326 research outputs found
Exact Separation Logic
Over-approximating (OX) program logics, such as separation logic (SL), are
used for verifying properties of heap-manipulating programs: all terminating
behaviour is characterised, but established results and errors need not be
reachable. OX function specifications are thus incompatible with true
bug-finding supported by symbolic execution tools such as Pulse and Pulse-X. In
contrast, under-approximating (UX) program logics, such as incorrectness
separation logic, are used to find true results and bugs: established results
and errors are reachable, but there is no mechanism for understanding if all
terminating behaviour has been characterised.
We introduce exact separation logic (ESL), which provides fully-verified
function specifications compatible with both OX verification and UX true
bug-funding: all terminating behaviour is characterised, and all established
results and errors are reachable. We prove soundness for ESL with mutually
recursive functions, demonstrating, for the first time, function
compositionality for a UX logic. We show that UX program logics require subtle
definitions of internal and external function specifications compared with the
familiar definitions of OX logics. We investigate the expressivity of ESL and,
for the first time, explore the role of abstraction in UX reasoning by
verifying abstract ESL specifications of various data-structure algorithms. In
doing so, we highlight the difference between abstraction (hiding information)
and over-approximation (losing information). Our findings demonstrate that,
expectedly, abstraction cannot be used as freely in UX logics as in OX logics,
but also that it should be feasible to use ESL to provide tractable function
specifications for self-contained, critical code, which would then be used for
both verification and true bug-finding
Remarks on the Levi core
We investigate a few aspects of the notion of Levi core, introduced by the
authors in a previous work: a basic finiteness question, the connections with
Kohn's algorithm and with Catlin's property (P)
Abstract embeddability ranks
We describe several ordinal indices that are capable of detecting, according
to various metric notions of faithfulness, the embeddability between pairs of
Polish spaces. These embeddability ranks are of theoretical interest but seem
difficult to estimate in practice. Embeddability ranks, which are easier to
estimate in practice, are embeddability ranks generated by Schauder bases.
These embeddability are inspired by the nonlinear indices \`a la Bourgain from
\cite{BLMS_FM}. In particular, we resolve a problem \cite[Problem
3.10]{BLMS_FM} regarding the necessity of additional set-theoretic axioms
regarding the main coarse universality result of \cite{BLMS_FM}.Comment: 10 page
Indeterminacy and the law of the excluded middle
This thesis is an investigation into indeterminacy in the foundations of mathematics and its possible consequences for the applicability of the law of the excluded middle (LEM). It characterises different ways in which the natural numbers as well as the sets may be understood to be indeterminate, and asks in what sense this would cease to support applicability of LEM to reasoning with them. The first part of the thesis reviews the indeterminacy phenomena on which the argument is based and argues for a distinction between two notions of indeterminacy: a) indeterminacy as applied to domains and b) indefiniteness as applied to concepts. It then addresses possible attempts to secure determinacy in both cases. The second part of the thesis discusses the advantages that an argument from indeterminacy has over traditional intuitionistic arguments against LEM, and it provides the framework in which conditions for the applicability of LEM can be explicated in the setting of indeterminacy. The final part of the thesis then applies these findings to concrete cases of indeterminacy. With respect to indeterminacy of domains, I note some problems for establishing a rejection of LEM based on the indeterminacy of the height of the set theoretic hierarchy. I show that a coherent argument can be made for the rejection of LEM based on the indeterminacy of its width, and assess its philosophical commitments. A final chapter addresses the notion of indefiniteness of our concepts of set and number and asks how this might affect the applicability of LEM
Filling gaps in Hardy fields
We show how to fill "countable" gaps in Hardy fields. We use this to prove
that any two maximal Hardy fields are back-and-forth equivalent.Comment: 57 p
Free dcpo-algebras via directed spaces
Directed spaces are natural topological extensions of dcpos in domain theory
and form a cartesian closed category. We will show that the D-completion of
free algebras over a Scott space , on the context of directed spaces,
are exactly the free dcpo-algebras over dcpo , which reveals the close
connection between directed powerspaces and powerdomains. By this result, we
provide a topological representation of upper, lower and convex powerdomains of
dcpos uniformly.Comment: 18 page
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
Programs as Diagrams: From Categorical Computability to Computable Categories
This is a draft of the textbook/monograph that presents computability theory
using string diagrams. The introductory chapters have been taught as graduate
and undergraduate courses and evolved through 8 years of lecture notes. The
later chapters contain new ideas and results about categorical computability
and some first steps into computable category theory. The underlying
categorical view of computation is based on monoidal categories with program
evaluators, called *monoidal computers*. This categorical structure can be
viewed as a single-instruction diagrammatic programming language called Run,
whose only instruction is called RUN. This version: improved text, moved the
final chapter to the next volume. (The final version will continue lots of
exercises and workouts, but already this version has severely degraded graphics
to meet the size bounds.)Comment: 150 pages, 81 figure
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