Propositional Logic Formulas Research Articles

Generating Java code pairing with ChatGPT

The Large Language Models (LLMs) like ChatGPT 3.5 have created a new era of automatic code generation. However, the existing research primarily focuses on generating simple code based on datasets (such as HumanEval, etc.). Most of approaches pay less attention to complex and practical code generation. Therefore, in this paper, we propose a new approach called “Xd-CodeGen” which can be used to generate large scale Java code. This approach is composed of four phases: requirement analysis, modeling, code generation, and code verification. In the requirement analysis phase, ChatGPT 3.5 is utilized to decompose and restate user requirements. To do so, a knowledge graph is developed to describe entities and their relationship in detail. Further, Propositional Projection Temporal Logic (PPTL) formulas are employed to define the properties of requirements. In the modeling phase, we use knowledge graphs to enhance prompts and generate UML class and activity diagrams for each sub-requirement using ChatGPT 3.5. In the code generation phase, based on established UML models, we make use of prompt engineering and knowledge graph to generate Java code. In the code verification phase, a runtime verification at code level approach is employed to verify generated Java code. Finally, we apply the proposed approach to develop a practical Java web project.

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.

Modal Logic for Relationships between Sets

In this article, we present a modal logic system that allows representing relationships between sets or classes of individuals defined by a specific property. We introduce two modal operators, [a] and <a>, which are used respectively to express “for all A” and “there exists an A”. Both the syntax and semantics of the system have two levels that avoid the nesting of the modal operator. The semantics is based on a variant of Kripke semantics, where the modal operators are indexed over propositional logic formulas (“pre-formulas” in the paper). Furthermore, we present a set of axioms and rules that govern the system and we prove that the logic is correct and complete with respect to Kripke models. In the final section of the article, we discuss potential future work. We consider the possibility of combining our operator with other modalities, such as necessity or knowledge. Additionally, as an example of the utility of our modal operator, we briefly analyze a conveniently adapted Barcan formula within the framework of our system. In summary, we propose combining our modal operator with other ones as a simpler, more compact, albeit less expressive way to address quantified modal logic.

Convolutional Tsetlin Machine-based Training and Inference Accelerator for 2-D Pattern Classification

The Tsetlin Machine (TM) is a machine learning algorithm based on an ensemble of Tsetlin Automata (TAs) that learns propositional logic expressions from Boolean input features. In this paper, the design and implementation of a Field Programmable Gate Array (FPGA) accelerator based on the Convolutional Tsetlin Machine (CTM) is presented. The accelerator performs classification of two pattern classes in 4 × 4 Boolean images with a 2 × 2 convolution window. Specifically, there are two separate TMs, one per class. Each TM comprises 40 propositional logic formulas, denoted as clauses, which are conjunctions of literals. Include/exclude actions from the TAs determine which literals are included in each clause. The accelerator supports full training, including random patch selection during convolution based on parallel reservoir sampling across all clauses. The design is implemented on a Xilinx Zynq XC7Z020 FPGA platform. With an operating clock speed of 40 MHz, the accelerator achieves a classification rate of 4.4 million images per second with an energy per classification of 0.6 μJ. The mean test accuracy is 99.9% when trained on the 2-dimensional Noisy XOR dataset with 40% noise in the training labels. To achieve this performance, which is on par with the original software implementation, Linear Feedback Shift Register (LFSR) random number generators of minimum 16 bits are required. The solution demonstrates the core principles of a CTM and can be scaled to operate on multi-class systems for larger images.

Neurosymbolic Reasoning and Learning with Restricted Boltzmann Machines

Knowledge representation and reasoning in neural networks has been a long-standing endeavour which has attracted much attention recently. The principled integration of reasoning and learning in neural networks is a main objective of the area of neurosymbolic Artificial Intelligence. In this paper, a neurosymbolic system is introduced that can represent any propositional logic formula. A proof of equivalence is presented showing that energy minimization in restricted Boltzmann machines corresponds to logical reasoning. We demonstrate the application of our approach empirically on logical reasoning and learning from data and knowledge. Experimental results show that reasoning can be performed effectively for a class of logical formulae. Learning from data and knowledge is also evaluated in comparison with learning of logic programs using neural networks. The results show that our approach can improve on state-of-the-art neurosymbolic systems. The theorems and empirical results presented in this paper are expected to reignite the research on the use of neural networks as massively-parallel models for logical reasoning and promote the principled integration of reasoning and learning in deep networks.

A Distributed Network-Based Runtime Verification of Full Regular Temporal Properties

As a lightweight method, runtime verification aims to check whether one program execution satisfies a desired property. For online runtime verification, the approach efficiency and property expressiveness are two key points restricting its wide application. In this paper, we propose a distributed network-based parallel runtime verification approach to verifying full regular temporal properties for a suitable subset of C (named by Xd-C) programs in an online manner. With this approach, an Xd-C program is translated into an equivalent Modeling, Simulation and Verification Language (MSVL) program, and a desired property is specified as a Propositional Projection Temporal Logic (PPTL) formula; during the program execution, segments of the generated state sequence are verified in parallel by distributed multi-core machines. Experimental results show that, our approach has a speedup of 2.5X-5.0X over the state-of-art runtime verification approaches and supports full regular temporal properties, meaning that our approach can not only take full advantage of computing and storage resources in a distributed network, but also support more expressive properties.

Measuring Inconsistency in Some Logics with Tense Operators

This paper starts the systematic study of inconsistency measures for propositional logics enriched with operators involving time. We use Prior’s operators for tense logic: H, G, P, and F; however, we apply different semantics to them. We define two logics. The first one, ATPL, allows formulas with the application of any of the four operators any number of times to propositional logic formulas. The semantics is given in terms of TPL structures. We then show how to measure the inconsistency of a set of ATPL formulas using appropriately modified versions of standard inconsistency measures for propositional logic. Additionally, these new measures are checked for the satisfaction of various properties. We also investigate a new concept of inconsistency that cannot be defined for propositional logic. Then, ATPL is extended to CTPL by applying the propositional connectives to ATPL formulas and measuring inconsistency is extended to sets of CTPL formulas.

Semantics modulo satisfiability with applications: function representation, probabilities and game theory

AbstractIn the context of propositional logics, we apply semantics modulo satisfiability—a restricted semantics which comprehends only valuations that satisfy some specific set of formulas—with the aim to efficiently solve some computational tasks. Three possible such applications are developed.We begin by studying the possibility of implicitly representing rational McNaughton functions in Łukasiewicz Infinitely-valued Logic through semantics modulo satisfiability. We theoretically investigate some approaches to such representation concept, called representation modulo satisfiability, and describe a polynomial algorithm that builds representations in the newly introduced system. An implementation of the algorithm, test results and ways to randomly generate rational McNaughton functions for testing are presented. Moreover, we propose an application of such representations to the formal verification of properties of neural networks by means of the reasoning framework of Łukasiewicz Infinitely-valued Logic.Then, we move to the investigation of the satisfiability of joint probabilistic assignments to formulas of Łukasiewicz Infinitely-valued Logic, which is known to be an NP-complete problem. We provide an exact decision algorithm derived from the combination of linear algebraic methods with semantics modulo satisfiability. Also, we provide an implementation for such algorithm for which the phenomenon of phase transition is empirically detected.Lastly, we study the game theory situation of observable games, which are games that are known to reach a Nash equilibrium, however, an external observer does not know what is the exact profile of actions that occur in a specific instance; thus, such observer assigns subjective probabilities to players actions. We study the decision problem of determining if a set of these probabilistic constraints is coherent by reducing it to the problems of satisfiability of probabilistic assignments to logical formulas both in Classical Propositional Logic and Łukasiewicz Infinitely-valued Logic depending on whether only pure equilibria or also mixed equilibria are allowed. Such reductions rely upon the properties of semantics modulo satisfiability. We provide complexity and algorithmic discussion for the coherence problem and, also, for the problem of computing maximal and minimal probabilistic constraints on actions that preserves coherence.Abstract prepared by Sandro Márcio da Silva Preto.E-mail: spreto@ime.usp.brURL:https://doi.org/10.11606/T.45.2021.tde-17062021-163257

Strategyproof social choice when preferences and outcomes may contain ties

The Gibbard-Satterthwaite theorem implies that all anonymous, Pareto-optimal, and single-valued social choice functions can be strategically manipulated. In this paper, we investigate whether there exist social choice correspondences (SCCs), that satisfy these conditions under various assumptions about how single alternatives are eventually selected from the choice set. These assumptions include even-chance lotteries as well as resolute choice functions and linear tie-breaking orderings unknown to the agents. We show that (i) all anonymous Pareto-optimal SCCs where ties are broken according to some linear tie-breaking ordering or by means of even-chance lotteries are manipulable, and that (ii) all pairwise Pareto-optimal SCCs are manipulable for any deterministic tie-breaking rule. These results are proved by reducing the statements to finite—yet very large—formulas in propositional logic, which are then shown to be unsatisfiable by a computer.

A tableau construction for finite linear-time temporal logic

This paper gives a tableau-based technique for converting formulas in finite propositional linear-time temporal logic (Finite LTL) into finite-state automata whose languages are the models of the given formula. Finite LTL differs from traditional LTL in that formulas are interpreted with respect to finite, rather than infinite, sequences of states; this fact means that traditional finite-state automata, rather than ω-automata such as those developed by Büchi and others, suffice for recognizing models of such formulas. The approach presented here works by associating with each state in the constructed automaton a Finite LTL formula satisfied by the sequences that can be accepted by the automaton when started in the given state. Such tableau-construction techniques are well-known in the setting of traditional LTL, where they are used to construct ω-automata; we adapt here for the setting of Finite LTL and finite-state automata. The resulting automata may be used as a basis for model checking, model synthesis, query checking and satisfiability testing.

Modal Logic S5 Satisfiability in Answer Set Programming

AbstractModal logic S5 has attracted significant attention and has led to several practical applications, owing to its simplified approach to dealing with nesting modal operators. Efficient implementations for evaluating satisfiability of S5 formulas commonly rely on Skolemisation to convert them into propositional logic formulas, essentially by introducing copies of propositional atoms for each set of interpretations (possible worlds). This approach is simple, but often results into large formulas that are too difficult to process, and therefore more parsimonious constructions are required. In this work, we propose to use Answer Set Programming for implementing such constructions, and in particular for identifying the propositional atoms that are relevant in every world by means of a reachability relation. The proposed encodings are designed to take advantage of other properties such as entailment relations of subformulas rooted by modal operators. An empirical assessment of the proposed encodings shows that the reachability relation is very effective and leads to comparable performance to a state-of-the-art S5 solver based on SAT, while entailment relations are possibly too expensive to reason about and may result in overhead.

Block Compression and Invariant Pruning for SAT-based Totally-Ordered HTN Planning

Translations into propositional logic are currently one of the most efficient techniques for solving Totally-Ordered HTN planning problems. The two current encodings both iterate over the maximum allowed depth of decomposition. Given this depth, they compute a tree that represents all possible decompositions up to this depth. Based on this tree, a formula in propositional logic is created. We show that much of the computed tree is actually useless as it cannot possibly belong to a solution. We provide a technique for removing (parts of) these useless structures using state invariants. We further show that is often not necessary to encode all leafs of this tree as separate timesteps, as the prior encodings did. Instead, we can compress the leafs into blocks and encode all leafs of a block at one timestep. We show that these changes provide an improvement over the state-of-the-art in HTN planning.

General information spaces: measuring inconsistency, rationality postulates, and complexity

AI systems often need to deal with inconsistent information. For this reason, since the early 2000s, some AI researchers have developed ways to measure the amount of inconsistency in a knowledge base. By now there is a substantial amount of research about various aspects of inconsistency measuring. The problem is that most of this work applies only to knowledge bases formulated as sets of formulas in propositional logic. Hence this work is not really applicable to the way that information is actually stored. The purpose of this paper is to extend inconsistency measuring to real world information. We first define the concept of general information space which encompasses various types of databases and scenarios in AI systems. Then, we show how to transform any general information space to an inconsistency equivalent propositional knowledge base, and finally apply propositional inconsistency measures to find the inconsistency of the general information space. Our method allows for the direct comparison of the inconsistency of different information spaces, even though the data is presented in different ways. We demonstrate the transformation on four general information spaces: a relational database, a graph database, a spatio-temporal database, and a Blocks world scenario, where we apply several inconsistency measures after performing the transformation. Then we review so-called rationality postulates that have been developed for propositional knowledge bases as a way to judge the intuitive properties of these measures. We show that although general information spaces may be nonmonotonic, there is a way to transform the postulates so they can be applied to general information spaces and we show which of the measures satisfy which of the postulates. Finally, we discuss the complexity of inconsistency measures for general information spaces.

Measuring inconsistency in some branching time logics

Branching time logics have been studied in computer science since the 1980s primarily to model the tree of computations for discrete transition systems. Inconsistency measures for propositional logic have been studied since the early 2000s by AI researchers. This paper introduces inconsistency measures for three branching time logics: ABTL, BBTL, and CBTL. In order to measure inconsistency properly in branching time logics, the semantics differs from the standard semantics using the construction of a canonical tree. ABTL extends propositional logic by three pairs of unary operators: next time, path, and eventually. BBTL adds a pair of binary Until operators that can be applied only to propositional logic formulas. CBTL adds the propositional connectives to BBTL formulas. Propositional logic inconsistency measures are extended to these logics and examples of the computation are given. These measures are shown to satisfy intuitively the concepts involved in the branching time logic operators unlike direct extensions of propositional logic inconsistency measures.

Induction and Exploitation of Subgoal Automata for Reinforcement Learning

In this paper we present ISA, an approach for learning and exploiting subgoals in episodic reinforcement learning (RL) tasks. ISA interleaves reinforcement learning with the induction of a subgoal automaton, an automaton whose edges are labeled by the task’s subgoals expressed as propositional logic formulas over a set of high-level events. A subgoal automaton also consists of two special states: a state indicating the successful completion of the task, and a state indicating that the task has finished without succeeding. A state-of-the-art inductive logic programming system is used to learn a subgoal automaton that covers the traces of high-level events observed by the RL agent. When the currently exploited automaton does not correctly recognize a trace, the automaton learner induces a new automaton that covers that trace. The interleaving process guarantees the induction of automata with the minimum number of states, and applies a symmetry breaking mechanism to shrink the search space whilst remaining complete. We evaluate ISA in several gridworld and continuous state space problems using different RL algorithms that leverage the automaton structures. We provide an in-depth empirical analysis of the automaton learning performance in terms of the traces, the symmetry breaking and specific restrictions imposed on the final learnable automaton. For each class of RL problem, we show that the learned automata can be successfully exploited to learn policies that reach the goal, achieving an average reward comparable to the case where automata are not learned but handcrafted and given beforehand.

Algebraic semantics for propositional superposition logic

We provide a new semantics and a slightly different formalisation for the propositional logic with superposition (PLS) introduced and studied in Tzouvaras [(2018). Propositional superposition logic. Logic Journal of the IGPL, 26(1), 149–190]. PLS results from Propositional Logic (PL) by adding a new binary connective construed as the ‘superposition operation’ and a few axioms about it. The original semantics used in the above paper was the so-called sentence choice semantics (SCS), based on choice functions for all pairs of classical formulas of PL. In contrast, the algebraic or Boolean-value choice semantics (BCS) developed in this paper is based on choice functions for pairs of elements of a Boolean algebra in which the classical sentences take truth values. The Boolean-value choice functions can be subject to similar constraints as those imposed on sentence choice functions. The new axiomatisation is based on the same set of axioms as the previous one but uses a new inference rule, called Rule of Analogy (RA), in place of the rule Salva Veritate (SV) of the previous systems. The Deduction Theorem fails in the systems containing the new rule. As a consequence the completeness theorems for them hold conditionally again, namely the systems are complete with respect to BCS if and only if every consistent set of sentences is extended to a consistent and complete set. Finally connections are established between tautologies of the semantics SCS and those of BCS.

Discussion on Propositional Logic Incorporating Set Thought into Discrete Mathematics

Discrete mathematics is a basic core course for computer and big data majors. It includes four parts: mathematical logic, set theory, algebraic structure, graph theory. They are relatively independent branches respectively, of which propositional logic is the first part of discrete mathematics teaching materials in mathematical logic. This chapter is difficult for students in the second half of this chapter because the basic concepts are not easy to understand thoroughly. In order to solve this problem, this paper integrates the set theory into the understanding and mastery of the connectives, truth tables and important equivalent formulas in propositional logic, thus breaking through the learning difficulties in this chapter of propositional logic, enabling students to find a sense of achievement in learning from the first chapter of discrete mathematics and stimulating students’ interest in learning this course.Discrete mathematics is a mathematical discipline that studies the structure of discrete quantities and their interrelationships. It is an important branch of modern mathematics, as well as a basic core course for computer major and big data major. Its theory is strong [2]. Learning this course can cultivate students’ logical reasoning ability, abstract thinking ability and meticulous generalization ability, and lay a foundation for other professional courses. Discrete mathematics includes four learning parts: mathematical logic, set theory, algebraic structure and graph theory. Among them, mathematical logic mainly studies the reasoning and formalization methods of discrete objects. Propositional logic is the first chapter in mathematical logic, and is also the first part of most discrete mathematics textbooks. This chapter is characterized by many concepts, abstraction, rigorous logic and many proof, which makes it difficult for students to learn [3]. Based on many years of teaching experience, this paper summarizes the root of the students’ key difficulties in learning this chapter, integrates the collective thought to teach students to compare and understand, and then solves all the learning difficulties in this chapter.

Measuring Inconsistency in Some Logics with Modal Operators

The first mention of the concept of an inconsistency measure for sets of formulas in first-order logic was given in 1978, but that paper presented only classifications for them. The first actual inconsistency measure with a numerical value was given in 2002 for sets of formulas in propositional logic. Since that time, researchers in logic and AI have developed a substantial theory of inconsistency measures. While this is an interesting topic from the point of view of logic, an important motivation for this work is also that some intelligent systems may encounter inconsistencies in their operation. This research deals primarily with propositional knowledge bases, that is, finite sets of propositional logic formulas. The goal of this paper is to extend the concept of inconsistency measure in a formal way to sets of formulas with the modal operators “necessarily” and “possibly” applied to propositional logic formulas. We use frames for the semantics, but in a way that is different from the way that frames are commonly used in modal logics, in order to facilitate measuring inconsistency. As a set of formulas may have different inconsistency measures for different frames, we define the concept of a standard frame that can be used for all finite sets of formulas in the language. We do this for two languages. The first language, AMPL, contains formulas where a prefix of operators is applied to a propositional logic formula. The second language, CMPL, adds connectives that can be applied to AMPL formulas in a limited way. We show how to extend propositional logic inconsistency measures to such sets of formulas. Finally, we define a new concept, weak inconsistency measure, and show how to compute it.

Verifying Properties of MapReduce-Based Big Data Processing

Big data techniques are widely used in various fields. To deal with large data sets efficiently, a new programming framework MapReduce has emerged. Thus, new verification challenges arise to improve the reliability of big data processing. In this article, MapReduce processes are implemented by modeling simulation and verification language programs. Then, several data properties such as data soundness, nonconflict, nonduplication, cooperation, and completeness are taken into account. Moreover, these properties are specified by propositional projection temporal logic formulas. To verify these properties, a runtime verification approach at code level based on unified model checking is employed. In addition, two case studies are conducted to demonstrate our approach: sparse matrix multiplication and tracking down suspected patients of an infectious disease.

An Effective Sharpsat Solution Method Based on Learnt Clause Minimization Approach

The problem of satisfiability of the propositional logic formula (SAT problem) in computer science is an important and difficult problem. Learning clauses often contain redundant literal in CDCL SAT solving, which may have a negative impact on the performance of the solver. To overcome this shortcoming, we show a new sharpSAT solver based on learning clause minimization is proposed. By recoding the CNF formula to reduce the storage space, and applying Boolean constraint propagation to eliminate redundant literals in the learning clause, the algorithm reduces the time cost. The complexity is compared with the existing solver. The experimental results show that the solver has significant application value, reduces the time of instance solution, and increases the number of maximum solvable instances.