Algebras Of Binary Relations Research Articles

The Algebra of Non-deterministic Programs: Demonic Operations, Orders and Axioms

Abstract Demonic composition, demonic refinement and demonic union are alternatives to the usual ‘angelic’ composition, angelic refinement (inclusion) and angelic (usual) union defined on binary relations. We first motivate both the angelic and the demonic via an analysis of the behaviour of non-deterministic programs, with the angelic associated with partial correctness and demonic with total correctness, both cases emerging from a richer algebraic model of non-deterministic programs incorporating both aspects. Zareckiĭ has shown that the isomorphism class of algebras of binary relations under angelic composition and inclusion is finitely axiomatized as the class of ordered semigroups. The proof can be used to establish that the same axiomatization applies to binary relations under demonic composition and refinement, and a further modification of the proof can be used to incorporate a zero element representing the empty relation in the angelic case and the full relation in the demonic case. For the signature of angelic composition and union, it is known that no finite axiomatization exists, and we show the analogous result for demonic composition and demonic union by showing that the same axiomatization holds for both. We show that the isomorphism class of algebras of binary relations with the ‘mixed’ signature of demonic composition and angelic inclusion has no finite axiomatization. As a contrast, we show that the isomorphism class of partial algebras of binary relations with the partial operation of constellation product and inclusion (also a ‘mixed’ signature) is finitely axiomatizable.

On matrices and K-relations

We show that the matrix query language MATLANG corresponds to a natural fragment of the positive relational algebra on K-relations. The fragment is defined by introducing a composition operator and restricting K-relation arities to 2. We then proceed to show that MATLANG can express all matrix queries expressible in the positive relational algebra on K-relations, when intermediate arities are restricted to 3. Thus we offer an analogue, in a model with numerical data, to the situation in classical logic, where the algebra of binary relations is equivalent to first-order logic with three variables.

On algebras of binary relations with conjunctive operations

On algebras of binary relations with conjunctive operations

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.

Relation algebras of Sugihara, Belnap, Meyer, and Church

Algebras introduced by, or attributed to, Sugihara, Belnap, Meyer, and Church are representable as algebras of binary relations with set-theoretically defined operations. They are definitional reducts or subreducts of proper relation algebras. The representability of Sugihara matrices yields sound and complete set-theoretical semantics for R-mingle.

A Restricted Fragment of the Lambek Calculus with Iteration and Intersection Operations

The Lambek calculus (a variant of intuitionistic linear logic initially introduced for mathematical linguistics) enjoys natural interpretations over the algebra of formal languages (L-models) and over the algebra of binary relations which are subsets of a given transitive relation (R-models). For both classes of models there are completeness theorems (Andreka and Mikulas [J. Logic Lang. Inf., 3, No. 1, 1-37 (1994)]; Pentus [Ann. Pure Appl. Logic, 75, Nos. 1/2, 179-213 (1995); Fund. Prikl. Mat., 5, No. 1, 193- 219 (1999)]). The operations of the Lambek calculus include product and two divisions, left and right. We consider an extension of the Lambek calculus with intersection and iteration (Kleene star). It is proved that this extension is incomplete both w.r.t. L-models and w.r.t. R-models. We introduce a restricted fragment, in which iteration is allowed only in denominators of division operations. For this fragment we prove completeness w.r.t. R-models. We also prove completeness w.r.t. L-models for the subsystem without product. Both results are strong completeness theorems, i.e., they establish equivalence between derivability from sets of hypotheses (finite or infinite) and semantic entailment from sets of hypotheses on a given class of models. Finally, we prove $$ {\Pi}_1^0 $$ -completeness of the algorithmic problem of derivability in the restricted fragment in question.

A framework for comparing query languages in their ability to express boolean queries

For any query language $\mathcal {F}$ , we consider three natural families of boolean queries. Nonemptiness queries are expressed as e ≠ ∅ with e an $\mathcal {F}$ expression. Emptiness queries are expressed as e = ∅. Containment queries are expressed as e1 ⊆ e2. We refer to syntactic constructions of boolean queries as modalities. In first order logic, the emptiness, nonemptiness and containment modalities have exactly the same expressive power. For other classes of queries, e.g., expressed in weaker query languages, the modalities may differ in expressiveness. We propose a framework for studying the expressive power of boolean query modalities. Along one dimension, one may work within a fixed query language and compare the three modalities. Here, we identify crucial query features that enable us to go from one modality to another. Furthermore, we identify semantical properties that reflect the lack of these query features to establish separations. Along a second dimension, one may fix a modality and compare different query languages. This second dimension is the one that has already received quite some attention in the literature, whereas in this paper we emphasize the first dimension. Combining both dimensions, it is interesting to compare the expressive power of a weak query language using a strong modality, against that of a seemingly stronger query language but perhaps using a weaker modality. We present some initial results within this theme. The two main query languages to which we apply our framework are the algebra of binary relations, and the language of conjunctive queries.

Allegories for database modelling

Allegories abstract and generalise (in the categorical framework) the algebra of binary relations. Arrows in an allegory enjoy a lot of properties and structure available for plain binary relations...

Computer Aided Verification of Relational Models by Strategic Rewriting

Binary relational algebra provides semantic foundations for major areas of computing, such as database design, state-based modeling and functional programming. Remarkably, static checking support in these areas fails to exploit the full semantic content of relations. In particular, properties such as the simplicity or injectivity of relations are not statically enforced in operations such as database queries, state transitions, or composition of functional components. When data models, their constraints and operations are represented by point-free binary relational expressions, proof obligations can be expressed as inclusions between relational expressions. We developed a typedirected, strategic term rewriting system that can be used to simplify relational proof obligations and ultimately reduce them to tautologies. Such reductions can be used to provide extended static checking for design contraints commonly found in software modeling and development.

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

We prove that the problems of representing a finite ordered complemented semigroup or finite lattice-ordered semigroup as an algebra of binary relations over a finite set are undecidable. In the case that complementation is taken with respect to a universal relation, this result can be extended to infinite representations of ordered complemented semigroups.

On the Satisfiability Problem for SPARQL Patterns

The satisfiability problem for SPARQL 1.0 patterns is undecidable in general, since the relational algebra can be emulated using such patterns. The goal of this paper is to delineate the boundary of decidability of satisfiability in terms of the constraints allowed in filter conditions. The classes of constraints considered are bound-constraints, negated bound- constraints, equalities, nonequalities, constant-equalities, and constant-nonequalities. The main result of the paper can be summarized by saying that, as soon as inconsistent filter conditions can be formed, satisfiability is undecidable. The key insight in each case is to find a way to emulate the set difference operation. Undecidability can then be obtained from a known undecidability result for the algebra of binary relations with union, composition, and set difference. When no inconsistent filter conditions can be formed, satisfiability is decidable by syntactic checks on bound variables and on the use of literals. Although the problem is shown to be NP-complete, it is experimentally shown that the checks can be implemented efficiently in practice. The paper also points out that satisfiability for the so-called ‘well-designed’ patterns can be decided by a check on bound variables and a check for inconsistent filter conditions.

Algorithms for Kleene algebra with converse

The equational theory generated by all algebras of binary relations with operations of union, composition, converse and reflexive transitive closure was studied by Bernátsky, Bloom, Ésik and Stefanescu in 1995. In particular, they obtained its decidability by using a particular automata construction.We show that deciding this equational theory is PSpace-complete, by providing a PSpace algorithm (the problem is easily shown to be PSpace-hard). We obtain other algorithms that are time-efficient in practice, despite not being PSpace.Our results use an alternative automata construction, inspired by the one from Bloom, Ésik and Stefanescu. We relate those two constructions by exhibiting a bisimulation between the resulting deterministic automata, and by showing how our construction results in more sharing between states, thus producing smaller automata.

Lower Semilattice-Ordered Residuated Semigroups and Substructural Logics

We look at lower semilattice-ordered residuated semigroups and, in particular, the representable ones, i.e., those that are isomorphic to algebras of binary relations. We will evaluate expressions (terms, sequents, equations, quasi-equations) in representable algebras and give finite axiomatizations for several notions of validity. These results will be applied in the context of substructural logics.

Undecidability of representability as binary relations

AbstractIn this article we establish the undecidability of representability and of finite representability as algebras of binary relations in a wide range of signatures. In particular, representability and finite representability are undecidable for Boolean monoids and lattice ordered monoids, while representability is undecidable for Jónsson's relation algebra. We also establish a number of undecidability results for representability as algebras of injective functions.

Axiomatizability of positive algebras of binary relations

We consider all positive fragments of Tarski’s representable relation algebras and determine whether the equational and quasiequational theories of these fragments are finitely axiomatizable in first-order logic. We also look at extending the signature with reflexive, transitive closure and the residuals of composition.

On the Semigroup Algebra of Binary Relations

The semigroup of binary relations on {1,…, n} with the relative product is isomorphic to the semigroup B n of n × n zero-one matrices with the Boolean matrix product. Over any field F, we prove that the semigroup algebra FB n contains an ideal K n of dimension (2 n − 1)2, and we construct an explicit isomorphism of K n with the matrix algebra M 2 n −1(F).

POSITIVE FRAGMENTS OF RELEVANCE LOGIC AND ALGEBRAS OF BINARY RELATIONS

We prove that algebras of binary relations whose similarity type includes intersection, union, and one of the residuals of relation composition form a nonfinitely axiomatizable quasivariety and that the equational theory is not finitely based. We apply this result to the problem of the completeness of the positive fragment of relevance logic with respect to binary relations.

Equationally distinct countable simple Q-relation algebras

Relation algebras were conceived by Tarski as the means to capture the algebra of binary relations. In this paper, we prove that a Maddux Style Representation preserves well-foundedness of relations, which is not in general true for a relation algebra isomorphism. This theorem enables us to construct equationally distinct countable simple Q-relation algebras using the method of forcing.

RELEVANCE LOGIC AND THE CALCULUS OF RELATIONS

Sound and complete semantics for classical propositional logic can be obtained by interpreting sentences as sets. Replacing sets with commuting dense binary relations produces an interpretation that turns out to be sound but not complete for R. Adding transitivity yields sound and complete semantics for RM, because all normal Sugihara matrices are representable as algebras of binary relations.

Algebras of Relations and Relevance Logic

We prove that algebras of binary relations whose similarity type includes intersection, composition, converse negation and the identity constant form a non-finitely axiomatizable quasivariety and that the equational theory is not finitely based. We apply this result to the problem of the completeness of relevant logic with respect to binary relations.