Roger D. Maddux

On binary constraint problems

The concepts of binary constraint satisfaction problems can be naturally generalized to the relation algebras of Tarski. The concept of path-consistency plays a central role. Algorithms for path-consistency can be implemented on matrices of relations and on matrices of elements from a relation algebra. We give an example of a 4-by-4 matrix of infinite relations on which on iterative local path-consistency algorithm terminates. We give a class of examples over a fixed finite algebra on which all iterative local algorithms, whether parallel or sequential, must take quadratic time. Specific relation algebras arising from interval constraint problems are also studied: the Interval Algebra, the Point Algebra, and the Containment Algebra.

The origin of relation algebras in the development and axiomatization of the calculus of relations

The calculus of relations was created and developed in the second half of the nineteenth century by Augustus De Morgan, Charles Sanders Peirce, and Ernst Schröder. In 1940 Alfred Tarski proposed an axiomatization for a large part of the calculus of relations. In the next decade Tarskis axiomatization led to the creation of the theory of relation algebras, and was shown to be incomplete by Roger Lyndons discovery of nonrepresentable relation algebras. This paper introduces the calculus of relations and the theory of relation algebras through a review of these historical developments.

Nonfinite axiomatizability results for cylindric and relation algebras

The set of equations which use only one variable and hold in all representable relation algebras cannot be derived from any finite set of equations true in all representable relation algebras. Similar results hold for cylindric algebras and for logic with finitely many variables. The main tools are a construction of nonrepresentable one-generated relation algebras, a method for obtaining cylindric algebras from relation algebras, and the use of relation algebras in defining algebraic semantics for first-order logic.

Relation-algebraic semantics

The first half is a tutorial on orderings, lattices, Boolean algebras, operators on Boolean algebras, Tarskis fixed point theorem, and relation algebras. In the second half, elements of a complete relation algebra are used as “meanings” for program statements. The use of relation algebras for this purpose was pioneered by de Bakker and de Roever in [10–12]. For a class of programming languages with program schemes, single μ-recursion, while-statements, if-then-else, sequential composition, and nondeterministic choice, a definition of “correct interpretation” is given which properly reflects the intuitive (or operational) meanings of the program constructs. A correct interpretation includes for each program statement an element serving as “input/output relation” and a domain element specifying that statements “domain of nontermination”. The derivative of Hitchcock and Park [17] is defined and a relation-algebraic version of the extension by de Bakker [8, 9] of the Hitchcock-Park theorem is proved. The predicate transformers wps(-) and wlps(-) are defined and shown to obey all the standard laws in [15]. The “law of the excluded miracle” is shown to hold for an entire language if it holds for that languages basic statements (assignment statements and so on). Determinism is defined and characterized for all the program constructs. A relation-algebraic version of the invariance theorem for while-statements is given. An alternative definition of intepretation, called “demonic”, is obtained by using “demonic union” in place of ordinary union, and “demonic composition” in place of ordinary relational composition. Such interpretations are shown to arise naturally from a special class of correct interpretations, and to obey the laws of wps(-).

The equational theory of CA 3 is undecidable

There is no algorithm for determining whether or not an equation is true in every 3-dimensional cylindric algebra. This theorem completes the solution to the problem of finding those values of a and P3 for which the equational theories of CA,, and RCA: are undecidable. (CAa and RCA: are the classes of a-dimensional cylindric algebras and representable 13-dimensional cylindric algebras. See [4] for definitions.) This problem was considered in [3]. It was known that RCAo = CAo and RCA1 = CA1 and that the equational theories of these classes are decidable. Tarski had shown that the equational theory of relation algebras is undecidable and, by utilizing connections between relation algebras and cylindric algebras, had also shown that the equational theories of CA,, and RCA: are undecidable whenever 4 is a member of every equivalence relation E c Tn x Tn having the following properties. (AO) , ..., E EF

Self‐Similarity and the Species‐Area Relationship

Self‐similar distributions of species across a landscape have been proposed as one potential cause of the well‐known species‐area relationship. The best known of these proposals is in the form of a probability rule for species occurrence. The application of this rule to the number of species occurring in primary well‐shaped rectangles within the landscape gives rise to a discrete power law for species‐area relationships. However, this result requires a specific scheme for bisecting the landscape to generate the rectangles. Some additional, more general consequences of the probability rule are presented here. These include the result that the number of species in a well‐shaped rectangle depends on its location, not just on its area. In addition, a self‐similar landscape contains well‐shaped rectangles that are, in fact, not self‐similar. The probability rule in general produces testable predictions about how and where species are distributed that are independent of the power law.

Algebraic Logic and Universal Algebra in Computer Science

Algebraic methods, in particular those of universal algebra and algebraic logic, are playing an increasingly important role in computer science, especially in the areas of algebraic specification of data types, relational data types, relational database theory, logic of programmes, functional and logic programming, and semantics of programming languages. To a large extent this work has been carried forward by computer scientists independent of the very active group of mathematicians who work in universal algebra and algebraic logic. A conference was held at Iowa State University in June 1988 to bring together leading researchers from both groups to identify areas of common interest. Addresses were given by Joel Berman, H.Peter Gumm, Bjarni Jonsson, Dexter Kozen, Istvan Nemeti, Vaughan Pratt, Dana Scott and Eric Wagner. The programme also included contributed papers and a round-table discussion of the role of algebra and logic in computer science.

RELATION ALGEBRAS OF EVERY DIMENSION

Conjecture (1) of [Ma83] is confirmed here by the following result: if 3 ≤ α ω , then there is a finite relation algebra of dimension α , which is not a relation algebra of dimension α + 1. A logical consequence of this theorem is that for every finite α ≥ 3 there is a formula of the form S ⊆ T (asserting that one binary relation is included in another), which is provable with α + 1 variables, but not provable with only α variables (using a special sequent calculus designed for deducing properties of binary relations).

Representations for Small Relation Algebras

There are eighteen isomorphism types of finite relation algebras with eight or fewer elements, and all of them are representable. We determine all the cardinalities of sets on which these algebras have representations.

Provability with finitely many variables

For every finite n ≥ 4 there is a logically valid sentence φn with the following properties: φn contains only 3 variables (each of which occurs many times); φn contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol); φn has a proof in first-order logic with equality that contains exactly n variables, but no proof containing only n − 1 variables. This result was first proved using the machinery of algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) first-order binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that φn has a proof with only n variables. To show that φn has no proof with only n − 1 variables we use alternative semantics in place of the usual, standard, set-theoretical semantics of first-order logic. §