Functional equations, constraints, definability of function classes, and functions of Boolean variables

The paper deals with classes of functions of several variables defined on an arbitrary set A and taking values in a possibly different set B. Definability of function classes by functional equations is shown to be equivalent to definability by relational constraints, generalizing a fact established by Pippenger in the case A = B = {0,1}. Conditions for a class of functions to be definable by constraints of a particular type are given in terms of stability under certain functional compositions. This leads to a correspondence between functional equations with particular algebraic syntax and relational constraints with certain invariance properties with respect to clones of operations on a given set. When A = {0,1} and B is a commutative ring, such B-valued functions of n variables are represented by multilinear polynomials in n indeterminates in B[X1,..., Xn], Functional equations are given to describe classes of field-valued functions of a specified bounded degree. Classes of Boolean and pseudo-Boolean functions are covered as particular cases

Definability of linear equation systems over groups and rings

Motivated by the quest for a logic for PTIME and recent insights that the descriptive complexity of problems from linear algebra is a crucial aspect of this problem, we study the solvability of linear equation systems over finite groups and rings from the viewpoint of logical (inter-)definability. All problems that we consider are decidable in polynomial time, but not expressible in fixed-point logic with counting. They also provide natural candidates for a separation of polynomial time from rank logics, which extend fixed-point logics by operators for determining the rank of definable matrices and which are sufficient for solvability problems over fields. Based on the structure theory of finite rings, we establish logical reductions among various solvability problems. Our results indicate that all solvability problems for linear equation systems that separate fixed-point logic with counting from PTIME can be reduced to solvability over commutative rings. Moreover, we prove closure properties for classes of queries that reduce to solvability over rings, which provides normal forms for logics extended with solvability operators. We conclude by studying the extent to which fixed-point logic with counting can express problems in linear algebra over finite commutative rings, generalising known results on the logical definability of linear-algebraic problems over finite fields

Function classes and relational constraints stable under compositions with clones

The general Galois theory for functions and relational constraints over arbitrary sets described in the authors' previous paper is refined by imposing algebraic conditions on relations

Homogeneous Structures: Model Theory meets Universal Algebra (online meeting)

The workshop "Homogeneous Structures: Model Theory meets Universal Algebra'' was centred around transferring recently obtained advances in universal algebra from the finite to the infinite. As it turns out, the notion of homogeneity together with other model-theoretic concepts like $\omega$-categoricity and the Ramsey property play an indispensable role in this endeavour

Linearly definable classes of Boolean functions

International audienceIn this paper we address the question "How many properties of Boolean functions can be defined by means of linear equations?" It follows from a result by Sparks that there are countably many such linearly definable classes of Boolean functions. In this paper, we refine this result by completely describing these classes. This work is tightly related with the theory of function minors and stable classes, a topic that has been widely investigated in recent years by several authors including Maurice Pouzet

Algebraic Characterization of FO for Scattered Linear Orderings

We prove that for the class of sets of words indexed by countable scattered linear orderings, there is an equivalence between definability in first-order logic, star-free expressions with marked product, and recognizability by finite aperiodic semigroups which satisfy some additional equation

Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201