670 research outputs found
On the Relative Power of Linear Algebraic Approximations of Graph Isomorphism
We compare the capabilities of two approaches to approximating graph isomorphism using linear algebraic methods: the invertible map tests (introduced by Dawar and Holm) and proof systems with algebraic rules, namely polynomial calculus, monomial calculus and Nullstellensatz calculus. In the case of fields of characteristic zero, these variants are all essentially equivalent to the Weisfeiler-Leman algorithms. In positive characteristic we show that the distinguishing power of the monomial calculus is no greater than the invertible map method by simulating the former in a fixed-point logic with solvability operators. In turn, we show that the distinctions made by this logic can be implemented in the Nullstellensatz calculus
Admissibility in Finitely Generated Quasivarieties
Checking the admissibility of quasiequations in a finitely generated (i.e.,
generated by a finite set of finite algebras) quasivariety Q amounts to
checking validity in a suitable finite free algebra of the quasivariety, and is
therefore decidable. However, since free algebras may be large even for small
sets of small algebras and very few generators, this naive method for checking
admissibility in \Q is not computationally feasible. In this paper,
algorithms are introduced that generate a minimal (with respect to a multiset
well-ordering on their cardinalities) finite set of algebras such that the
validity of a quasiequation in this set corresponds to admissibility of the
quasiequation in Q. In particular, structural completeness (validity and
admissibility coincide) and almost structural completeness (validity and
admissibility coincide for quasiequations with unifiable premises) can be
checked. The algorithms are illustrated with a selection of well-known finitely
generated quasivarieties, and adapted to handle also admissibility of rules in
finite-valued logics
Limitations of Game Comonads via Homomorphism Indistinguishability
Abramsky, Dawar, and Wang (2017) introduced the pebbling comonad for
k-variable counting logic and thereby initiated a line of work that imports
category theoretic machinery to finite model theory. Such game comonads have
been developed for various logics, yielding characterisations of logical
equivalences in terms of isomorphisms in the associated co-Kleisli category. We
show a first limitation of this approach by studying linear-algebraic logic,
which is strictly more expressive than first-order counting logic and whose
k-variable logical equivalence relations are known as invertible-map
equivalences (IM). We show that there exists no finite-rank comonad on the
category of graphs whose co-Kleisli isomorphisms characterise IM-equivalence,
answering a question of \'O Conghaile and Dawar (CSL 2021). We obtain this
result by ruling out a characterisation of IM-equivalence in terms of
homomorphism indistinguishability and employing the Lov\'asz-type theorems for
game comonads established by Dawar, Jakl, and Reggio (2021). Two graphs are
homomorphism indistinguishable over a graph class if they admit the same number
of homomorphisms from every graph in the class. The IM-equivalences cannot be
characterised in this way, neither when counting homomorphisms in the natural
numbers, nor in any finite prime field.Comment: Minor corrections in Section
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
When Can Matrix Query Languages Discern Matrices?
We investigate when two graphs, represented by their adjacency matrices, can be distinguished by means of sentences formed in MATLANG, a matrix query language which supports a number of elementary linear algebra operators. When undirected graphs are concerned, and hence the adjacency matrices are real and symmetric, precise characterisations are in place when two graphs (i.e., their adjacency matrices) can be distinguished. Turning to directed graphs, one has to deal with asymmetric adjacency matrices. This complicates matters. Indeed, it requires to understand the more general problem of when two arbitrary matrices can be distinguished in MATLANG. We provide characterisations of the distinguishing power of MATLANG on real and complex matrices, and on adjacency matrices of directed graphs in particular. The proof techniques are a combination of insights from the symmetric matrix case and results from linear algebra and linear control theory
- âŠ