Related papers: Models for Metamath
Among the various forms of reasoning studied in the context of artificial intelligence, qualitative reasoning makes it possible to infer new knowledge in the context of imprecise, incomplete information without numerical values. In this…
Complex system design often proceeds in an iterative fashion, starting from a high-level model and adding detail as the design matures. This process can be assisted by metamodeling techniques that automate some model manipulations and check…
We consider sets/relations/computations defined by *Elementary Inference Systems* I, which are obtained from Smullyan's *elementary formal systems* using Gentzen's notation for inference rules, and proof trees for atoms P(t_1,...,t_n),…
The aim of this project is to attach a geometric structure to the ring of integers. It is generally assumed that the spectrum $\mathrm{Spec}(\mathbb{Z})$ defined by Grothendieck serves this purpose. However, it is still not clear what…
There is an increasing interest in applying recent advances in AI to automated reasoning, as it may provide useful heuristics in reasoning over formalisms in first-order, second-order, or even meta-logics. To facilitate this research, we…
This paper presents a sequent calculus and a dual domain semantics for a theory of definite descriptions in which these expressions are formalised in the context of complete sentences by a binary quantifier $I$. $I$ forms a formula from two…
Several queries and scores have recently been proposed to explain individual predictions over ML models. Given the need for flexible, reliable, and easy-to-apply interpretability methods for ML models, we foresee the need for developing…
In this paper, we unify the study of classical and non-classical algebra-valued models of set theory, by studying variations of the interpretation functions for identity and set-membership. Although, these variations coincide with the…
Unlike some other formal systems, the proof system Metamath has no built-in concept of "decimal number" in the sense that arbitrary digit strings are not recognized by the system without prior definition. We present a system of theorems and…
With infinitely many high-quality data points, infinite computational power, an infinitely large foundation model with a perfect training algorithm and guaranteed zero generalization error on the pretext task, can the model be used for…
The notion of class is ubiquitous in computer science and is central in many formalisms for the representation of structured knowledge used both in knowledge representation and in databases. In this paper we study the basic issues…
We present a categorical theory of the composition methods in finite model theory -- a key technique enabling modular reasoning about complex structures by building them out of simpler components. The crucial results required by the…
We consider the foundational relation between arithmetic and set theory. Our goal is to criticize the construction of standard arithmetic models as providing grounds for arithmetic truth (even in a relative sense). Our method is to…
We show that including degrees of a particular kind of provability in the search target for any theorem-prover in sufficiently powerful formal systems over finite-sized statements preserves well-definition and a sufficient consistency while…
In physics, density matrices are used to represent mixed states, i.e. probabilistic mixtures of pure states. This concept has previously been used to model lexical ambiguity. In this paper, we consider metaphor as a type of lexical…
Much mathematical writing exists that is, explicitly or implicitly, based on set theory, often Zermelo-Fraenkel set theory (ZF) or one of its variants. In ZF, the domain of discourse contains only sets, and hence every mathematical object…
Conceptual models as representations of real-world systems are based on diverse techniques in various disciplines but lack a framework that provides multidisciplinary ontological understanding of real-world phenomena. Concurrently, systems…
We use fast-growing finite and infinite sequences of natural numbers and more complicated constructs to define models of hypercomputation and interpret non-arithmetic predicates, with the strongest extensions reaching full second order…
We clarified the connection between measurements and partitions, and discussed the meaning of semiotics for measurements based on functions. The terms of property and relation quantity were defined by our understanding of partitions and…
The Unified Modeling Language is extensible, and so can be regarded as a family of languages. Implicitly or explicitly, any particular UML model should be accompanied by a definition of the particular UML family member used for the model.…