540 research outputs found
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
The Temporal Logic of two dimensional Minkowski spacetime is decidable
We consider Minkowski spacetime, the set of all point-events of spacetime
under the relation of causal accessibility. That is, can access if an electromagnetic or (slower than light) mechanical signal could be
sent from to . We use Prior's tense language of
and representing causal accessibility and its converse relation. We
consider two versions, one where the accessibility relation is reflexive and
one where it is irreflexive.
In either case it has been an open problem, for decades, whether the logic is
decidable or axiomatisable. We make a small step forward by proving, for the
case where the accessibility relation is irreflexive, that the set of valid
formulas over two-dimensional Minkowski spacetime is decidable, decidability
for the reflexive case follows from this. The complexity of either problem is
PSPACE-complete.
A consequence is that the temporal logic of intervals with real endpoints
under either the containment relation or the strict containment relation is
PSPACE-complete, the same is true if the interval accessibility relation is
"each endpoint is not earlier", or its irreflexive restriction.
We provide a temporal formula that distinguishes between three-dimensional
and two-dimensional Minkowski spacetime and another temporal formula that
distinguishes the two-dimensional case where the underlying field is the real
numbers from the case where instead we use the rational numbers.Comment: 30 page
Algebraic foundations for qualitative calculi and networks
A qualitative representation is like an ordinary representation of a
relation algebra, but instead of requiring , as
we do for ordinary representations, we only require that , for each in the algebra. A constraint
network is qualitatively satisfiable if its nodes can be mapped to elements of
a qualitative representation, preserving the constraints. If a constraint
network is satisfiable then it is clearly qualitatively satisfiable, but the
converse can fail. However, for a wide range of relation algebras including the
point algebra, the Allen Interval Algebra, RCC8 and many others, a network is
satisfiable if and only if it is qualitatively satisfiable.
Unlike ordinary composition, the weak composition arising from qualitative
representations need not be associative, so we can generalise by considering
network satisfaction problems over non-associative algebras. We prove that
computationally, qualitative representations have many advantages over ordinary
representations: whereas many finite relation algebras have only infinite
representations, every finite qualitatively representable algebra has a finite
qualitative representation; the representability problem for (the atom
structures of) finite non-associative algebras is NP-complete; the network
satisfaction problem over a finite qualitatively representable algebra is
always in NP; the validity of equations over qualitative representations is
co-NP-complete. On the other hand we prove that there is no finite
axiomatisation of the class of qualitatively representable algebras.Comment: 22 page
There is no finite-variable equational axiomatization of representable relation algebras over weakly representable relation algebras
We prove that any equational basis that defines RRA over wRRA must contain
infinitely many variables. The proof uses a construction of arbitrarily large
finite weakly representable but not representable relation algebras whose
"small" subalgebras are representable.Comment: To appear in Review of Symbolic Logi
First-order axiomatisations of representable relation algebras need formulas of unbounded quantifier depth
We prove that RRA, the class of all representable relation algebras, cannot
be axiomatised by any first-order theory of bounded quantifier depth. The proof
uses of significant modification of the standard rainbow construction. We also
discuss and correct a strategy proposed elsewhere for proving that RRA cannot
be axiomatised by any first-order theory using only finitely many variables.Comment: v2 adds arXiv link to another pape
Temporal Logic of Minkowski Spacetime
We present the proof that the temporal logic of two-dimensional Minkowski
spacetime is decidable, PSPACE-complete. The proof is based on a type of
two-dimensional mosaic. Then we present the modification of the proof so as to
work for slower-than-light signals. Finally, a subframe of the
slower-than-light Minkowski frame is used to prove the new result that the
temporal logic of real intervals with during as the accessibility relation is
also PSPACE-complete
A homotopy-theoretic view of Bott-Taubes integrals and knot spaces
We construct cohomology classes in the space of knots by considering a bundle
over this space and "integrating along the fiber" classes coming from the
cohomology of configuration spaces using a Pontrjagin-Thom construction. The
bundle we consider is essentially the one considered by Bott and Taubes, who
integrated differential forms along the fiber to get knot invariants. By doing
this "integration" homotopy-theoretically, we are able to produce integral
cohomology classes. We then show how this integration is compatible with the
homology operations on the space of long knots, as studied by Budney and Cohen.
In particular we derive a product formula for evaluations of cohomology classes
on homology classes, with respect to connect-sum of knots.Comment: 32 page
Finite Representability of Semigroups with Demonic Refinement
Composition and demonic refinement of binary relations are
defined by \begin{align*} (x, y)\in (R;S)&\iff \exists z((x, z)\in R\wedge (z,
y)\in S)
R\sqsubseteq S&\iff (dom(S)\subseteq dom(R) \wedge
R\restriction_{dom(S)}\subseteq S)
\end{align*} where and
denotes the restriction of to pairs where
.
Demonic calculus was introduced to model the total correctness of
non-deterministic programs and has been applied to program verification.
We prove that the class of abstract
structures isomorphic to a set of binary relations ordered by demonic
refinement with composition cannot be axiomatised by any finite set of
first-order formulas. We provide a fairly simple, infinite,
recursive axiomatisation that defines . We prove that a
finite representable structure has a representation over a
finite base. This appears to be the first example of a signature for binary
relations with composition where the representation class is non-finitely
axiomatisable, but where the finite representations for finite representable
structures property holds
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