25 research outputs found
Algebras of multiplace functions for signatures containing antidomain
We define antidomain operations for algebras of multiplace partial functions.
For all signatures containing composition, the antidomain operations and any
subset of intersection, preferential union and fixset, we give finite
equational or quasiequational axiomatisations for the representation class. We
do the same for the question of representability by injective multiplace
partial functions. For all our representation theorems, it is an immediate
corollary of our proof that the finite representation property holds for the
representation class. We show that for a large set of signatures, the
representation classes have equational theories that are coNP-complete.Comment: 33 pages. Added brief discussion of square algebra
The algebra of functions with antidomain and range
We give complete, finite quasiequational axiomatisations for algebras of unary partial functions under the operations of composition, domain, antidomain, range and intersection. This completes the extensive programme of classifying algebras of unary partial functions under combinations of these operations. We look at the complexity of the equational theories and provide a nondeterministic polynomial upper bound. Finally we look at the problem of finite representability and show that finite algebras can be represented as a collection of unary functions over a finite base set provided that intersection is not in the signature
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
Domain and Antidomain Semigroups
Abstract. We axiomatise and study operations for relational domain and antidomain on semigroups and monoids. We relate this approach with previous axiomatisations for semirings, partial transformation semi-groups and dynamic predicate logic.
Domain and range for angelic and demonic compositions
We give finite axiomatizations for the varieties generated by representable
domain--range algebras when the semigroup operation is interpreted as angelic
or demonic composition, respectively
Free Kleene algebras with domain
First we identify the free algebras of the class of algebras of binary
relations equipped with the composition and domain operations. Elements of the
free algebras are pointed labelled finite rooted trees. Then we extend to the
analogous case when the signature includes all the Kleene algebra with domain
operations; that is, we add union and reflexive transitive closure to the
signature. In this second case, elements of the free algebras are 'regular'
sets of the trees of the first case. As a corollary, the axioms of domain
semirings provide a finite quasiequational axiomatisation of the equational
theory of algebras of binary relations for the intermediate signature of
composition, union, and domain. Next we note that our regular sets of trees are
not closed under complement, but prove that they are closed under intersection.
Finally, we prove that under relational semantics the equational validities of
Kleene algebras with domain form a decidable set.Comment: 22 pages. Some proofs expande
The finite representation property for composition, intersection, domain and range
We prove that the nite representation property holds for rep-
resentation by partial functions for the signature consisting of composition,
intersection, domain and range and for any expansion of this signature by the
antidomain, xset, preferential union, maximum iterate and opposite opera-
tions. The proof shows that, for all these signatures, the size of base required
is bounded by a double-exponential function of the size of the algebra. This
establishes that representability of nite algebras is decidable for all these
signatures. We also give an example of a signature for which the nite repre-
sentation property fails to hold for representation by partial functions
Algebras of partial functions
This thesis collects together four sets of results, produced by investigating modifications, in four distinct directions, of the following. Some set-theoretic operations on partial functions are chosenācomposition and intersection are examplesāand the class of algebras isomorphic to a collection of partial functions, equipped with those operations, is studied. Typical questions asked are whether the class is axiomatisable, or indeed finitely axiomatisable, in any fragment of first-order logic, what computational complexity classes its equational/quasiequational/first-order theories lie in, and whether it is decidable if a finite algebra is in the class. The first modification to the basic picture asks that the isomorphisms turn any existing suprema into unions and/or infima into intersections, and examines the class so obtained. For composition, intersection, and antidomain together, we show that the suprema and infima conditions are equivalent. We show the resulting class is axiomatisable by a universal-existential-universal sentence, but not axiomatisable by any existential-universal-existential theory. The second contribution concerns what happens when we demand partial functions on some finite base set. The finite representation property is essentially the assertion that this restriction that the base set be finite does not restrict the algebras themselves. For composition, intersection, domain, and range, plus many supersignatures, we prove the finite representation property. It follows that it is decidable whether a finite algebra is a member of the relevant class. The third set of results generalises from unary to āmultiplaceā functions. For the signatures investigated, finite equational or quasiequational axiomatisations are obtained; similarly when the functions are constrained to be injective. The finite representation property follows. The equational theories are shown to be coNP-complete. In the last section we consider operations that may only be partial. For most signatures the relevant class is found to be recursively, but not finitely, axiomatisable. For others, finite axiomatisations are provided