Propositional Logic Research Articles

On the role of logical separability in knowledge compilation

Knowledge compilation is an alternative solution to address demanding reasoning tasks with high complexity via converting knowledge bases into a suitable target language. The notion of logical separability, proposed by Levesque, offers a general explanation for the tractability of clausal entailment for two remarkable languages: decomposable negation normal form and prime implicates. It is interesting to explore what role logical separability plays in problem tractability. In this paper, we apply the notion of logical separability to a number of reasoning problems within the context of propositional logic: satisfiability checking (CO), clausal entailment checking (CE), model counting (CT), model enumeration (ME) and forgetting (FO), as well as their dual tasks, contributing to several recursive procedures. We provide the corresponding logical separability based properties: CO-logical separability, CE-logical separability, CT-logical separability, ME-logical separability and their duals. Based on these properties, we then identify four novel normal forms: CO-LSNNF, CE-LSNNF, CT-LSNNF and ME-LSNNF, as well as their dual languages. We show that each of them is the necessary and sufficient condition under which the corresponding procedure is correct. We finally integrate the above normal forms into the knowledge compilation map.

Combining Intuitionistic and Classical Propositional Logic: Gentzenization and Craig Interpolation

Combining Intuitionistic and Classical Propositional Logic: Gentzenization and Craig Interpolation

The existential fragment of second-order propositional intuitionistic logic is undecidable

The provability problem in intuitionistic propositional second-order logic with existential quantifier and implication ( ∃ , → ) is proved to be undecidable in presence of free type variables (constants). This contrasts with the result that inutitionistic propositional second-order logic with existential quantifier, conjunction and negation is decidable.

A novel way to build expert systems with infinite-valued attributes

<abstract><p>An expert system is a computer program that uses the knowledge of an expert to solve problems in a specific domain. Expert systems are used in a wide variety of fields, such as medicine, financial diagnosis and engineering. The attributes of an expert system are the characteristics of the problems that the system can solve. In traditional expert systems, attributes typically have a finite number of possible values. However, in scenarios where an attribute can assume a value from an infinite (or significantly large finite) set, the expert system cannot be represented using propositional logic. Until now, no method had been identified to implement such a system on a Computer Algebra System. Here, we break new ground by presenting a model that not only addresses this gap but also provides a fresh perspective on previous results. In fact, these prior results can be viewed as specific instances within the broader framework of our proposed solution. In this paper, we put forth an algebraic approach for the development of expert systems capable of handling attributes with infinite values, thereby expanding the problem-solving capacity of these systems.</p></abstract>

CCN+: A neuro-symbolic framework for deep learning with requirements

For their outstanding ability of finding hidden patterns in data, deep learning models have been extensively applied in many different domains. However, recent works have shown that, if a set of requirements expressing inherent knowledge about the problem at hand is given, then neural networks often fail to comply with them. This represents a major drawback for deep learning models, as requirements compliance is normally considered a necessary condition for standard software deployment. In this paper, we propose a novel neuro-symbolic framework able to make any neural network compliant by design to a given set of requirements over the output space expressed in full propositional logic. This framework, called CCN+, integrates the requirements into the output layer of the neural network by applying multiple inference rules that ensure compliance with the requirements and adapts the standard binary cross-entropy loss function to the requirement output layer. As a result, not only the outputted predictions are guaranteed to be compliant with the requirements, but the neural network itself learns how to exploit the domain knowledge expressed by the requirements to get better performance. We conduct an extensive experimental evaluation of CCN+ on 19 real-world multi-label classification datasets with propositional logic requirements, including a challenging dataset for autonomous driving. Our experimental analysis confirms that CCN+ is able to outperform both its neural counterparts and the state-of-the-art models.

АНАЛІЗ ПРИРОДНОМОВНИХ ЗНАНЬ В УМОВАХ НЕВИЗНАЧЕНОСТІ НА ОСНОВІ ДЕСКРИПТИВНИХ ЛОГІК

The article overviews the means for describing and formally analyzing natural- language text knowledge under uncertainty. We consider a family of classic attribute languages and logics based on them, their properties, problems, and solution tools. We also give an overview of propositional n-valued logics and fuzzy logics, their syntax, and semantics. Based on the considered logical constructions, we propose syntax and set-theoretic interpretation of n-valued description logic ALCQn that provides means for describing concept intersection, union, complement, value restrictions, and qualitative and quantitative constraints. We consider the means for solving key problems of reasoning over such logics: executability, augmentation, equivalence, and disjunctivity. As an algorithm for calculating executability degree, we consider an extension of the tableau algorithm often used for first-order logic with solving simple numerical constraints. We prove that the algorithm is terminal, complete, and non-contradictory. We also provide several applications for the provided formal representation in natural language processing, including extending results of machine learning models, combining knowledge from multiple sources, and formally describing uncertain facts.

On the quantified version of the Belnap-Dunn modal logic

We develop a quantified version of the propositional modal logic $\mathsf{BK}$ from an article by Odintsov and Wansing, which is based on the (non-modal) Belnap-Dunn system; we denote this version by $\mathsf{QBK}$. First, by using the canonical model method we prove that $\mathsf{QBK}$, as well as some important extensions of it, is strongly complete with respect to a suitable possible world semantics. Then we define translations (in the spirit of Gödel-McKinsey-Tarski) that faithfully embed the quantified versions of Nelson's constructive logics into suitable extensions of $\mathsf{QBK}$. In conclusion, we discuss interpolation properties for $\mathsf{QBK}$-extensions. Bibliography: 21 titles.

Theory of ∆ conditional randomized truth degree in Gödel n-valued propositional logic system of adding ∆ operator

Theory of ∆ conditional randomized truth degree in Gödel n-valued propositional logic system of adding ∆ operator

Сognitive Dimension of the Word-forming Nest with Dialectal Vertices Yeden / Yeden (“єде́н / є́ден”): Logical Propositions

In this article, a current task of modern derivatolofical studies is addressed – the cognitive structuring of derived words in denumerative word formation, grouped according to word-forming semantics, considering motivational relationships through the prism of divergent-convergent properties of the content of the generative word – the vertex of the word-formation nest. The analysis was carried out on the basis of a word-forming nest with variational vertices belonging to the dialect layer of the lexicon – yeden /yeden (“еден / еден”). The specified motivational words are characterized by a polysemantic continuum, so they are characterized by differences in creative meanings both directly and indirectly deep into the word-forming nest. Such complex units of word-formation have the status of being divergent. Motivating relations in the dyad «the vertex of the word-forming nest – the denumerative» can record mental knowledge in the aspect of the linguo-cognitive approach to the analysis of linguistic phenomena: the genetic codes of the word-formation, preserved in the derivative, as well as the semantic increments present in the meaning structure of the derivatives are revealed. Analyzing the cognitive semantics within word-forming values, the elements of frame are established, the components of which are derivative units of modification and transposition types of derivation, namely: subframe, cognitive slot, verbalized slot, propositional structure, and two-component logical proposition. The prospects of the research consist in the systematіc cognitive interpretation of derivatives based on various word-formation units, which lead the formatives that belong not only to the core, but also to the periphery of the motivational base, which will make it possible to comprehensively model the semantic positions of derived words, or more broadly, the mental manifestation of the word-formation level Ukrainian language.

A Supraclassical Probabilistic Entailment Relation

The paper presents an original supraclassical nontrivial plausible entailment relation $\vapprox$ that employs Kolmogorov's probability theory. Its crucial feature is the primitiveness of a conditional probability, which one calculates with the help of the method of truth tables for classical propositional logic. I study the properties of the entailment relation in question. In particular, I show that while being supraclassical, i.e., all classical entailments and valid formulas are $\vapprox$-valid, but not vice versa, it is not trivial and enjoys the same form of inconsistency as classical entailment ⊧ does. I specify the place of the proposed probability entailment relation in certain classifications of nonclassical entailment relations. In particular, I use Douven's analysis of some probabilistic entailment relations that contains dozens of properties that are crucial for any probabilistic entailment relation, as well as Hlobil's choosing your nonmonotonic logic: shopper’s guide, due to the fact that $\vapprox$ is not monotonic, and Cobreros, Egré, Ripley, van Rooij's entailment relations for tolerant reasoning. At last, I perform a comparative analysis of classical, the proposed, and some other entailment relations closely related to the latter: those introduced by Bocharov, Markin, Voishvillo, Degtyarev, Ivlev, where the last two entailment relations are based on the so-called principle of reverse deduction, which is an intuitively acceptable way to connect classical and probabilistic entailment relations.

G. Boole, Not J. M. Keynes, Is the Founder of the Logical Approach to Probability

A fundamental and basic error committed by all economists and philosophers doing work on Keynes is their failure to realize that the original, logical theory of probability, with the consequent emphasis placed on relational, propositional logic, imprecise probability and interval valued approaches to decision making, was developed by George Boole and not by J. M. Keynes.
 All one needs to do is read chapters I, II and X to XVII of Keynes’s 1921 A Treatise on Probability and compare Keynes’s chapters with chapters I, XI, XII and XVI to XXI of Boole’s The Laws of Thought. A shorter route to this conclusion is to skip all but chapter I of Boole and compare it with chapter I of Keynes. Why no social scientist, behavioral scientist, economist, or philosopher has been able to do this simple task is a question that needs to be investigated by psychologists and psychiatrists looking for problems related to cognitive dissonance.
 A careful reader of chapters I and II of Keynes’s A Treatise on Probability will soon realize that Keynes is talking explicitly about Boole’s logical, objective, probability relation. This relation has absolutely nothing to do with Plato’s theory of forms. How do we come to know this relationship? Boole states that we come to know this relationship through both observation (empirical) and reflection (intuition or perception).
 This paper shows that there is a deep Boole-Keynes connection that has been overlooked in the literature on Keynes. Keynes built his logical theory of probability on Boole. One can extend Hishiyama’s 1969 conjecture, about Keynes’s A Treatise on Probability not having been read by any economist, to also conclude that Boole’s The Laws of Thought was not read by any other economist except Keynes.

Interpretable land cover classification with modal decision trees

ABSTRACT Land cover classification (LCC) refers to the task of classifying each pixel in satellite/aerial imagery by predicting a label carrying information about its nature. Despite the importance of having transparent, symbolic decision models, in the recent literature, LCC has been mainly approached with black-box functional models, that are able to leverage the spatial dimensions within the data. In this article, we argue that standard symbolic decision models can be extended to perform a form of spatial reasoning that is adequate for LCC. We propose a generalization of a classical decision tree learning model, based on replacing propositional logic with a modal spatial logic, and provide a CART-like learning algorithm for it. We evaluate its performance at five different LCC tasks, showing that this technique leads to classification models whose performances are superior to those of their propositional counterpart, and at least comparable with those of non-symbolic ones. Ultimately, we show that spatial decision trees and random forests are able to extract complex, but interpretable spatial patterns.

DAG-Based Formal Modeling of Spark Applications with MSVL

Apache Spark is a high-speed computing engine for processing massive data. With its widespread adoption, there is a growing need to analyze its correctness and temporal properties. However, there is scarce research focused on the verification of temporal properties in Spark programs. To address this gap, we employ the code-level runtime verification tool UMC4M based on the Modeling, Simulation, and Verification Language (MSVL). To this end, a Spark program S has to be translated into an MSVL program M, and the negation of the property P specified by a Propositional Projection Temporal Logic (PPTL) formula that needs to be verified is also translated to an MSVL program M1; then, a new MSVL program “M and M1” can be compiled and executed. Whether program S violates the property P is determined by the existence of an acceptable execution of “M and M1”. Thus, the key issue lies in how to formalize model Spark programs using MSVL programs. We previously proposed a solution to this problem—using the MSVL functions to perform Resilient Distributed Datasets (RDD) operations and converting the Spark program into an MSVL program based on the Directed Acyclic Graph (DAG) of the Spark program. However, we only proposed this idea. Building upon this foundation, we implement the conversion from RDD operations to MSVL functions and propose, as well as implement, the rules for translating Spark programs to MSVL programs based on DAG. We confirm the feasibility of this approach and provide a viable method for verifying the temporal properties of Spark programs. Additionally, an automatic translation tool, S2M, is developed. Finally, a case study is presented to demonstrate this conversion process.

Algorithmic predictions and pre-emptive violence: artificial intelligence and the future of unmanned aerial systems

AbstractThe military rationale of a pre-emptive strike is predicated upon the calculation and anticipation of threat. The underlying principle of anticipation, or prediction, is foundational to the operative logic of AI. The deployment of predictive, algorithmically driven systems in unmanned aerial systems (UAS) would therefore appear to be all but inevitable. However, the fatal interlocking of martial paradigms of pre-emption and models of predictive analysis needs to be questioned insofar as the irreparable decisiveness of a pre-emptive military strike is often at odds with the probabilistic predictions of AI. The pursuit of a human right to protect communities from aerial threats needs to therefore consider the degree to which algorithmic auguries—often erroneous but nevertheless evident in the prophetic mechanisms that power autonomous aerial apparatuses—essentially authorise and further galvanise the long-standing martial strategy of pre-emption. In the context of unmanned aerial systems, this essay will outline how AI actualises and summons forth “threats” through (i) the propositional logic of algorithms (their inclination to yield actionable directives); (ii) the systematic training of neural networks (through habitually biased methods of data-labelling); and (iii) a systemic reliance on models of statistical analysis in the structural design of machine learning (which can and do produce so-called “hallucinations”). Through defining the deterministic intentionality, systematic biases and systemic dysfunction of algorithms, I will identify how individuals and communities—configured upon and erroneously flagged through the machinations of so-called “black box” instruments—are invariably exposed to the uncertainty (or brute certainty) of imminent death based on algorithmic projections of “threat”.

A TERMINATING INTUITIONISTIC CALCULUS

Abstract A terminating sequent calculus for intuitionistic propositional logic is obtained by modifying the R $\supset $ rule of the labelled sequent calculus $\mathbf {G3I}$ . This is done by adding a variant of the principle of a fortiori in the left-hand side of the premiss of the rule. In the resulting calculus, called ${\mathbf {G3I}}_{\mathbf {t}}$ , derivability of any given sequent is directly decidable by root-first proof search, without any extra device such as loop-checking. In the negative case, the failed proof search gives a finite countermodel to the sequent on a reflexive, transitive, and Noetherian Kripke frame. As a byproduct, a direct proof of faithfulness of the embedding of intuitionistic logic into Grzegorcyk logic is obtained.

Quantitative Robustness for Signal Temporal Logic With Time-Freeze Quantifiers

Signal Temporal Logic (STL) is a variant of Metric Temporal Logic (MTL) which can express intricate temporal requirements over signals, and has found wide adoption for expressing requirements over complex control systems models. A key factor in the success of STL has been that of quantitative robustness, and the development of efficient algorithms for computing the robustness values over traces. The real-valued quantitative robustness of a signal with respect to an STL property quantifies the degree of satisfaction or violation of the property by the signal. In this work we introduce a notion of robustness for a more expressive logic, Timed Signal Temporal Logic (TSTL), which can be seen as Timed Propositional Temporal Logic (TPTL) with predicates defined over real-valued signals. This logic can express many natural engineering requirements that STL cannot. We also develop algorithms for computing this robustness value over traces in the pointwise semantics. While the robustness computation for general TSTL formulae is PSPACEhard due to the PSPACE-hardness of the monitoring problem for TPTL, for the special case of one variable TSTL, a fragment of TSTL which is still more expressive than STL, we develop an optimized algorithm which computes robustness in time linear in the length of the trace. Finally, we experimentally validate the tractability of our algorithms with our prototype tool in Matlab.

The connection of mathematics and logic in the structure of axiomatized and formalized theories

The aim of the research is to study the relationship between logic and mathematics in the structure of axiomatized and formalized scientific theories. The object of the study is the explication of this connection and its explanation. The subject of the study is syntactic and semantic views on the structure of scientific theories, the relationship between logic and mathematics has not been studied in detail in them. In the syntactic view, the structure of the theory is understood as a linguistic construct build from various logical propositions of the theoretical level, correspondence propositions and observation propositions. The structure of the theory does not take into account the variety of model representations of the theory that generate a variety of language constructs. The semantic view overcomes this disadvantage, and in it the structure of the theory is presented as a hierarchy of models: from axioms to theoretical-level models, experimental models and data models. The structure of the theory, the connection of logic and mathematics were studied using comparative analysis, methods of interpretive analysis and reconstruction of scientific theories. The methods made it possible to explicate mathematical concepts in the structure of the theory and correlate them with logic and natural language. Comparative analysis has shown that in the syntactic view, the connection between logic and mathematics lies in the fact that mathematical concepts of physics are interpreted in the language of logic of first-order predicates with equality. The connection between mathematical concepts is provided by the axiomatic method, which serves as a means of formalizing concepts. Mathematics comes down to logic. In the semantic approach, in order to identify the connection between mathematics and logic, it was necessary to reconstruct the structure of non-relativistic quantum mechanics. With the help of the set-theoretic predicate of Suppes, its axioms were determined, the connection between mathematical structures, postulates of the theory, axioms, and observable quantities was established. Logic and mathematics are related to each other in such a way that metamathematics or linguistics is a part of mathematics. Mathematics includes set theory and model theory, i.e. mathematical logic. The connection of mathematical formalisms with phenomena and with natural language remains problematic, and there is this drawback in the syntactic approach. The novelty lies in the fact that the research contributes to the methodology and logic of science, to the explanation of the connection between logic and mathematics in scientific theory, which was illustrated by various examples from various fields of physics.

Defining Logical Systems via Algebraic Constraints on Proofs

Abstract We present a comprehensive programme analysing the decomposition of proof systems for non-classical logics into proof systems for other logics, especially classical logic, using an algebra of constraints. That is, one recovers a proof system for a target logic by enriching a proof system for another, typically simpler, logic with an algebra of constraints that act as correctness conditions on the latter to capture the former; e.g. one may use Boolean algebra to give constraints in a sequent calculus for classical propositional logic to produce a sequent calculus for intuitionistic propositional logic. The idea behind such forms of decomposition is to obtain a tool for uniform and modular treatment of proof theory and to provide a bridge between semantics logics and their proof theory. The paper discusses the theoretical background of the project and provides several illustrations of its work in the field of intuitionistic and modal logics: including, a uniform treatment of modular and cut-free proof systems for a large class of propositional logics; a general criterion for a novel approach to soundness and completeness of a logic with respect to a model-theoretic semantics; and a case study deriving a model-theoretic semantics from a proof-theoretic specification of a logic.

Frege and the Logic of the Historical Proposition

Abstract This article argues that history played a larger role in the thought of Gottlob Frege than has usually been acknowledged. Frege’s logical writings frequently employed statements about the past as examples that included references to historical persons. Frege also described history as a science and argued that historical propositions could support valid inferences and reliably identify historical persons and events. But Frege’s eternalist theory of reference, designed primarily for formal concepts and objects, struggled to accommodate such propositions. Identifying an objective referent for the subjectivity of historical actors was particularly problematic. The article suggests that Frege’s writings are interesting for the philosopher of history for at least two reasons: first, his work is clarificatory when considering the key features that historical propositions must have to count as objective knowledge, and second, it foreshadowed the issues with historicity that analytical philosophy experienced in the twentieth century. It concludes that the problems Frege raised exposed the need for a new concept of inter-subjectivity to replace his own Platonic foundation for objectivity.

Some Logics in the Vicinity of Interpretability Logics

In this paper we shall define semantically some families of propositional modal logics related to the interpretability logic \(\mathbf{IL}\). We will introduce the logics \(\mathbf{BIL}\) and \(\mathbf{BIL}^{+}\) in the propositional language with a modal operator \(\square\) and a binary operator \(\Rightarrow\) such that \(\mathbf{BIL}\subseteq\mathbf{BIL}^{+}\subseteq\mathbf{IL}\). The logic \(\mathbf{BIL}\) is generated by the relational structures \(\left<X,R,N\right>\), called basic frames, where \(\left<X,R\right>\) is a Kripke frame and \(\left<X,N\right>\) is a neighborhood frame. We will prove that the logic \(\mathbf{BIL}^{+}\) is generated by the basic frames where the binary relation \(R\) is definable by the neighborhood relation \(N\) and, therefore, the neighborhood semantics is suitable to study the logic \(\mathbf{BIL}^{+}\) and its extensions. We shall also study some axiomatic extensions of \(\mathsf{\mathbf{BIL}}\) and we will prove that these extensions are sound and complete with respect to a certain classes of basic frames. Finally, we prove that the logic BIL+ and some of its extensions are complete respect with the class of neighborhood frames.