Formal verification is an essential task to ensure the functional correctness of circuits. While several formal verification methods exist to ensure the functional correctness of circuits, they do not provide tight upper bounds for space and time complexities. Therefore, Polynomial Formal Verification (PFV) has been introduced to tackle this problem. In prior work, it has been shown that binary circuits with constant cutwidth can be verified in linear time using Answer Set Programming (ASP). Extending binary logic verification to Multi-Valued Logic (MVL) presents challenges due to the computational complexity of MVL gates and their encodings. In this paper, we propose Linear Formal Verification (LFV) of MVL circuits as a subclass of PFV. Additionally, we prove that MVL circuits with constant cutwidth can be verified in linear time and space. Finally, we evaluate circuits with constant cutwidth in terms of the upper bound of the cutwidth and verification time under different logic levels to confirm our theoretical findings.
The purpose of this work is to explore the connection between Tarski's notion of logical invariance under permutations and the universe of finitely supported sets, both in the classical Fraenkel-Mostowski axiomatic setting and in recent approaches involving invariant sets. We demonstrate that such finitely supported entities naturally qualify as logical notions in Tarski's sense, thereby illustrating the structural harmony between symmetry principles in logic and set theory.
This study presents an innovative approach to optimize delivery success in the online retail industry. The research's core is a machine learning (ML) model that predicts order delivery success, which is made explainable using SHAP (Shapley Additive exPlanations) values. The main novelty of the study is the transformation of SHAP values into Spherical Fuzzy Sets (SFS) and their integration with the TOPSIS method, creating a powerful decision-making mechanism. This mechanism optimizes monthly delivery plans through two scenarios: multi-package delivery, which increases efficiency, and immediate delivery, which enhances customer satisfaction. ML and SHAP serve as tools in this process, while SFS-TOPSIS integration is the main goal of decision-making. This hybrid approach strikes an optimal balance between prediction accuracy, model explainability, and decision-making flexibility. As a result, this study presents a novel methodology that enables the development of data-driven, understandable, and feasible delivery strategies in the online retail sector.
This study introduces the concepts of intuitionistic fuzzy prime ideal, prime intuitionistic fuzzy ideal, intuitionistic fuzzy 2-absorbing, 2absorbing intuitionistic fuzzy ideal, intuitionistic fuzzy primary ideal, and primary intuitionistic fuzzy ideal on a lattice. Then, intuitionistic fuzzy 2-absorbing primary ideals of a lattice are characterized. The transitions among these concepts and their relationships with the ideals of a lattice are discussed. Finally, these concepts are examined on a lattice formed by the Cartesian product of lattices.
A vague graph (VG) is one of the versatile application tools in the field of mathematics, which allows the user to easily describe the vague relation between any objects. VGs are beneficial to give more precision and flexibility to the system as compared to the classical model (i.e.,) crisp theory. The topological index (TI) of graph has a wide range of applications in theoretical chemistry, network design, computer science, etc. In chemical graph theory, the wiener index (also wiener number) introduced by Harry Wiener, is a topological index of a molecular, defined as the sum of the lengths of the shortest paths between all pairs of nodes in the chemical graph. Many topological indices exist only in the crisp but it's new to the VG-environment. The main aim of this research work is to define the topological indices in VGs. Here, indices described in VGs index (MWI), Schultz index (SI), and Gutman index (GI) with illustrations. Various examples are given for each of these topological indices and the necessary condition for the equality of connectivity index (CI) , WI is introduced. We have shown that if two graphs are isomorphic, then their WI are equal. Finally, we give an application of the WI, which is used as a simple screening model based on risk factors in patients to find the most effective factor in cancer.
