Undecidability of representability for lattice-ordered semigroups and ordered complemented semigroups

Šemrl

2021

Algebra Univers.

The motivation for using demonic calculus for binary relations stems from the behaviour of demonic turing machines, when modelled relationally. Relational composition (; ) models sequential runs of two programs and demonic refinement ($$\sqsubseteq $$ ⊑ ) arises from the partial order given by modeling demonic choice ($$\sqcup $$ ⊔ ) of programs (see below for the formal relational definitions). We prove that the class $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) of abstract $$(\le , \circ )$$ ( ≤ , ∘ ) structures isomorphic to a set of binary relations ordered by demonic refinement with composition cannot be axiomatised by any finite set of first-order $$(\le , \circ )$$ ( ≤ , ∘ ) formulas. We provide a fairly simple, infinite, recursive axiomatisation that defines $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) . We prove that a finite representable $$(\le , \circ )$$ ( ≤ , ∘ ) 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 representation property holds for finite structures.

Section: Finite Axiomatisability and Representabilitymentioning

confidence: 75%

Finite representability of semigroups with demonic refinement

Šemrl

2021

Algebra Univers.

“…Based on results in [HJ12,Neu16] these results may be strengthened further. ; e 0 = e 0 , so if B has a finite base X then there must be points x ∈ X where (x, x) ∈ e B 0 , and when negation is in the signature, since it is complementation relative to X × X, every reflexive edge (x, x) belongs to some element of B.…”

mentioning

confidence: 90%

“…A proper S-structure A over the base X is an S-structure where each element of A is a binary relation over X, where ≤ is set inclusion ⊆, 0 is the empty relation, •, + denote ∩, ∪, respectively, 1 = X × X (these proper structures are often called square), negation is complement in X × X (called universal complementation in [Neu16] ) and 3 Decidablity.…”

mentioning

confidence: 99%

Decidability of Equational Theories for Subsignatures of Relation Algebra

Lecture Notes in Computer Science

2018

Let S be a subset of the signature of relation algebra. Let R(S) be the closure under isomorphism of the class of proper S-structures and let F (S) be the closure under isomorphism of the class of proper S-structures over finite bases. Based on previous work, we prove that membership of R(S) is undecidable when S ⊇ {•, +, ; }, S ⊇ {•, , ; } or ∈ S ⊇ {≤, −, ; }, and for any of these signatures S if converse is excluded from S then membership of F (S) is also undecidable, for finite S-structures. We prove that the equational theories of F (S) and R(S) are undecidable when S includes composition and the signature of boolean algebra. If all operators in S are positive and it does not include negation, or if it can define neither domain, range nor composition, then the equational theory of either class is decidable. Open cases for decidability of the equational theory of R(S) are when S can define negation but not meet (or join) and either domain, range or composition. Open cases for the decidability of the equational theory of F (S) are (i) when S can define negation, converse and either domain, range or composition, and (ii) when S contains negation but not meet (or join) and either domain, range or composition.

“…A number of results in the field suggests that the good behaviour of the ordered semigroups does not extend to the rest of the angelic signatures. R(∩, ; ) is finitely axiomatisable, but not finitely representable [9,10], R(∪, ; ) is not finitely axiomatisable [11] and the representation problem for R(∪, ∩, ; ) is undecidable [10], implying both non-finite-axiomatisability and failure of finite representation property. These results are summarised in Table I, along with the contributions of this paper.…”

Section: B Related Workmentioning

confidence: 99%

Demonic Lattices and Semilattices in Relational Semigroups with Ordinary Composition

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

Šemrl

2021

Relation algebra and its reducts provide us with a strong tool for reasoning about nondeterministic programs and their partial correctness. Demonic calculus, introduced to model the behaviour of a machine where the demon is in control of nondeterminism, has also provided us with an extension of that reasoning to total correctness.We formalise the framework for relational reasoning about total correctness in nondeterministic programs using semigroups with ordinary composition and demonic lattice operations. We show that the class of representable demonic join semigroups is not finitely axiomatisable and that the representation class of demonic meet semigroups does not have the finite representation property for its finite members.For lattice semigroups (with composition, demonic join and demonic meet) we show that the representation problem for finite algebras is undecidable, moreover the finite representation problem is also undecidable. It follows that the representation class is not finitely axiomatisable, furthermore the finite representation property fails.