Step by step – Building representations in algebraic logic

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

Fix a finite ordinal \(n\geq 3\) and let \(\alpha\) be an arbitrary ordinal. Let \(\mathsf{CA}_n\) denote the class of cylindric algebras of dimension \(n\) and \(\sf RA\) denote the class of relation algebras. Let \(\mathbf{PA}_{\alpha}(\mathsf{PEA}_{\alpha})\) stand for the class of polyadic (equality) algebras of dimension \(\alpha\). We reprove that the class \(\mathsf{CRCA}_n\) of completely representable \(\mathsf{CA}_n\)s, and the class \(\sf CRRA\) of completely representable \(\mathsf{RA}\)s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety \(\sf V\) between polyadic algebras of dimension \(n\) and diagonal free \(\mathsf{CA}_n\)s. We show that that the class of completely and strongly representable algebras in \(\sf V\) is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class \(\sf CRRA\) is not closed under \(\equiv_{\infty,\omega}\). In contrast, we show that given \(\alpha\geq \omega\), and an atomic \(\mathfrak{A}\in \mathsf{PEA}_{\alpha}\), then for any \(n<\omega\), \(\mathfrak{Nr}_n\mathfrak{A}\) is a completely representable \(\mathsf{PEA}_n\). We show that for any \(\alpha\geq \omega\), the class of completely representable algebras in certain reducts of \(\mathsf{PA}_{\alpha}\)s, that happen to be varieties, is elementary. We show that for \(\alpha\geq \omega\), the the class of polyadic-cylindric algebras dimension \(\alpha\), introduced by Ferenczi, the completely representable algebras (slightly altering representing algebras) coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension \(n\) and \(\mathsf{PA}_{\alpha}\)s for \(\alpha\) an infinite ordinal, proving negative results for the first and positive ones for the second.

In this paper we prove that the class of all relation algebras which are isomorphic to relation algebras of the form φ(L) for some semigroupL (so called semigroup relation algebras) is not first-order axiomatizable.

The article is devoted to the historiography of the application of classical logic algebra in the models of technical systems. This problem is analyzed in both the historical and contemporary contexts. Particular attention is given to a section of discrete mathematics, the theory of reliability, synthesis and complexity of control systems, in particular, mathematical modeling of the synthesis of reliable functions and functional circuits from unreliable gates. The technical implementation of reliable logic algebra circuits in the logic function devices, using an unreliable element base, is one of the topical areas in applied logic algebra

A class of nonclassical relation algebras that correspond to Post logics is introduced and a method of algebraization of those logics is proposed. Relational semantics for Post logics leads to a Rasiowa-Sikorski style proof system for Post logics. A logic LPo intended to provide a formal tool to verify equations in Post relation algebras is defined. Two kinds of rules for the relational logic are defined: decomposition rules enabling the decomposition of relational formulas into some simpler formulas, depending on symbols of relational operations occurring in the formulas; and specific rules, which correspond to semantical postulates assumed in the models of the relational logic. The rules apply to finite sequences of formulas. As a result of application of a rule, a family of new sequences is obtained. >

The central result of this paper is that every pair-dense relation algebra is completely representable. A relation algebra is said to be pair-dense if every nonzero element below the identity contains a pair. A pair is the relation algebraic analogue of a relation of the form {(a, a), (b, b)} (with a = b allowed). In a simple pair-dense relation algebra, every pair is either a point (an algebraic analogue of {(a, a)}) or a twin (a pair which contains no point). In fact, every simple pair-dense relation algebra 2( is completely representable over a set U iS | Ul = K + 2R, where K iS the number of points of 2( and A is the number of twins of 2( . A relation algebra is said to be point-dense if every nonzero element below the identity contains a point. In a point-dense relation algebra every pair is a point, so a simple point-dense relation algebra 2( is completely representable over U iS IUI = c, where K iS the number of points of 2( . This last result actually holds for semiassociative relation algebras, a class of algebras strictly containing the class of relation algebras. It follows that the relation algebra of all binary relations on a set U may be characterized as a simple complete point-dense semiassociative relation algebra whose set of points has the same cardinality as U . Semiassociative relation algebras may not be associative, so the equation (x; y); z = x; (y; z) may fail, but it does hold if any one of x, y, or z is 1. In fact, any rearrangement of parentheses is possible in a term of the form x0; ...; X(X_1 X in case one of the XKSS is 1. This result is proved in a general setting for a special class of groupoids.

Consider a set I which, in the text which follows, will play the role of a universal set. The different subsets of I form a set on which may be defined three operations: union, intersection and complementation. Endowed with these, the set of subsets of I then constitutes an ‘algebra’, usually known as an “algebra of sets” or an “algebra of classes”. From the definitions of set, subset and those of the three operations above the diverse properties of this algebra can be demonstrated. The more important of these properties will be stated later. They will be justified with the aid of Venn diagrams. However, by way of an exercise the reader can carry out rigorous abstract demonstrations based on the definitions in Chapter 1. It will be shown by an example (relationship E9 below) which type of demonstration can then be made.


 
 
 Extending a construction of Andreka, Givant, and Nemeti (2019), we construct some finite vector spaces and use them to build finite non-representable relation algebras. They are simple, measurable, and persistently finite, and they validate arbitrary finite sets of equations that are valid in the variety RRA of representable relation algebras. It follows that there is no finitely axiomatisable class of relation algebras that contains RRA and validates every equation that is both valid in RRA and preserved by completions of relation algebras. Consequently, the variety generated by the completions of representable relation algebras is not finitely axiomatisable. This answers a question of Maddux (2018). 
 
 

The paper explores categorical interconnections between lattice-valued relational systems and algebras of Fitting’s lattice-valued modal logic. We define lattice-valued Boolean systems, and then we study adjointness and co-adjointness of functors defined on them. As a result, we get a duality for algebras of lattice-valued logic. Following this duality result, we establish a duality for algebras of lattice-valued modal logic.

A Galois correspondence between cylindric set algebras and permutation groups is presented in this paper.Moreover, the Galois connection is used to help establish two important algebraic properties for certain classes of finite-dimensional cylindric algebras, namely the amalgamation property and the property that epimorphisms are surjective.The importance of the amalgamation property (AP for short) in algebraic logic has been recognized for a long time.In [16] the connection with the interpolation property of first-order logic is discussed.The positive amalgamation results from the author's thesis [2] and their extensions announced in [3] are cited in 2.4 of [16].These results are established, with a slight improvement, in 4 below.The key to this study is a lattice anti-isomorphism between subgroups of the symmetric group 5 nd subalgebras of the full set algebra 31 (a, p) when 0</i<a + l<.This Galois theory is developed in 2.It is applied in 3 to show that the algebra 9(o, p), 0</i<a + l<w, is homogeneous and the variety it generates has enough injectives.The announcement [3] dealt with the case ju < a.In this treatment of Galois theory we also include the results of H. Andrka and I. Nmeti which show that the theory works for p = a + 1, but for no larger p.In 5 the property (ES) that all epimorphisms are surjective is established for certain varieties of CAa's.It has been shown by I. Nmeti (see [14]) that the ES property for classes of CAa's is related to Beth's definability result.The question of when the ES property holds was raised in [8, p. 311, Problem 10].To put the results in this paper in perspective, perhaps an additional remark will be useful.Many of the results assert that some property (that depends on a and p.) holds if /i < a + 1 < (o.Upon reading a preliminary draft of the paper, Andrka and Nmeti showed that most of the main results could not be improved in the sense that examples show that the properties fail for a > p. > a + 1.In particular, this applies to Theorems 2.2, 2.8, 2.9, 3.7, 3.8(2), Corollaries 3.10, 5.5 and Lemma 5.2(2).Essentially the only result where the value of p is not known to be the best possible is in Theorem 2.5.

Typed query languages for databases containing queries

The motivation for this paper comes from the following sources. First, one can observe that the two major concepts underlying the methods of reasoning with incomplete information are the concept of degree of truth of a piece of information and the concept of approximation of a set of information items. We shall refer to the theories employing the concept of degree of truth as to theories of fuzziness and to the theories employing the concept of approximation as to theories of roughness (see [6] for a survey). The algebraic structures relevant to these theories are residuated lattices ([7], [12], [13], [16], [17], [18], [22], [23]) and Boolean algebras with operators ([19], [21], [10], [11]), respectively. Residuated lattices provide an arithmetic of degrees of truth and Boolean algebras equipped with the appropriate operators provide a method of reasoning with approximately determined information. Both classes of algebras have a lattice structure as a basis. Second, both theories of fuzziness and theories of roughness develop generalizations of relation algebras to algebras of fuzzy relations [20] and algebras of rough relations ([4], [5], [9]), respectively. In both classes a lattice structure is a basis. Third, not necessarily distributive lattices with modal operators, which can be viewed as most elementary approximation operators, are recently developed in [24] (distributive lattices with operators are considered in [14] and [25]). With this background, our aim is to begin a systematic study of the classes of algebras that have the structure of a (not necessarily distributive) lattice and, moreover, in each class there are some operators added to the lattice which are relevant for binary relations. Our main interest is in developing relational representation theorems for the classes of lattices with operators under consideration. More precisely, we wish to guarantee that each algebra of our classes is isomorphic to an algebra of binary relations on a set. We prove the theorems of that form by suitably extending the Urquhart representation theorem for lattices ([26]) and the representation theorems presented in [1]. The classes defined in the paper are the parts which put together lead to what might be called lattice-based relation algebras. Our view is that these algebras would be the weakest structures relevant for binary relations. All the other algebras of binary relations considered in the literature would then be their signature and/or axiomatic extensions.Throughout the paper we use the same symbol for denoting an algebra or a relational system and their universes.KeywordsBinary RelationDistributive LatticeRepresentation TheoremResiduated LatticeRelation 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.

We formulate some natural conditions which should be satisfied in an extension of the relational algebra from the usual relations to tables with null values, and we explain the motivation behind these conditions. Roughly speaking, our conditions say that the extended algebra makes it possible to correctly compute the true tuples in the result of applying a relational expression (query) to tables with nulls (database state), and that the computation can be carried out recursively, following the structure of the expression. We prove that these conditions are exactly equivalent to other conditions proposed earlier by the author and T. Imielinski. We give a simple proof of the correctness of the extension of the relational algebra to tables with marked nulls, where the nulls are treated as if they were regular values, and which supports the operations of projection, positive selection, union, join and renaming of attributes. We also show that the result of the naive evaluation of such an expression (query) is equal to the response to the query as defined --- in a proof-theoretic framework --- by Reiter.

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 α and β for which the equational theories of CAα and RCAβ are undecidable. (CAα and RCAβ are the classes of α-dimensional cylindric algebras and representable β-dimensional cylindric algebras. See [4] for definitions.) This problem was considered in [3]. It was known that RCA0 = CA0 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 CAα and RCAβ are undecidable whenever 4 ≤ α and 3 ≤ β. (Tarski's argument also applies to some varieties K ⊆ RCAβ with 3 ≤ β and to any variety K such that RCAα ⊆ K ⊆ CAα and 4 ≤ α.)Thus the only cases left open in 1961 were CA2, RCA2 and CA3. Shortly there-after Henkin proved, in one of Tarski's seminars at Berkeley, that the equational theory of CA2 is decidable, and Scott proved that the set of valid sentences in a first-order language with only two variables is recursive [11]. (For a more model-theoretic proof of Scott's theorem see [9].) Scott's result is equivalent to the decidability of the equational theory of RCA2, so the only case left open was CA3.

The consistency problem for a class of algebraic structures asks for an algorithm to decide, for any given conjunction of equations, whether it admits a non-trivial satisfying assignment within some member of the class. For the variety of all groups, this is the complement of the triviality problem, shown undecidable by by Adyan [Algorithmic unsolvability of problems of recognition of certain properties of groups. (Russian) Dokl. Akad. Nauk SSSR (N.S.) 103 (1955) 533–535] and Rabin [Recursive unsolvability of group theoretic problems, Ann. of Math. (2) 67 (1958) 172–194]. For the class of finite groups, it amounts to the triviality problem for profinite completions, shown undecidable by Bridson and Wilton [The triviality problem for profinite completions, Invent. Math. 202 (2015) 839–874]. We derive unsolvability of the consistency problem for the class of (finite) modular lattices and various subclasses; in particular, the class of all subspace lattices of finite-dimensional vector spaces over a fixed or arbitrary field of characteristic [Formula: see text] and expansions thereof, e.g. the class of subspace ortholattices of finite-dimensional Hilbert spaces. The lattice results are used to prove unsolvability of the consistency problem for (finite) rings with unit and (finite) representable relation algebras. These results in turn apply to equations between simple expressions in Grassmann–Cayley algebra and to functional and embedded multivalued dependencies in databases.