First-order Predicate Logic Research Articles

Heterogeneous-branch integration framework: Introducing first-order predicate logic in Logical Reasoning Question Answering

The logical reasoning question-answering is a critical task in natural language processing, as it equips models with human-like logical reasoning intelligence. Existing approaches focus on extracting and leveraging the hidden logical structures within text. However, previous works explore partial logical relationships and neglect the holistic extraction within the text. Moreover, they struggle to fully model logical connections, including long-distance dependencies and local topology information. To address these issues, we propose a novel heterogeneous-branch integration framework. Our framework is based on first-order predicate logic theory and consists of three primary components. First, we construct two heterogeneous logical graphs to model logical relationships within and between propositions. Second, we propose a novel Graph-Masked Transformer with a novel graph-masked multi-head attention mechanism to enable distant node interactions and local sparse relationship modeling. Third, we propose a novel multi-branch fusion module to integrate information from multiple sources and generate answer predictions. The proposed heterogeneous-branch integration framework outperforms the VDGN method by 2.73% in accuracy on the ReClor dataset and 2.15% on the LogiQA dataset. Our code and models will be made available at https://github.com/starry-y/HBI.

Exclusive Disjunctions With Three Disjuncts from First-Order Predicate Calculus

First-order predicate logic seems to be incompatible with the way people understand embedded exclusive disjunctions with three disjuncts. In classical logic, an exclusive disjunction with three disjuncts holds when the three disjuncts hold. However, it is hard to note that for people. The theory of mental models can explain this fact. According to that theory, individuals tend to process embedded exclusive disjunctions with three disjuncts intuitively. Thus, they only consider possible situations in which just one of the disjuncts is the case. The present paper tries to explain this problem within first-order predicate logic. The main point is that, in the latter logic, in inferences having, as its first premise, an embedded exclusive disjunction with three disjuncts, and, as its second premise, the first disjunct of that very exclusive disjunction, it is possible to infer none of the other two disjuncts.

Requirement patterns in deductive verification of poST Programs

Process-oriented programming is one of the approaches to developing control software. A process-oriented program is defined as a sequence of processes. Each process is represented by a set of named states containing program code that define the logic of the process's behavior. Program execution is sequential execution of each of these processes in their current states at every iteration of the control cycle. Processes can interact through changing each other's states and shared variables. The paper expands a method for classifying temporal requirements for process-oriented programs in order to simplify and automate the deductive verification of such programs. The method consists of the following steps. At the first step, the requirements are formalized in a specialized language DV-TRL, a variant of typed first-order predicate logic with a set of interpreted types and predicate and functional symbols, that reflect specific concepts of control systems in a process-oriented paradigm. At the second step, the formalized requirements are divided into classes, each of which is defined by a pattern — a parametric formula of the DV-TRL language. The correctness conditions generated for process-oriented programs regarding requirements satisfying the same pattern have the same proof scheme. At the third step, appropriate proof schemes are developed. In our paper, we first give a brief introduction to the poST language, a process-oriented extension to the ST language of the IEC 61131-3 standard. Next, the DV-TRL language is defined. We also provide a collection of natural language requirements for several control systems. Then we define patterns that fully cover all the requirements of this collection. For each of these patterns we give an example of a formalized requirement from the collection and describe a scheme for proving the correctness conditions for this pattern. Statistics on the distribution of requirements from the collection across patterns reveals the most popular patterns. We also analyzed related works.

Semantics of Belief Change Operators for Intelligent Agents

This paper summarises several contributions to the theory of belief change by the authors’ dissertation thesis. First, a relational characterization of belief revision for Tarskian logics is considered, encompassing first-order predicate logic, description logic, modal logics and many monotonic logics with model-theoretic semantics. Those logics where total preorders are the standard semantics for revision are characterized. The second contribution considered is a theory of belief revision that builds upon the idea that agents are limited in what the outcome of a revision is. Furthermore, advancements in principles for iterated belief contraction given in the thesis are outlined.

DimBridge: Interactive Explanation of Visual Patterns in Dimensionality Reductions with Predicate Logic.

Dimensionality reduction techniques are widely used for visualizing high-dimensional data. However, support for interpreting patterns of dimension reduction results in the context of the original data space is often insufficient. Consequently, users may struggle to extract insights from the projections. In this paper, we introduce DimBridge, a visual analytics tool that allows users to interact with visual patterns in a projection and retrieve corresponding data patterns. DimBridge supports several interactions, allowing users to perform various analyses, from contrasting multiple clusters to explaining complex latent structures. Leveraging first-order predicate logic, DimBridge identifies subspaces in the original dimensions relevant to a queried pattern and provides an interface for users to visualize and interact with them. We demonstrate how DimBridge can help users overcome the challenges associated with interpreting visual patterns in projections.

Reasoning about logical systems in the Coq proof assistant

The theory of institutions provides an abstract mathematical framework for specifying logical systems and their semantic relationships. Institutions are based on category theory and have deep roots in a well-developed branch of algebraic specification. However, there are no machine-assisted proofs of correctness for institution-theoretic constructions—chiefly satisfaction conditions for institutions and their (co)morphisms—making them difficult to incorporate into mainstream formal methods. This paper therefore provides the details of our approach to formalizing a fragment of the theory of institutions in the Coq proof assistant. We instantiate this framework with the institutions FOPEQ for first-order predicate logic and EVT for the Event-B specification language, and define some institution-independent constructions, all of which serve as an illustration and evaluation of the overall approach.

A novel temporal moment retrieval model for apron surveillance video

This study addresses the problem of retrieving a specific moment from an untrimmed apron surveillance video through a query sentence. Existing methods cannot tackle it effectively by using sliding windows over the entire video, which suffers from exhaustively enumerated candidates. Here temporal and spatial characters of events are abstracted to corresponding heuristic rules in first-order predicate logic. Temporal characters are exploited to skip unimportant frames while potential time boundaries are deduced from spatial characters. Meanwhile, an object-detection model is designed to obtain spatial characters of potential time boundaries, which is compared with pre-defined heuristic rules for detection. If the obtained spatial characters do not match the heuristic rules, a multiple-time-scales searching module is devised to localize the boundaries. This method is evaluated on TACoS and Apron Surveillance Video (ASV) dataset. Results show that it outperforms the state-of-the-art methods on TACoS and accurately retrieves target events on ASV in one minute.

Non-Classical Approaches to Logic and Quantification as a Means for Analysis of Classroom Argumentation and Proof in Mathematics Education Research

Background: While it is usually taken for granted that logic taught in the mathematics classroom should consist of elements of classical propositional or first-order predicate logic, the situation may differ when referring to students’ discursive productions. Objectives: The paper aims to highlight how classical logic cannot grasp some epistemic aspects, such as evolution over time, uncertainty, and quantification on blurred domains, because it is specifically tailored to capture the set-theoretic language and to validate, rather than to consider epistemic aspects. The aim is to show that adopting classical and non-classical lenses might lead to different results in analysis. Design: Nyaya pragmatic and empiricist logic, with Peircean non-standard quantification, both linked by the concept of free logic, are used as theoretical lenses in analysing two paradigmatic examples of classroom argumentation. Setting and Participants: excerpts from a set of data collected by prof. Paolo Boero from the University of Genoa during research activities in a secondary school mathematical class. Methodology: The examples are discussed by adopting a hermeneutic approach. Results: The analysis shows that different logical lenses can lead to varying interpretations of students’ behaviour in argumentation and presenting proof in mathematics and that the adopted non-classical lenses expand the range of possible explanations of students’ behaviour. Conclusion: In mathematics education research, the need to consider an epistemic dimension in the analysis of classroom argumentation and proof production leads to the necessity to consider and combine logical tools in a way specific to the discipline, which might differ from those usually required in mathematics.

Физическая культура в E-learning

Forming the need for a healthy lifestyle, the transfer of knowledge, skills and abilities in the field of physical culture and sports are an urgent pedagogical problem in the transition from the “classical” forms of organization of the educational process to E-learning. The solution of this problem by means of well-known programming languages is impossible due to the insufficient descriptive potential of the latter, in particular, the lack of the possibility of representing human knowledge, probable or possible future events, actions, results. The purpose of our research is to develop theoretical provisions and to verify in practice the possibilities of formalizing the presentation of the individual educational goal of the student and the conscious need for a healthy lifestyle as its basic component in declarative languages, as well as its implementation by means of the educational platform of the university. The researcher considers the possibility of describing the individual educational goal of a student, including a conscious need for a healthy lifestyle as its basic component, gaining knowledge, forming and developing skills in the field of physical culture and sports, as well as the possibility of building an individual educational trajectory of its achievements in the organization of E-learning at the university in the language of first-order predicate logic. A number of features of the predicate logic language that are essential for the problem under consideration are singled out. A pedagogical experiment was conducted on the basis of the educational platform of Voronezh State Pedagogical University. The study involved 158 students and undergraduates. The descriptions of individual educational goals of the student, formalized by the language of logic of predicates, suitable for implementation by means of computer technology are obtained, on the basis of which the optimal trajectory for achieving this goal is calculated for each participant. The practical implementation of the educational process was carried out, data on a conscious need for a healthy lifestyle, gaining knowledge, forming and developing skills in the field of physical culture and sports were obtained. The obtained results were compared with similar studies and a conclusion was made about the ways to improve the suggested approach to the formation of a conscious need for students to lead a healthy lifestyle, gain knowledge, form and develop their skills in the field of physical culture and sports in the context of the transition from “classical” forms of organization of the educational process to E-learning.

An Approach for Verification of Secure Access Control Using Security Pattern

According to OWASP-2021, more than 3,00,000 web applications have been detected for unauthenticated and unauthorised access leading to a breach of security trust. Security patterns are commonly used in web applications to address the problem of broken access. Web developers are not experts in implementing security patterns. Therefore, it is necessary to verify that the security pattern has been applied, specifying the original intent of the security pattern. In this paper, an approach has been proposed that analyses the behavioural aspect of security patterns to verify that it meets the security requirement of the web application. The proposed approach extracts the class diagram’s structural properties, relations, associations, and security-related constraints and verifies it using the first-order predicate logic. Experiments have been conducted using class diagrams of security patterns to detect instances of broken access control early in the design phase. The proposed approach will help minimise the risk of unauthenticated and unauthorised access to a web application.

ON DATABASE QUERIES OVER A WEAKLY O-MINIMAL DOMAIN WITH A SMALL COUNTABLE SPECTRUM

In the relational database model introduced by E.F. Codd, the database state is understood as a finite set of relationships between elements. The names of the relationships and their arnts (locations) are fixed and called the database schema. The individual information stored in the relationships of this schema is called the state of the database. Although relational databases were devised for finite sets of data, it is often convenient to assume that there is an infinite domain. We consider relational databases organized over an ordered domain with some additional relations – a typical example is the set of rational numbers together with the relation of linear order and binary operation of addition. If the first-order predicate logic language is used as the query language, then queries can use both database relationships and domain relationships, with variables changing throughout the domain. In the focus of our study are the first-order (FO) queries that are invariant under order-preserving permutations – such queries are called order-generic. It was discovered that for some domains order-generic FO queries fail to express more than pure order queries. Here we prove the collapse result theorem over a weakly o-minimal domain having convexity rank 1 and a small countable spectrum.

On Three Levels of Abstractness in Peirce’s Beta Graphs

Peirce’s beta graphs are roughly equivalent to our first-order predicate logic. However, Bellucci and Pietarinen have recently argued that the beta graphs are not well-equipped to handle asymmetric relative terms. I survey four proposed solutions to the problem and find them all wanting. I offer a fifth solution according to which Peirce’s beta graphs function at three different levels of abstractness from natural language. I diagnose the problem of asymmetric relative terms as arising when we transition from the first to the second level of abstractness. Making grammatical information encoded in natural language explicit at the first level of abstractness and interpreting graphs at the second level of abstractness as shorn of grammatical information resolves the problem. The solution is both well-motivated by Peirce’s own commitments and increases the expressiveness of the beta graphs.

A General Glivenko–Gödel Theorem for Nuclei

Glivenko's theorem says that, in propositional logic, classical provability of a formula entails intuitionistic provability of double negation of that formula. We generalise Glivenko's theorem from double negation to an arbitrary nucleus, from provability in a calculus to an inductively generated abstract consequence relation, and from propositional logic to any set of objects whatsoever. The resulting conservation theorem comes with precise criteria for its validity, which allow us to instantly include G\"odel's counterpart for first-order predicate logic of Glivenko's theorem. The open nucleus gives us a form of the deduction theorem for positive logic, and the closed nucleus prompts a variant of the reduction from intuitionistic to minimal logic going back to Johansson.

An efficient automatic accident detection system using inertial measurement through machine learning techniques for powered two wheelers

The two-wheeler accidents in most populated and developing countries have become vulnerable and six accidents happen every hour on average. This paper proposes an efficient automatic accident detection system that attempts to detect the occurrences of the accidents in powered two-wheelers (PTW) automatically using vehicle-dependent parameters and the physiological parameters of the rider in real-time. The proposed system builds an accident detection system in PTW using three steps namely, critical event detection system, accident detection system, and severity assessment system. The critical event detection system reads the accelerometer sensor values from the On-Board Diagnostic (OBD) unit mounted on the PTW and classifies the state of the vehicle as normal, fall-like, and fall through the enhanced decision tree algorithm. The enhanced decision tree algorithm uses a tanh function to calculate entropy values. The rules are extracted to fix the threshold by pruning the decision tree to identify the fall of the vehicle and the rider. Due to the unstable nature of PTW and the rider, a novel Adaptive Sequence Window algorithm (ASW) is proposed to substantiate and validate the occurrence of accidents based on the sequence of states identified. Once the accident is detected, the Decision Support System (DSS) running on the OBD mounted on the PTW decides the severity of the accident by combining the three parameters namely fall of the vehicle, fall of the rider, and pulse rate of the rider using the first-order predicate logic rules. The enhanced decision tree algorithm outperforms the other classifiers such as naïve Bayes, artificial neural network, and recurrent neural network with an accuracy of 99.8%. The OBD unit mounted on the PTW and the rider’s helmet is used to detect the occurrence of accidents automatically along with its severity with less time. The ASW algorithm enables the system to detect the fall of the vehicle and rider within five minutes and prevents false positives. Further, the information can be communicated based on the severity of the accident to the emergency medical service for quick response.

On the Convexity of a Fragment of Pure Set Theory with Applications within a Nelson-Oppen Framework

The Satisfiability Modulo Theories (SMT) issue concerns the satisfiability of formulae from multiple background theories, usually expressed in the language of first-order predicate logic with equality. SMT solvers are often based on variants of the Nelson-Oppen combination method, a solver for the quantifier-free fragment of the combination of theories with disjoint signatures, via cooperation among their decision procedures. When each of the theories to be combined by the Nelson-Oppen method is convex (that is, any conjunction of its literals can imply a disjunction of equalities only when it implies at least one of the equalities) and decidable in polynomial time, the running time of the combination procedure is guaranteed to be polynomial in the size of the input formula. In this paper, we prove the convexity of a fragment of Zermelo-Fraenkel set theory, called Multi-Level Syllogistic, most of whose polynomially decidable fragments we have recently characterized.

Charge prediction modeling with interpretation enhancement driven by double-layer criminal system

With the rapid development of artificial intelligence and the increasing demand for legal intelligence, using AI methods to predict legal judgments has become a hot spot in recent years. Charge prediction is one of the core tasks of Legal Judgment Prediction (LJP). It aims to predict charge from complicated legal facts, so as to help the court make judgments or provide legal professional guidance to non-professionals. In the field of legalAI, interpretability is crucial compared to others. Reasonable interpretability can eliminate hidden dangers such as gender discrimination and provide support for judges’ decisions. However, how to add the legal theory framework to the modeling to improve the interpretability is a challenge, which has few researches at present. To address this problem, we use Double-layer Criminal System as a guide to build Charge Prediction modeling called DCSCP which aims to predict charges in the criminal law of China. In general, our characteristic is to achieve multi-granularity inference of legal charges by obtaining the subjective and objective elements from the fact descriptions of legal cases. Specifically, our approach is performed in two steps: (1) extract the objective elements from the fact description and use them to generate candidate charges to achieve coarse-grained prediction; (2) extract the subjective elements from the fact description, and design the first-order predicate logic inference to realize the fine-grained charge inference in combination with the candidate charges. Experimental results show that our DCSCP can provide interpretable predictions, and it can maintain performance compared to other state-of-the-art charge prediction models.

First-order logic as a constraint satisfaction problem

In this paper, we discourse an analysis of classical first-order predicate logic as a constraint satisfaction problem, CSP. First, we will offer our general framework for CSPs, and then apply it to first-order logic. We claim it would function as a new semantics, constraint semantics, for logic. Then, we prove the soundness and completeness theorems with respect to the constraint semantics. The latter theorem, which will be proven by a proof-search method, implies the cut-elimination theorem. Furthermore, using the constraint semantics, we make a new proof of the Craig interpolation theorem. Also, we will provide feasible algorithms to generate interpolants for some decidable fragments of first-order logic: the propositional logic and the monadic fragments. The algorithms, reflecting a ‘projection’ of an indexed relation, will show how to transform given formulas syntactically to obtain interpolants.

Algorithmic properties of first-order superintuitionistic logics of finite Kripke frames in restricted languages

Abstract We consider the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate superintuitionistic logics of finite Kripke frames on the logics’ algorithmic properties. By a finite frame we mean a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that positive fragments of many predicate superintuitionistic logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, $\varPi ^0_1$-hard—in languages with three individual variables and a single monadic predicate letter; this applies to the logics of finite frames of the predicate counterparts of propositional logics lying between the intuitionistic logic and the logic of the weak law of the excluded middle.

Analogical Reasoning With Deep Learning-Based Symbolic Processing

The authors propose analogical reasoning systems based on first-order predicate logic using deep learning. The proposed systems consist of a model combining recursive neural networks and Word2Vec. When unknown data is input in this trained model, similar outputs to those of the trained data are obtained. Previous studies on symbolic inference using deep learning have focused on deductive and inductive inferences, and they require rule templates that are manually created and provided to the models in advance. However, it is not enough to infer unknown rules from a large amount of data on the Internet. Analogical reasoning is a mode of inference that attempts to solve a new case based on similarity. Thus, it enables inference of rules that were challenging to comprehend in previous studies. As a result of the experiments, unknown rules can be efficiently inferred from known rules contained in the knowledge bases. Using the proposed systems, the authors can extract unknown rules from five of the eight datasets consisting of three types of knowledge bases described in Prolog. The proposed method is the first case of analogical reasoning based on the first-order predicate logic using deep learning. Furthermore, the models used in the proposed systems have shown high robustness when using Word2Vec. For this reason, it is suggested that the practical applications of the proposed systems enable efficient inferences from a large amount of data that include noisy data on the Internet.

A Logical Reconstruction of Leibniz’s Argument for His Complete Concept Conception of the Nature of Substance in Discours §8

Abstract This paper develops a valid reconstruction in first-order predicate logic of Leibniz’s argument for his complete concept definition of substance in §8 of the Discours de Métaphysique. Following G. Rodriguez-Pereyra, it construes the argument as resting on two substantial premises, the “merely verbal” Aristotelian definition and Leibniz’s concept containment theory of truth, and it understands the resulting “real” definition as saying not that an entity is a substance iff its complete concept contains every predicate of that entity, but iff its complete concept contains every predicate of any subject to which that concept is truly attributable. An account is suggested of why Leibniz criticises the Aristotelian definition as merely nominal and how he takes his own definition to overcome this shortcoming: while on the Aristotelian basis the predication relation could generate endless chains, so that substances as endpoints of predication would be impossible, Leibniz’s definition reveals lowest species as such endpoints, which he therefore identifies with individual substances. Since duplicate lowest species make no sense, the Identity of Indiscernibles for substances follows. The reading suggests a Platonist interpretation according to which substances do not so much have but are individual essences, natures or forms.