Predicate Logic Research Articles

Theoretical analysis and implementation of abstract argumentation frameworks with domain assignments

A representational limitation of current argumentation frameworks is their inability to deal with sets of entities and their properties, for example to express that an argument is applicable for a specific set of entities that have a certain property and not applicable for all the others. In order to address this limitation, we recently introduced Abstract Argumentation Frameworks with Domain Assignments (AAFDs), which extend Abstract Argumentation Frameworks (AAFs) by assigning to each argument a domain of application, i.e., a set of entities for which the argument is believed to apply. We provided formal definitions of AAFDs and their semantics, showed with examples how this model can support various features of commonsense and non-monotonic reasoning, and studied its relation to AAFs. In this paper, aiming to provide a deeper insight into this new model, we present more results on the relation between AAFDs and AAFs and the properties of the AAFD semantics, and we introduce an alternative, more expressive way to define the domains of arguments using logical predicates. We also offer an implementation of AAFDs based on Answer Set Programming (ASP) and evaluate it using a range of experiments with synthetic datasets.

EXTENDED FRAMES AND SEPARATIONS OF LOGICAL PRINCIPLES

AbstractWe aim at developing a systematic method of separating omniscience principles by constructing Kripke models for intuitionistic predicate logic $\mathbf {IQC}$ and first-order arithmetic $\mathbf {HA}$ from a Kripke model for intuitionistic propositional logic $\mathbf {IPC}$ . To this end, we introduce the notion of an extended frame, and show that each IPC-Kripke model generates an extended frame. By using the extended frame generated by an IPC-Kripke model, we give a separation theorem of a schema from a set of schemata in $\mathbf {IQC}$ and a separation theorem of a sentence from a set of schemata in $\mathbf {HA}$ . We see several examples which give us separations among omniscience principles.

A NATURAL DEDUCTION SYSTEM FOR ORTHOMODULAR LOGIC

AbstractOrthomodular logic is a weakening of quantum logic in the sense of Birkhoff and von Neumann. Orthomodular logic is shown to be a nonlinear noncommutative logic. Sequents are given a physically motivated semantics that is consistent with exactly one semantics for propositional formulas that use negation, conjunction, and implication. In particular, implication must be interpreted as the Sasaki arrow, which satisfies the deduction theorem in this logic. As an application, this deductive system is extended to two systems of predicate logic: the first is sound for Takeuti’s quantum set theory, and the second is sound for a variant of Weaver’s quantum logic.

On Minimal Cut Sets Representation with Binary Decision Diagrams

Since their introduction in form of a canonical representation of logical functions, the Binary Decision Diagrams (BDDs) gained a wide acceptance in numerous industrial applications. This paper summarizes the properties of BDD representation of Minimal Cut Sets (MCS) of Fault Tree (FT) models most typically encountered in nuclear energetics. Cut sets from MCS are defined as paths from the top BDD node to terminal nodes in the BDD, on which a quantitative and qualitative FT analysis (FTA) is performed. The core of the FTA on the BDDs is performed with help of two fundamental algorithms, one for conditional probability evaluation and another for the selection of cut sets. The accuracy of conditional probability evaluation represents an essential feature for an unbiased quantitative analysis, such as the top event probability or the determination of event importance measures. The cut set selection algorithm is shown in a generic version introducing logical predicates for its selection criteria. As it is known, the efficiency of depicted algorithms depends only on the number of BDD nodes used for the FT representation. In order to appraise the compactness of the BDD representation of FT models, their characteristics have herein been evaluated on several real-life models from the Nuclear Power Plant Krško. The extraordinariness of the compactness of the BDD representation reflects in its ability to implement advanced dynamic analysis (i.e. what-if) of FT models. The efficiency of such an approach is recognized by commercial vendors upgrading their FT Tools to new versions by implementing BDD based algorithms.

Logical Satisfiability of Counterfactuals for Faithful Explanations in NLI

Evaluating an explanation's faithfulness is desired for many reasons such as trust, interpretability and diagnosing the sources of model's errors. In this work, which focuses on the NLI task, we introduce the methodology of Faithfulness-through-Counterfactuals, which first generates a counterfactual hypothesis based on the logical predicates expressed in the explanation, and then evaluates if the model's prediction on the counterfactual is consistent with that expressed logic (i.e. if the new formula is \textit{logically satisfiable}). In contrast to existing approaches, this does not require any explanations for training a separate verification model. We first validate the efficacy of automatic counterfactual hypothesis generation, leveraging on the few-shot priming paradigm. Next, we show that our proposed metric distinguishes between human-model agreement and disagreement on new counterfactual input. In addition, we conduct a sensitivity analysis to validate that our metric is sensitive to unfaithful explanations.

Definiteness Effect in the PP

This article demonstrates that abessive PPs impose the same type of definiteness restriction on their complements that existential predicates impose on their subjects. The definiteness effect (DE) in PPs is accounted for in the framework of the DE theory of Szabolcsi (1986a,b, 1992), who derives the DE from the incompatibility of a presuppositional subject and a logical predicate of existence that is present in a wide class of predicates (including verbs meaning ‘(cause to) come to exist in a particular fashion’ and nominal predicates meaning ‘(non)existence at a particular location’). The analysis points out this predicate of existence in the small clause complements of abessive Ps.

FIRST-ORDER FRIENDLINESS

Abstract In this note we study a counterpart in predicate logic of the notion of logical friendliness, introduced into propositional logic in [15]. The result is a new consequence relation for predicate languages with equality using first-order models. While compactness, interpolation and axiomatizability fail dramatically, several other properties are preserved from the propositional case. Divergence is diminished when the language does not contain equality with its standard interpretation.

On Finite Domains Based Slices in the Structure of Superintuitionistic Predicate Logics, Preview

The article presents a series of closely related papers that together make up a single piece of work, devoted to the description of a hierarchy of slices in the structure of superintuitionistic predicate logics and making probably the first attempt to transfer to the predicate case Hosoi's hierarchy of (finite) slices for the structure of superintuitionistic propositional logics.

Pando: Enhanced Data Skipping with Logical Data Partitioning

With enormous volumes of data, quickly retrieving data that is relevant to a query is essential for achieving high performance. Modern cloud-based database systems often partition the data into blocks and employ various techniques to skip irrelevant blocks during query execution. Several algorithms, often based on historical properties of a workload of queries run over the data, have been proposed to tune the physical layout of data to reduce the number of blocks accessed. The effectiveness of these methods at skipping blocks depends on what metadata is stored and how well the physical data layout aligns with the queries. Existing work on automatic physical database design misses significant opportunities in skipping blocks because it ignores logical predicates in the workload that exhibit strongly correlated results. In this paper, we present Pando which enables significantly better block skipping than past methods by informing physical layout decisions with correlation-aware logical partitioning. Across a range of benchmark and real-world workloads, Pando attains up to 2.8X reduction in the number of blocks scanned and up to 2.3X speedup in end-to-end query execution time over the state-of-the-art techniques.

Algebraizations of Propositional Logic and Monadic Logic

In this paper, we introduce the language systems of propositional logic (LSPL), which involves no variables, and monadic predicate logic (LSMPL), which consists of predicates applied to single variables. We review the validity and dedication related to (LSPL) and (LSMPL) with their properties. After that, we investigate the connection between Boolean algebras with (LSPL) and (LSMPL) to make algebraizations methods out of logic.

On The Substitutivity of Variables

This paper examines the constraints for the decidability of a logical system byanalyzing the theories of undecidability in formal logic and recursive func-tions. The proof of the diagonal lemma requires the implicit premise that allformulas are closed under avariable composition, i.e., the composition of vari-ables in the formulas signatures. A variable composition is representable infirst-order logic if∀ψ∀k(⊢ψ(k)↔∃y(ψ∧y≡k))can be derived, e.g., froma semantic definition of the quantifiers. A further condition is the assump-tion of an interpretation providing the existence of what will be defined as anindeterminate variable signaturewith aconstant interpretation.A recursion theorem for formulas in analogy to the recursion theorem ofKleene (1943, pp.52-53) will be proved which covers the diagonal lemma as aspecial case. The chosen notation and reasoning intend to make the necessaryconditions for its provability explicit. It will be proved that the representabilityof variable composition and the existence of an indeterminate variable signa-ture to represent a constant interpretation are consequences of the recursiontheorem, i.e., equivalent to the existence of fixed points∀ψ∃φ(⊢φ↔ψ(⌜φ⌝)).These results give a reason why the negation of the diagonal lemma canbe proved for a predicative logic that contradicts these premises but holdsthe explicit condition of the diagonal lemma, i.e., a language of this logic iscapable of representing all computable functions, as has been shown as anon-expectable result in Solte 2020. The paper concludes with an outline ofa decidable structure of computable functions. I.e., it is possible to provide aninterpretation of a predicative logic without undecidability in this structure

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

Application of logical modeling for the analysis and classification of medical data for the purpose of diagnosis.

The subject of the research is a logical approach to data analysis and the development of software tools capable of identifying hidden patterns, even with a limited amount of data. The input data consists of indicators of the diagnosis of patients, their diagnoses and the experience of doctors obtained in the course of medical practice. The research method is the development of software tools based on systems of multivalued predicate logic for the analysis of patient data. This approach considers the source data as a set of general rules, among which it is possible to distinguish those rules that are sufficient to explain all the observed data. These rules, in turn, are generative for the area under consideration and help to better understand the nature of the objects under study. The novelty of the study lies in the use of multivalued logic to analyze a limited amount of medical data of patients in order to determine the most likely diagnosis with a given accuracy. The proposed approach makes it possible to detect hidden patterns in the symptoms and results of patient examinations, classify them and identify unique signs of various forms of gastritis. Unlike neural networks, logical analysis is transparent and does not require training on large amounts of data. The conclusions of the study show the possibility of such an approach for diagnosis with a lack of information, as well as the offer of alternatives if the required accuracy of diagnosis is not achieved.

Generations of Reason: A Family's Search for Meaning in Post-Newtonian England

Generations of Reason: A Family's Search for Meaning in Post-Newtonian England

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.

A Completeness Proof for a Regular Predicate Logic with Undefined Truth Value

We provide a sound and complete proof system for an extension of Kleene’s ternary logic to predicates. The concept of theory is extended with, for each function symbol, a formula that specifies when the function is defined. The notion of “is defined” is extended to terms and formulas via a straightforward recursive algorithm. The “is defined” formulas are constructed so that they themselves are always defined. The completeness proof relies on the Henkin construction. For each formula, precisely one of the formula, its negation, and the negation of its “is defined” formula is true on the constructed model. Many other ternary logics in the literature can be reduced to ours. Partial functions are ubiquitous in computer science and even in (in)equation solving at schools. Our work was motivated by an attempt to precisely explain, in terms of logic, typical informal methods of reasoning in such applications.

Implementing associative memories by Echo State Network for the applications of natural language processing

This paper presents Echo State Network (ESN) based associative memories and their applications to English words. Among the papers describing ESN applications in general, some papers deal with natural language processing (NLP). Those NLP papers utilize an ESN’s property of temporal signal learning to translate an English sentence into its corresponding predicate logic formula. In practical NLP applications, input sentences often include misspelled or undefined words. To cope with such problems, we constructed a training algorithm to realize an ESN-based associative memory. This algorithm can be used in two ways: auto-associative one when the input and output patterns are the same in each pair of the training data, and hetero-associative one otherwise. In this paper, we firstly show the basic performance of an ESN-based auto-associative memory by applying it to two-dimensional images. We made sure that the performance is improved by averaging the result of multiple runs. Secondly, we describe the performances of ESN-based auto- and hetero-associative memories when being applied to English words. The former is to recall correct English words from incomplete spelling ones, and the latter is to chain two different English words of an input and its corresponding output. The former performance in recalling 26 incomplete words can be improved by averaging the outputs of multiple runs, and the improved success rate of correct words is around 65%. The latter performance in chaining 26 words attains the success rate of 83.4%.

A user-guided reduction concept lattice and its algebraic structure

Concept lattice has been widely studied and it is still attracting researchers in many fields by virtue of its well structured knowledge representation. However, it is more obvious that the number of concept lattice nodes generated from the initial formal context is enormous and results in more storage and lower efficiency of building, especially with the increasing of data to deal with. Meanwhile, users are not interested in the whole intents composed of all the attribute sets and it is a time consuming and troublesome job for them to acquire interesting knowledge. In order to deal with these deficiencies, we propose a constrained concept lattice structure referred to as a reduction concept lattice (RCL) guided by user preferences. This type of lattice structure not only can effectively reduce storage space and construction time, but also help to extract the knowledge that users are interested in. In this study, the definition of RCL is first given based on users’ three types of interesting background knowledge described by predicate logic, together with some RCL’s examples to illustrate this type of structure. Next, an algebraic structure of RCL is defined and its completeness is proved, and accordingly a new way is provided for reduction of concept lattice based on user preferences.

The nonarithmeticity of the predicate logic of primitive recursive realizability

The notion of primitive recursive realizability was introduced by S. Salehi as a kind of semantics for the language of basic arithmetic using primitive recursive functions. It is of interest to study the corresponding predicate logic. D. A. Viter proved that the predicate logic of primitive recursive realizability by Salehi is not arithmetical. The technically complex proof combines the methods used by the author of this article in the study of predicate logics of constructive arithmetic theories and the results of M. Ardeshir on the translation of the intuitionistic predicate logic into the basic predicate logic. The purpose of this article is to present another proof of Viter's result by directly transferring the methods used earlier in proving the nonarithmeticity of the predicate logic of recursive realizability.

The nonarithmeticity of the predicate logic of primitive recursive realizability

Понятие примитивно рекурсивной реализуемости было введено С. Салехи как своего рода семантика для языка базисной арифметики с использованием примитивно рекурсивных функций. Представляет интерес изучение соответствующей логики предикатов. Д. А. Витер доказал, что предикатная логика примитивно рекурсивной реализуемости Салехи не является арифметической. Технически сложное доказательство сочетает в себе методы, использованные автором настоящей статьи при изучении предикатных логик конструктивных арифметических теорий, и результаты М. Ардешира по переводу интуиционистской логики предикатов в базисную логику предикатов. Цель этой статьи - представить другое доказательство результата Витера путем прямого переноса методов, использованных ранее в доказательстве неарифметичности предикатной логики рекурсивной реализуемости. Библиография: 19 наименований.