges
Mechanised metamathematics : an investigation of first-order logic and set theory in constructive type theory
In this thesis, we investigate several key results in the canon of metamathematics, applying the contemporary perspective of formalisation in constructive type theory and mechanisation in the Coq proof assistant. Concretely, we consider the central completeness, undecidability, and incompleteness theorems of first-order logic as well as properties of the axiom of choice and the continuum hypothesis in axiomatic set theory. Due to their fundamental role in the foundations of mathematics and their technical intricacies, these results have a long tradition in the codification as standard literature and, in more recent investigations, increasingly serve as a benchmark for computer mechanisation. With the present thesis, we continue this tradition by uniformly analysing the aforementioned cornerstones of metamathematics in the formal framework of constructive type theory. This programme offers novel insights into the constructive content of completeness, a synthetic approach to undecidability and incompleteness that largely eliminates the notorious tedium obscuring the essence of their proofs, as well as natural representations of set theory in the form of a second-order axiomatisation and of a fully type-theoretic account. The mechanisation concerning first-order logic is organised as a comprehensive Coq library open to usage and contribution by external users.In dieser Doktorarbeit werden einige Schlüsselergebnisse aus dem Kanon der Metamathematik untersucht, unter Verwendung der zeitgenössischen Perspektive von Formalisierung in konstruktiver Typtheorie und Mechanisierung mit Hilfe des Beweisassistenten Coq. Konkret werden die zentralen Vollständigkeits-, Unentscheidbarkeits- und Unvollständigkeitsergebnisse der Logik erster Ordnung sowie Eigenschaften des Auswahlaxioms und der Kontinuumshypothese in axiomatischer Mengenlehre betrachtet. Aufgrund ihrer fundamentalen Rolle in der Fundierung der Mathematik und ihrer technischen Schwierigkeiten, besitzen diese Ergebnisse eine lange Tradition der Kodifizierung als Standardliteratur und, besonders in jüngeren Untersuchungen, eine zunehmende Bedeutung als Maßstab für Mechanisierung mit Computern. Mit der vorliegenden Doktorarbeit wird diese Tradition fortgeführt, indem die zuvorgenannten Grundpfeiler der Methamatematik uniform im formalen Rahmen der konstruktiven Typtheorie analysiert werden. Dieses Programm ermöglicht neue Einsichten in den konstruktiven Gehalt von Vollständigkeit, einen synthetischen Ansatz für Unentscheidbarkeit und Unvollständigkeit, der großteils den berüchtigten, die Essenz der Beweise verdeckenden, technischen Aufwand eliminiert, sowie natürliche Repräsentationen von Mengentheorie in Form einer Axiomatisierung zweiter Ordnung und einer vollkommen typtheoretischen Darstellung. Die Mechanisierung zur Logik erster Ordnung ist als eine umfassende Coq-Bibliothek organisiert, die offen für Nutzung und Beiträge externer Anwender ist
On Role Logic
We present role logic, a notation for describing properties of relational
structures in shape analysis, databases, and knowledge bases. We construct role
logic using the ideas of de Bruijn's notation for lambda calculus, an encoding
of first-order logic in lambda calculus, and a simple rule for implicit
arguments of unary and binary predicates. The unrestricted version of role
logic has the expressive power of first-order logic with transitive closure.
Using a syntactic restriction on role logic formulas, we identify a natural
fragment RL^2 of role logic. We show that the RL^2 fragment has the same
expressive power as two-variable logic with counting C^2 and is therefore
decidable. We present a translation of an imperative language into the
decidable fragment RL^2, which allows compositional verification of programs
that manipulate relational structures. In addition, we show how RL^2 encodes
boolean shape analysis constraints and an expressive description logic.Comment: 20 pages. Our later SAS 2004 result builds on this wor
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