1,411 research outputs found
Monoids with tests and the algebra of possibly non-halting programs
We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, if-then-else and while-do defined in terms of a Boolean algebra of conditions. It has previously been shown that there is no finite axiomatisation of algebras of partial functions under these operations alone, and this holds even if one restricts attention to transformations (representing halting programs) rather than partial functions, and omits while-do from the signature. In the halting case, there is a natural āfixā, which is to allow composition of halting programs with conditions, and then the resulting algebras admit a finite axiomatisation. In the current setting such compositions are not possible, but by extending the notion of if-then-else, we are able to give finite axiomatisations of the resulting algebras of (partial) functions, with while-do in the signature if the state space is assumed finite. The axiomatisations are extended to consider the partial predicate of equality. All algebras considered turn out to be enrichments of the notion of a (one-sided) restriction semigrou
Stone-Type Dualities for Separation Logics
Stone-type duality theorems, which relate algebraic and
relational/topological models, are important tools in logic because -- in
addition to elegant abstraction -- they strengthen soundness and completeness
to a categorical equivalence, yielding a framework through which both algebraic
and topological methods can be brought to bear on a logic. We give a systematic
treatment of Stone-type duality for the structures that interpret bunched
logics, starting with the weakest systems, recovering the familiar BI and
Boolean BI (BBI), and extending to both classical and intuitionistic Separation
Logic. We demonstrate the uniformity and modularity of this analysis by
additionally capturing the bunched logics obtained by extending BI and BBI with
modalities and multiplicative connectives corresponding to disjunction,
negation and falsum. This includes the logic of separating modalities (LSM), De
Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics
extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as
corollaries soundness and completeness theorems for the specific Kripke-style
models of these logics as presented in the literature: for DMBI, the
sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene
BI (connecting our work to Concurrent Separation Logic), this is the first time
soundness and completeness theorems have been proved. We thus obtain a
comprehensive semantic account of the multiplicative variants of all standard
propositional connectives in the bunched logic setting. This approach
synthesises a variety of techniques from modal, substructural and categorical
logic and contextualizes the "resource semantics" interpretation underpinning
Separation Logic amongst them
Semigroups with if-then-else and halting programs
The "ifāthenāelse" construction is one of the most elementary programming commands, and its abstract laws have been widely studied, starting with McCarthy. Possibly, the most obvious extension of this is to include the operation of composition of programs, which gives a semigroup of functions (total, partial, or possibly general binary relations) that can be recombined using ifāthenāelse. We show that this particular extension admits no finite complete axiomatization and instead focus on the case where composition of functions with predicates is also allowed (and we argue there is good reason to take this approach). In the case of total functions ā modeling halting programs ā we give a complete axiomatization for the theory in terms of a finite system of equations. We obtain a similar result when an operation of equality test and/or fixed point test is included
Algebras for Agent Norm-Regulation
An abstract architecture for idealized multi-agent systems whose behaviour is
regulated by normative systems is developed and discussed. Agent choices are
determined partially by the preference ordering of possible states and
partially by normative considerations: The agent chooses that act which leads
to the best outcome of all permissible actions. If an action is non-permissible
depends on if the result of performing that action leads to a state satisfying
a condition which is forbidden, according to the norms regulating the
multi-agent system. This idea is formalized by defining set-theoretic
predicates characterizing multi-agent systems. The definition of the predicate
uses decision theory, the Kanger-Lindahl theory of normative positions, and an
algebraic representation of normative systems.Comment: 25 page
Canonical extensions and ultraproducts of polarities
J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra
with operators has evolved into an extensive theory of canonical extensions of
lattice-based algebras. After reviewing this evolution we make two
contributions. First it is shown that the failure of a variety of algebras to
be closed under canonical extensions is witnessed by a particular one of its
free algebras. The size of the set of generators of this algebra can be made a
function of a collection of varieties and is a kind of Hanf number for
canonical closure. Secondly we study the complete lattice of stable subsets of
a polarity structure, and show that if a class of polarities is closed under
ultraproducts, then its stable set lattices generate a variety that is closed
under canonical extensions. This generalises an earlier result of the author
about generation of canonically closed varieties of Boolean algebras with
operators, which was in turn an abstraction of the result that a first-order
definable class of Kripke frames determines a modal logic that is valid in its
so-called canonical frames
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