Related papers: Second-Order Algebraic Theories
This paper proposes a new category theoretic account of equationally axiomatizable classes of algebras. Our approach is well-suited for the treatment of algebras equipped with additional computationally relevant structure, such as ordered…
We define an extension of predicate logic, called Binding Logic, where variables can be bound in terms and in propositions. We introduce a notion of model for this logic and prove a soundness and completeness theorem for it. This theorem is…
Monadic second order logic and linear temporal logic are two logical formalisms that can be used to describe classes of infinite words, i.e., first-order models based on the natural numbers with order, successor, and finitely many unary…
Rational word languages can be defined by several equivalent means: finite state automata, rational expressions, finite congruences, or monadic second-order (MSO) logic. The robust subclass of aperiodic languages is defined by: counter-free…
We consider two-variable first-order logic on finite words with a fixed number of quantifier alternations. We show that all languages with a neutral letter definable using the order and finite-degree predicates are also definable with the…
We introduce a natural Turing-complete extension of first-order logic FO. The extension adds two novel features to FO. The first one of these is the capacity to add new points to models and new tuples to relations. The second one is the…
We analyse the pseudofinite monadic second order theory of words over a fixed finite alphabet. In particular we present an axiomatisation of this theory, working in a one-sorted first order framework. The analysis hinges on the fact that…
In this paper, our aim is to briefly survey and articulate the logical and philosophical foundations of using (first-order) logic to represent (probabilistic) knowledge in a non-technical fashion. Our motivation is three fold. First, for…
We present syntactic characterisations for the union closed fragments of existential second-order logic and of logics with team semantics. Since union closure is a semantical and undecidable property, the normal form we introduce enables…
We present a mathematical framework for mapping second-order logic relations onto a simple state vector algebra. Using this algebra, basic theorems of set theory can be proven in an algorithmic way, hence by an expert system. We illustrate…
We contribute to the refined understanding of the language-logic-algebra interplay in the context of first-order properties of countable words. We establish decidable algebraic characterizations of one variable fragment of FO as well as…
This book can be seen either as a text on theorem proving that uses techniques from general algebra, or else as a text on general algebra illustrated and made concrete by practical exercises in theorem proving. The book considers several…
First-Order Logic (FOL) is widely regarded as one of the most important foundations for knowledge representation. Nevertheless, in this paper, we argue that FOL has several critical issues for this purpose. Instead, we propose an…
This paper is the second in a series of three, the aim of which is to construct algebraic geometry over a free metabelian Lie algebra $F$. For the universal closure of free metabelian Lie algebra of finite rank $r \ge 2$ over a finite field…
Multi-model databases are designed to store, manage, and query data in various models, such as relational, hierarchical, and graph data, simultaneously. In this paper, we provide a theoretical basis for querying categorical databases. We…
Categorical universal algebra can be developed either using Lawvere theories (single-sorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how…
We study the question of whether, for a given class of finite graphs, one can define, for each graph of the class, a linear ordering in monadic second-order logic, possibly with the help of monadic parameters. We consider two variants of…
Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a general representation theorem for a wide class of…
In the setting of modern mathematical logic and model theory, classification theory has been one of the landmark achievements of the field. Likewise, the classification of UHF-algebras and AF-algebras were substantial contributions to the…
We introduce a class of singular partial differential equations, the second-order hyperbolic Fuchsian systems, and we investigate the associated initial value problem when data are imposed on the singularity. First of all, we analyze a…