Squares in Fork Arrow Logic

We investigate amalgamation properties of relational type algebras. Besides purely algebraic interest, amalgamation in a class of algebras is important because it leads to interpolation results for the logic corresponding to that class (cf. [15]). The multi-modal logic corresponding to relational type algebras became known under the name of “arrow logic” (cf. [18, 17]), and has been studied rather extensively lately (cf. [10]). Our research was inspired by the following result of Andréka et al. [1].Let K be a class of relational type algebras such that(i) composition is associative,(ii) K is a class of boolean algebras with operators, and(iii) K contains the representable relation algebras RRA.Then the equational theory of K is undecidable.On the other hand, there are several classes of relational type algebras (e.g., NA, WA denned below) whose equational (even universal) theories are decidable (cf. [13]). Composition is not associative in these classes. Theorem 5 indicates that also with respect to amalgamation (a very weak form of) associativity forms a borderline. We first recall the relevant definitions.

In this paper we present PLATO, a tool designed for theory presentation construction, theorem proving and formula derivation. It was motivated by equational algebraic theory manipulation needs, nevertheless its aim is software development by program calculation. PLATO has applications in algebra boolean algebra, relation algebra, etc. and mathematical logic as a general framework for theorem proving under different logics . In the field of software development, we show its application in program calculi construction and in program development under these calculi. We will consider a programming calculus as being a theory presentation within a given logic. Thus, the task of constructing new calculi can be thought of as one of building new theory presentations. This is true for, in particular, programming calculi based on Relation Algebras, Functional Programming, Fork Algebras, Type Theory and the like. We will consider a theory presentation as a pair , where T is a similarity type, i.e., declaration of operation symbols with their arity, and 2 is a finite set of formulas constituting an axiomatization for that theory.

Complexity of equations valid in algebras of relations part I: Strong non-finitizability

Complexity of equations valid in algebras of relations part II: Finite axiomatizations

We consider classes of relation algebras expanded with new operations based on the formation of ordered pairs. Examples for such algebras are pairing (or projection) algebras of algebraic logic and fork algebras of computer science. It is proved by Sain and Németi [38] that there is no `strong' representation theorem for all abstract pairing algebras in most set theories including ZFC as well as most non-well-founded set theories. Such a `strong' representation theorem would state that every abstract pairing algebra is isomorphic to a set relation algebra having projection elements which are defined with the help of the real (set theoretic) pairing function. Here we show that, by choosing an appropriate (non-well-founded) set theory as our metatheory, pairing algebras and fork algebras admit such `strong' representation theorems.

Richard Routley and Robert K. Meyer introduced a ternary relational semantics for various relevance logics in the early 1970s. Johan van Benthem and Yde Venema introduced “arrow logic” in the early 1990s and about the same time I showed how a variation of the Routley–Meyer semantics could be used to provide an interpretation of Tarski’s axioms for relation algebras. In this paper I explore the relationships between the van Benthem–Venema semantics for arrow logics, and the Routley–Meyer semantics for relevance logic, and conclude with a comparison between van Benthem’s version of the semantics for arrow logic aimed at relation algebras, and my own version of the Routley–Meyer semantics which I used to give a representation of relation algebras (but at a type level higher than Tarski’s original intended interpretation of an element as a relation, for me it is a set of relations). In the process I show how van Benthem’s semantics for arrow logic can be just slightly tweaked (just one additional constraint) so as to give a representation of relation algebras.

At the end of Chapter 4 of the RelMiCS book [11] an application of fork algebras as the basis for a calculus for program construction is outlined. In this paper we make a detailed presentation of the calculus as well as present some examples. We present a methodology for program construction based on the first-order theory of fork algebras. In this theory we will describe program design strategies, for instance case analysis, trivialization, divide-and-conquer and others. Using these strategies, from generic specifications (i.e., parameterized specifications) we will derive parametric algorithms. We will also provide conditions that will help in finding the parameters of the generic algorithms from the parameters in the specifications. We assume the reader is acquainted with the terminology and notation for relation and fork algebras, as well as with their basic properties as they were presented in the RelMiCS book [11].

The development of programs from first-order specifications has as its main difficulty that of dealing with universal quantifiers. This work is focused in that point, i.e., in the construction of programs whose specifications involve universal quantifiers. This task is performed within a relational calculus based on fork algebras. The fact that first-order theories can be translated into equational theories in abstract fork algebras suggests that such work can be accomplished in a satisfactory way. Furthermore, the fact that these abstract algebras are representable guarantees that all properties valid in the standard models are captured by the axiomatization given for them, allowing the reasoning formalism to be shifted back and forth between any model and the abstract algebra. In order to cope with universal quantifiers, a new algebraic operation — relational implication — is introduced. This operation is shown to have deep significance in the relational statement of first-order expressions involving universal quantifiers. Several algebraic properties of the relational implication are stated showing its usefulness in program calculation. Finally, a non-trivial example of derivation is given to asses the merits of the relational implication as an specification tool, and also in calculation steps, where its algebraic properties are clearly appropriate as transformation rules.KeywordsBinary RelationEquational TheoryAbstract AlgebraProgram ConstructionLational ImplicationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper deals with relation, cylindric and polyadic equality algebras. First of all it addresses a problem of B. Jónsson. He asked whether relation set algebras can be expanded by finitely many new operations in a “reasonable” way so that the class of these expansions would possess a finite equational base. The present paper gives a negative answer to this problem: Our main theorem states that whenever Rs+ is a class that consists of expansions of relation set algebras such that each operation of Rs+ is logical in Jónsson's sense, i.e., is the algebraic counterpart of some (derived) connective of first-order logic, then the equational theory of Rs+ has no finite axiom systems. Similar results are stated for the other classes mentioned above. As a corollary to this theorem we can solve a problem of Tarski and Givant [87], Namely, we claim that the valid formulas of certain languages cannot be axiomatized by a finite set of logical axiom schemes. At the same time we give a negative solution for a version of a problem of Henkin and Monk [74] (cf. also Monk [70] and Németi [89]).Throughout we use the terminology, notation and results of Henkin, Monk, Tarski [71] and [85]. We also use results of Maddux [89a].Notation. RA denotes the class of relation algebras, Rs denotes the class of relation set algebras and RRA is the class of representable relation algebras, i.e. the class of subdirect products of relation set algebras. The symbols RA, Rs and RRA abbreviate also the expressions relation algebra, relation set algebra and representable relation algebra, respectively.For any class C of similar algebras EqC is the set of identities that hold in C, while Eq1C is the set of those identities in EqC that contain at most one variable symbol. (We note that Henkin et al. [85] uses the symbol EqC in another sense.)

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. 1

The foundation of an algebraic theory of binary relations was laid by C. S. Peirce, building on earlier work of Boole and De Morgan. The basic universe of discourse of this theory is a collection of binary relations over some set, and the basic operations on these relations are those of forming unions, complements, relative products (i.e., compositions), and converses (i.e., inverses). There is also a distinguished relation, the identity relation. Other operations and distinguished relations studied by Peirce are definable in terms of the ones just mentioned. Such an algebra of relations is called a set relation algebra. A modern development of this theory as a theory of abstract relation algebras, axiomatized by a finite set of equations, was undertaken by Tarski and his students and colleagues, beginning around 1940. In 1942, Tarski proved that all of classical mathematics could be developed within the framework of the equational theory of relation algebras. Indeed, he created a general method for interpreting into the equational theory of relation algebras first-order theories that are strong enough to form a basis for the development of mathematics, in particular, set theories and number theories. As a consequence, he established that the equational theories of relation algebras and of set relation algebras are undecidable (see [10] and [11], and see [2] or [11], in particular Chapter 8, for unexplained terminology). They were the first known examples of undecidable equational theories. As was pointed out in [11], Tarski’s proof actually shows more. Namely, any class of relation algebras that contains the full set relation algebra on some infinite set (i.e., the set relation algebra whose universe consists of all binary relations on the infinite set) or, equivalently, that contains all set relation algebras on infinite sets must have an undecidable equational theory.

In this paper, we propose an arrow decision logic (ADL) for relational information systems (RIS). The logic combines the main features of decision logic (DL) and arrow logic (AL). DL represents and reasons about knowledge extracted from decision tables based on rough set theory, whereas AL is the basic modal logic of arrows. The semantic models of DL are functional information systems (FIS). ADL formulas, on the other hand, are interpreted in RIS. RIS , which not only specifies the properties of objects, but also the relationships between objects. We present a complete axiomatization of ADL and discuss its application to knowledge representation in multicriteria decision analysis.

We solve a problem of Jonsson [12] by showing that the class ℛ of (isomorphs of) algebras of binary relations, under the operations of relative product, conversion, and intersection, and with the identity element as a distinguished constant, is not axiomatizable by a set of equations. We also show that the set of equations valid in ℛ is decidable, and in fact the set of equations true in the class of all positive algebras of relations is decidable.

Built on the foundations laid by Peirce, Schröder, and others in the 19th century, the modern development of relation algebras started with the work of Tarski and his colleagues [21, 22]. They showed that relation algebras can capture strong first‐order theories like ZFC, and so their equational theory is undecidable. The less expressive class WA of weakly associative relation algebras was introduced by Maddux [7]. Németi [16] showed that WA's have a decidable universal theory. There has been extensive research on increasing the expressive power of WA by adding new operations [1, 4, 11, 13, 20]. Extensions of this kind usually also have decidable universal theories. Here we give an example – extending WA's with set‐theoretic projection elements – where this is not the case. These “logical” connectives are set‐theoretic counterparts of the axiomatic quasi‐projections that have been investigated in the representation theory of relation algebras [22, 6, 19]. We prove that the quasi‐equational theory of the extended class PWA is not recursively enumerable. By adding the difference operator D one can turn WA and PWA to discriminator classes where each universal formula is equivalent to some equation. Hence our result implies that the projections turn the decidable equational theory of “WA + D ” to non‐recursively enumerable (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Navigation is at the core of most XML processing tasks. The W3C endorsed navigation language XPath is part of XPointer (for creating links between elements in (different) XML documents), XSLT (for transforming XML documents) and XQuery (for, indeed, querying XML documents). Navigation in an XML document tree is the task of moving from a given node to another node by following a path specified by a certain formula. Hence formulas in navigation languages denote paths, or stated otherwise binary relations between nodes. Binary relations can be expressed in XPath or with first or second order formulas in two free variables. The problem with all of these formalisms is that they are not compositional in the sense that each subexpression also specifies a binary relation. This makes a mathematical study of these languages complicated because one has to deal with objects of different sorts. Fortunately there exists an algebraic formalism which is created solely to study binary relations. This formalism goes back to logic pioneers as de Morgan, Peirce and Schröder and has been formalized by Tarski as relation algebras [7]. (Cf., [5] for a monograph on this topic, and [8] for a database oriented introduction). A relation algebra is a boolean algebra with three additional operations. In its natural representation each element in the domain of the algebra denotes a binary relation. The three extra operations are a constant denoting the identity relation, a unary conversion operation, and a binary operation denoting the composition of two relations. The elements in the algebra denote first order definable relations. Later Tarski and Ng added the Kleene star as an additional operator, denoting the transitive reflexive closure of a relation [6].We will show that the formalism of relation algebras is very well suited for defining navigation paths in XML documents. One of its attractive features is that it does not contain variables, a feature shared by XPath 1.0 and the regular path expressions of [1]. The connection between relation algebras and XPath was first made in [4].The aim of this talk is to show that relation algebras (possibly expanded with the Kleene star) can serve as a unifying framework in which many of the proposed navigation languages can be embedded. Examples of these embeddings are 1 Every Core XPath definable path is definable using composition, union and the counterdomain operator ~ with semantics ~R = {(x,x)|not ∃ y: xRy}. 2 Every first order definable path is definable by a relation algebraic expression. 3 Every first order definable path is definable by a positive relation algebraic expression which may use the Kleene star. 4 The paths definable by tree walk automata and certain tree walk automata with pebbles can be characterized by natural fragments of relation algebras with the Kleene star. All these results hold restricted to the class of finite unranked sibling ordered trees. The main open problem is the expressive power of relation algebras expanded with the Kleene star, interpreted on this class of models. Is this formalism equally expressive as binary first order logic with transitive closure of binary formulas? Whether the latter is equivalent to binary monadic second order logic is also open [2,3]. So in particular we do not know whether each regular tree language can be defined in relation algebras with the Kleene star.KeywordsBinary RelationBoolean AlgebraTransitive ClosureOrder LogicRelation AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.