Related papers: A Binary Quantifier for Definite Descriptions in I…
We define a model of predicate logic in which every term and predicate, open or closed, has an absolute denotation independently of a valuation of the variables. For each variable a, the domain of the model contains an element [[a]] which…
Interpretability and explainability are among the most important challenges of modern artificial intelligence, being mentioned even in various legislative sources. In this article, we develop a method for extracting immediately human…
We consider grammar-restricted exact learning of formulas and terms in finite variable logics. We propose a novel and versatile automata-theoretic technique for solving such problems. We first show results for learning formulas that…
Automatic structures are first-order structures whose universe and relations can be represented as regular languages. It follows from the standard closure properties of regular languages that the first-order theory of an automatic structure…
Inductive and coinductive specifications are widely used in formalizing computational systems. Such specifications have a natural rendition in logics that support fixed-point definitions. Another useful formalization device is that of…
We advocate the use of de Bruijn's universal abstraction $\lambda^\infty$ for the quantification of schematic variables in the predicative setting and we present a typed $\lambda$-calculus featuring the quantifier $\lambda^\infty$…
Let $F$ be the set of functions from an infinite set, $S$, to an ordered ring, $R$. For $f$, $g$, and $h$ in $F$, the assertion $f = g + O(h)$ means that for some constant $C$, $|f(x) - g(x)| \leq C |h(x)|$ for every $x$ in $S$. Let $L$ be…
We discuss a new approach to functional interpretations based on uniform quantification and relativization. The uniform quantification in the background permits a more penetrating analysis of principles related to collection and…
The purpose of this paper is to give an easy to understand with step-by-step explanation to allow interested people to fully appreciate the power of natural deduction for first-order logic. Natural deduction as a proof system can be used to…
We introduce a variant of free logic (i.e., a logic admitting terms with nonexistent referents) that accommodates truth-value gluts as well as gaps. Employing a suitable expansion of the Belnap-Dunn four-valued logic, we specify a…
In this paper a constructive formalization of quantifier elimination is presented, based on a classical formalization by Tobias Nipkow. The formalization is implemented and verified in the programming language/proof assistant Agda. It is…
Definite descriptions are phrases of the form 'the $x$ such that $\varphi$', used to refer to single entities in a context. They are often more meaningful to users than individual names alone, in particular when modelling or querying data…
Binary logic programs can be obtained from ordinary logic programs by a binarizing transformation. In most cases, binary programs obtained this way are less efficient than the original programs. (Demoen, 1992) showed an interesting example…
We introduce a method for computing immediately human interpretable yet accurate classifiers from tabular data. The classifiers obtained are short Boolean formulas, computed via first discretizing the original data and then using feature…
We describe two systems for supporting beginner students in acquiring basic skills in expressing statements in the formalism of first-order predicate logic; the first, called "math dictations", presents users with the task of formalizing a…
The definitional equality of an intensional type theory is its test of type compatibility. Today's systems rely on ordinary evaluation semantics to compare expressions in types, frustrating users with type errors arising when evaluation…
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…
Nominalistic Logic (NL) is a new presentation of Paul Gilmore's Intensional Type Theory (ITT) as a sequent calculus together with a succinct nominalization axiom (N) that permits names of predicates as individuals in certain cases. The…
Dependence logic provides an elegant approach for introducing dependencies between variables into the object language of first-order logic. In [1] generalized quantifiers were introduced in this context. However, a satisfactory account was…
We present a systematic approach for achieving fairness in a binary classification setting. While we focus on two well-known quantitative definitions of fairness, our approach encompasses many other previously studied definitions as special…