Logic prototyping: approach and use cases

In this article, we present an approach to prototyping complex systems and processes using classical predicate logic. The prototype is built by the interpreter based on a logical description of the properties and/or behavior of the designed system. The description contains the definitions of the prototype elements and the constraints that the correct prototype must satisfy. Definitions are used to build a prototype, and constraints are used to analyze it and check the required properties. Definitions are interpreted using direct logic inference, constraints are only checked on the resulting model. A wider class of formulas is used than in well-known logical languages. Computable logical and denotational semantics are defined for them. In the process of building a prototype, logical errors of uncertainty, redefinition of functions, and contradictions are diagnosed. We are given examples of prototype descriptions used for semantic program analysis, space training, transport system design.

Model theory of monadic predicate logic with the infinity quantifier

This paper establishes model-theoretic properties of $$\texttt {M} \texttt {E} ^{\infty }$$ M E ∞ , a variation of monadic first-order logic that features the generalised quantifier $$\exists ^\infty $$ ∃ ∞ (‘there are infinitely many’). We will also prove analogous versions of these results in the simpler setting of monadic first-order logic with and without equality ($$\texttt {M} \texttt {E} $$ M E and $$\texttt {M} $$ M , respectively). For each logic $$\texttt {L} \in \{ \texttt {M} , \texttt {M} \texttt {E} , \texttt {M} \texttt {E} ^{\infty }\}$$ L ∈ { M , M E , M E ∞ } we will show the following. We provide syntactically defined fragments of $$\texttt {L} $$ L characterising four different semantic properties of $$\texttt {L} $$ L -sentences: (1) being monotone and (2) (Scott) continuous in a given set of monadic predicates; (3) having truth preserved under taking submodels or (4) being truth invariant under taking quotients. In each case, we produce an effectively defined map that translates an arbitrary sentence $$\varphi $$ φ to a sentence $$\varphi ^\mathsf{p}$$ φ p belonging to the corresponding syntactic fragment, with the property that $$\varphi $$ φ is equivalent to $$\varphi ^\mathsf{p}$$ φ p precisely when it has the associated semantic property. As a corollary of our developments, we obtain that the four semantic properties above are decidable for $$\texttt {L} $$ L -sentences.

A Constructive Treatment to Elemental Life Forms through Mathematical Philosophy

The quest to understand the natural and the mathematical as well as philosophical principles of dynamics of life forms are ancient in the human history of science. In ancient times, Pythagoras and Plato, and later, Copernicus and Galileo, correctly observed that the grand book of nature is written in the language of mathematics. Platonism, Aristotelian logism, neo-realism, monadism of Leibniz, Hegelian idealism and others have made efforts to understand reasons of existence of life forms in nature and the underlying principles through the lenses of philosophy and mathematics. In this paper, an approach is made to treat the similar question about nature and existential life forms in view of mathematical philosophy. The approach follows constructivism to formulate an abstract model to understand existential life forms in nature and its dynamics by selectively combining the elements of various schools of thoughts. The formalisms of predicate logic, probabilistic inference and homotopy theory of algebraic topology are employed to construct a structure in local time-scale horizon and in cosmological time-scale horizon. It aims to resolve the relative and apparent conflicts present in various thoughts in the process, and it has made an effort to establish a logically coherent interpretation.

Boolean Logic, Fregean Logic, Wittgensteinian Logic and the Processing of Natural Language

Gottlob Frege (1848-1925) transformed the field of logic from what it had remained since the days of Aristotle. Regarded as the founder of modern logic and much of modern philosophy, Frege laid the foundations of predicate logic, first-order predicate calculus and quantificational logic – formal systems central to computer science and mathematics. Frege was not satisfied with the ambiguity and imprecision of ordinary language. He created a new ‘formula language’ with elaborate symbols and definite rules, focused on conceptual content rather than rhetorical style, which he called Begriffsschrift – a formal language for 'pure thought'. Before Frege, George Boole (1815-1864) created what later became known as ‘Boolean logic’ which is fundamental to operations of computer science today. An application of Wittgensteinian logic could help filter authentic information from information disorder (non-information, off-information, mal-information and mis-information). Wittgensteinian logic applied in natural language processing technology (NLP), if possible and via automation, could transform the quality of information online. Many challenges remain.

Mining predicate rules without minimum support threshold

Association rule mining (ARM) is used for discovering frequent itemsets for interesting relationships of associative and correlative behaviors within the data. This gives new insights of great value, both commercial and academic. The traditional ARM techniques discover interesting association rules based on a predefined minimum support threshold. However, there is no known standard of an exact definition of minimum support and providing an inappropriate minimum support value may result in missing important rules. In addition, most of the rules discovered by these traditional ARM techniques refer to already known knowledge. To address these limitations of the minimum support threshold in ARM techniques, this study proposes an algorithm to mine interesting association rules without minimum support using predicate logic and a property of a proposed interestingness measure (g measure). The algorithm scans the database and uses g measure’s property to search for interesting combinations. The selected combinations are mapped to pseudo-implications and inference rules of logic are used on the pseudo-implications to produce and validate the predicate rules. Experimental results of the proposed technique show better performance against state-of-the-art classification techniques, and reliable predicate rules are discovered based on the reliability differences of the presence and absence of the rule’s consequence.