Computer Science and Mathematics

Abstract: Circumscription is a classical non-monotonic formalism in which selected atoms are minimized while other atoms are fixed or allowed to vary. For propositional clause theories, checking whether a candidate interpretation is a circumscription model amounts to a global minimality test. We study this checking problem through the minimal reduct of the candidate interpretation. The reduct turns the global test into a residual entailment problem; we then decompose that entailment problem along the collapsed negative dependency graph. The checker verifies source components over their ancestor scopes, contracts atoms whose obligations have been certified, and records certificate fragments that refer back to clauses of the original input theory. We give two exact local certification strategies: a direct SAT check and a MUS-based extraction procedure. Experiments on solved random 3CNF instances and industrial CNF instances show that the decomposition-based checker agrees with the global reduct baseline and that its certificates can be replayed. The MUS variant produces much smaller supports, but it also spends more time on extraction.

Abstract: AI-enabled infrastructure systems increasingly govern access to emergency services, disaster relief, and utility restoration, yet they routinely produce inequitable outcomes even when allocation algorithms apply procedurally neutral rules. The standard explanation locates the cause inside the algorithm. This paper argues instead that inequity arises from the interaction between the algorithm and the physical environment in which it operates: network topology, resource locations, and demand distribution jointly constrain what any policy can achieve, and when those constraints are sufficiently binding, ethical infeasibility is structural rather than algorithmic. We introduce a constraint-based formulation that embeds ethical requirements into the feasible region, and a hierarchical Irreducible Infeasible Subsystem (IIS) procedure that attributes infeasibility to rule design, algorithmic choice, or physical infrastructure. We further establish the Structural Infeasibility Theorem, deriving closed-form bounds on inter-group disparity across all feasible policies. The framework was applied to zone-decomposable infrastructure allocation problems generally, with a metropolitan ambulance-dispatch system serving as a concrete instantiation. The study delivers four findings. First, the minimum-service violation may not be caused by the allocation algorithm itself; rather, it may arise from the physical layout of the infrastructure. Second, the observed efficiency–equity trade-off may not be an unavoidable feature of equitable allocation, but may instead reflect the difficulty of achieving equity within an underbuilt system. Third, before new infrastructure is added, improvements in equity may represent harm redistribution rather than harm reduction. Fourth, the IIS certificate can be translated into a concrete capital-investment requirement, showing what physical change may be needed to restore ethical feasibility.

Abstract: Let F(x, n) denote the formula ∃ab∀i_≤n∃swpq∀jν∃eg {(s + w)² + 3w + s = 2i ∧ <[j = w ∧ ν = q] ν [j = 3i ∧ ν = p + q] ∨ [j = s ∧ (ν = p ∨ (i = n ∧ ν = q + x))] ν [j = 3i + 1 ∧ ν = pq] → a = ν + e + ejb ∧ ν + g = jb>} from J. P. Jones’ article published in 1978. From the results of this article, it follows that the set {n ∈ N : ¬F(n, n)} is non-recursively enumerable and co-recursively enumerable. We prove that the set W = {n ∈ N : ∃p, q ∈ N ((2n = (p + q)(p + q + 1) + 2q) ∧ ∀(x0, . . . , xp) ∈ Np+1 ∃(y0, . . . , yp) ∈ {0, . . . , q}p+1 ((∀j, k ∈ {0, . . . , p} (xj + 1 = xk ⇒ yj + 1 = yk)) ∧ (∀i, j, k ∈ {0, . . . , p} (xi · xj = xk ⇒ yi · yj = yk))))} is not recursively enumerable. We prove that the set N \W is recursively enumerable. Let β : N³ → N denote Gödel’s β function. For x₁, x₂, x₃ ∈ N, β(x₁, x₂, x₃) equals the remainder after integer division of x₁ by 1 + (x₃ + 1) · x₂. We prove that the set W consists of all n ∈ N such that ∀u, v ∈ N ∃a, b, p, q ∈ N ((2n = (p + q)(p + q + 1) + 2q) ∧ ∀i, j, k ∈ {0, . . . , p} ((β(a, b, i) ⩽ q) ∧ (β(u, v, j) + 1 = β(u, v, k) ⇒ β(a, b, j) + 1 = β(a, b, k)) ∧ (β(u, v, i) · β(u, v, j) = β(u, v, k) ⇒ β(a, b, i) · β(a, b, j) = β(a, b, k)))) The above formula can be easily translated into a formula in Peano arithmetic.

Abstract: This paper focuses on the optimization of engineering decision-making under uncertain environments. Engineering decision-making requires optimizing the input of production materials and the selection of equipment and processes under the constraints of cost and expected return to minimize costs and maximize production benefits. As an efficient formal verification technique, model checking provides a new approach to solve this problem. Traditional model checking mainly focuses on qualitative verification, while quantitative model checking techniques (such as probabilistic and possibilistic model checking) have been developed gradually, among which possibilistic model checking is more suitable for systems with fuzzy uncertainty. However, existing possibilistic model checking techniques have obvious defects: first, they only target closed systems and do not consider the interaction between the system and the external environment; second, the simple information aggregation method leads to information desynchronization and information loss; third, they cannot model and verify systems with incomplete information. Model checking technology based on possibilistic decision processes considers uncertain action selection and initially solves the problem of modeling and verification of open systems. The author has introduced the idea of quality constraints into possibilistic temporal logic to solve the problems of information desynchronization and information loss in possibilistic model checking; moreover, the author has established the theories of Intuitionistic Fuzzy Kripke Structure (IFKS) and Intuitionistic Fuzzy Computation Tree Logic (IFCTL), which can model and verify systems with incomplete information. To improve the usability and accuracy of engineering decisions, this paper will draw on the ideas and methods of uncertain selection of decision behaviors, quality constraints, and incompleteness modeling, extend IFKS to Weighted Intuitionistic Fuzzy Kripke Structure (WIFKS), induce IFCTL to Intuitionistic Fuzzy Decision Tree Logic (IFDTL), propose an algorithm for solving IFDTL model checking problems, and present a solution algorithm for multi-attribute engineering decision-making based on IFDTL model checking, along with its correctness proof and complexity analysis. Finally, a case study of Qinling health-preserving tourism planning is given to verify the rationality and efficiency of the proposed method, providing a new formal solution for uncertain engineering decision-making.

Abstract: The Classical Propositional Calculus CPC (zero-order logic, classical propositional logic), is the most fundamental two-valued logical system. Next, the Intuitionistic Propositional Calculus IPC differs from the CPC among others, that in IPC some laws of CPC are invalid (among others, the law of excluded middle and the law of double strong negation). Another difference is such that in IPC the principle of indirect proof (proof by contradiction) is rejected. In this paper, inconsistency (in the absolute sense i.e. Post’s sense) of the Classical Propositional Calculus is proved. From the inconsistency of CPC it follows immediately that the Intuitionistic Propositional Calculus is inconsistent in the absolute sense (Post’s sense), too.

Abstract: Mathematics, as actually practiced, operates as a federated system: practitioners work within autonomous domain-specific axiomatizations (geometry, algebra, analysis) and construct explicit bridges only when cross-domain reasoning is required. This organization is not accidental; it is a structural adaptation that safeguards local decidability and algorithmic efficiency. Yet the dominant foundational narrative still operates on the Compiler Myth—the belief that all mathematics must theoretically compile down into ZFC set theory to achieve rigor. We argue that this monolithic reductionism confuses representational universality with logical priority. Embedding a decidable (tame) domain into an undecidable (wild) one does not clarify foundations; it imposes a crippling epistemic overhead. It buries efficient, domain-specific decision procedures under general proof search and destroys the native structural immunities of the object. We introduce the Decidability Threshold — a litmus test based on Negation, Representability, and Discrete Unboundedness — to explain why mathematicians instinctively isolate tame domains from wild ones. Finally, we distinguish the Mathematician (builder of formal systems) from the Scientist (consumer modeling empirical reality). We argue that federalism is not a failure of unification, but the primary safeguard preventing the scientist from inadvertently importing uncomputable, undecidable paradoxes into physical theories. We show that for empirical applications, syntactic safety is insufficient; valid scientific modeling must be strictly confined to the constructively computable sub-fragments of these domains.

Abstract: This paper investigates the structure of fuzzy Lie subalgebras, with particular emphasis on isomorphisms and nilpotency. Building on two prior conference contributions, one of which established foundational results on fuzzy bases of Lie algebras, we develop here a more complete and unified treatment of these themes. We introduce a notion of isomorphism between fuzzy Lie subalgebras based on the transfer principle via t-cut sets, and we prove that isomorphic fuzzy Lie subalgebras necessarily share the same nilpotency measure. The central contribution of the paper is a fuzzy measure of nilpotency N(μ)∈[0,1], defined for any non-constant fuzzy Lie subalgebra μ of a Lie algebra g. This invariant equals 1 precisely when μ is fuzzy nilpotent, and decreases as the subalgebra departs from nilpotency. We show that nilpotency of the underlying Lie algebra implies N(μ)=1, but that the converse fails in general, as witnessed by an explicit counterexample.

Abstract: This work traces a philosophical and mathematical thread from ancient Greek mathematics to modern foundational logic. The Greeks maintained a sharp distinction between ἀριθμητική (arithmetic as the theoretical science of numbers) and λογιστική (calculation as a practical art), while also separating arithmetic from formal logic. This separation, grounded in ontological and epistemological considerations, allowed Greek mathematics to avoid the foundational crises that would emerge two millennia later. The development of formal logic in the late nineteenth and early twentieth centuries—particularly through the work of Frege, Russell, and Hilbert—sought to unify arithmetic and logic within a single syntactic framework. Gödel's incompleteness theorems (1931) demonstrated the impossibility of this project, showing that any consistent, recursively axiomatizable theory strong enough to encode arithmetic must be incomplete and cannot prove its own consistency. Furthermore, phenomena such as Tarski's undefinability of truth and the existence of non-standard models demonstrate that pure syntax faces a total epistemological collapse. This work argues that these metamathematical limits can be synthesized into a "Semantic Necessity Theorem": a complete, consistent, arithmetically strong theory cannot be purely syntactic. The Greek separation of arithmetic from formal logic thus appears not merely as a historical curiosity, but as a mathematically prescient framework that anticipates the structural necessity of ontology in modern mathematics.

Abstract: In this exercise, we will discuss quantum entanglement in an intuitionistic context and its evolution. This requires a definition of the tensor product, as well as the introduction of a Hamiltonian.

Abstract: Being either true or false, 1 or 0, the standard logic and the Boolean algebra traditionality never rotate. Thus, it can only account for polarization states but not superposition states. This paper proves the Boolean rotation theorem through complexifications. This result allows us to formulate polarization spinors as well as superpositions spinors. It provides a new understanding of the Riemann sphere of two-state systems. It also provides an alternative solution to the measurement of wavefunctions, which accounts for both the U-process and the R-process. The work reported in this paper formulates the Penrose twistor geometry of the polarization spinors and the superposition spinors.

Abstract: This paper investigates cooperative advertising decisions in production–retailing chan-nels for seasonal products under demand seasonality. We develop analytical game-theoretic models to examine how advertising cooperation influences channel coor-dination and profit distribution between manufacturers and retailers. Two channel struc-tures are considered: a single-manufacturer–single-retailer channel and a sin-gle-manufacturer channel with two competing retailers. For each structure, Stackelberg and Nash equilibrium settings are analyzed and compared. Our results show that coop-erative advertising can serve as an effective coordination mechanism by increasing adver-tising intensity and improving channel efficiency. Retailers always benefit from manu-facturer-supported advertising through cost sharing and higher profitability, whereas the manufacturer’s incentive to participate depends on whether demand expansion out-weighs shared advertising costs. Importantly, we demonstrate that channel leadership plays a critical role: the Stackelberg equilibrium consistently dominates the Nash equilib-rium in terms of total channel profit. This study contributes to the cooperative advertising literature by explicitly incorporating demand seasonality and competing retailers, and by clarifying when cooperative advertising leads to Pareto improvements in seasonal supply chains.

Abstract: In the framework of (fuzzy) Multi-Criteria Decision-Making, we propose a method that1 allows the decision maker to subjectively approach the problem by suitably modifying the decision matrix. We consider a decision problem related to a random quantity X with set of values {x₁, x₂, . . . , x_n}, and a set of properties {C₁, C₂, . . . , Cm}of X. In this setting, the properties C_j are the criteria of the decision problem, the alternatives represent the events A_i = (X= x_i), for i= 1, . . . , n, and the criteria’s weights w_j, for j= 1, 2, . . . , m, are seen as the probabilities of the events “C_j is relevant with respect to the decision problem”. For each i= 1, . . . , n and j= 1, 2, . . . , m, we interpret the scores aij as membership functions representing “how much alternative A_i satisfies criterion C_j”. By adopting the interpretation of membership functions as suitable conditional probabilities, together with the theory of logical operations among conditional events, we allow logical operations among criteria and consistently apply this interpretation to the corresponding scores. In particular, when considering the complement, conjunction, and disjunction of criteria, the resulting scores are the coherent) previsions of the respective compound conditionals within the framework of conditional random quantities.

Abstract: This paper introduces Paraconsistent-Lib, an open-source, easy-to-use Python library for building PAL2v algorithms in reasoning and decision-making systems. Paraconsistent-Lib is designed as a general-purpose library of PAL2v standard calculations, presenting three types of results: paraconsistent analysis in one of the 12 classical lattice PAL2v regions, paraconsistent analysis node (PAN) outputs, and a decision output. With Paraconsistent-Lib, well-known PAL2v algorithms such as Para-analyzer, ParaExtrCTX, PAL2v Filter, paraconsistent analysis network (PANnet), and paraconsistent neural network (PNN) can be written in stand-alone or network form, reducing complexity, code size, and bugs, as two examples presented in this paper. Given its stable state, Paraconsistent-Lib is an active development to respond to user-required features and enhancements received on GitHub.

Abstract: Integration science is an advancement of cognitive science. This paper opens a new topic called metalogic geometry that aims to integrate metalogic with the mathematical twistor theory. We first revisit Gödel methods used in metalogic including Gödel numbering, expressibility, definability, self-referential statement, and proof. Second, it revisits the core ideas of the twistor theory. To follow the Penrose idea: Light rays as twistors, we define the notions of Gödel ray and Penrose cone. By the expressibility and definability, a pair of Gödel numbers compose a Gödel ray (or Tarski ray). A family of Gödel rays composes a Penrose cone. The intersection point of a Penrose cone yields a Gödel point. Gödel rays are projected as twistors. Each Gödel point is projected to a Riemann celestial sphere. Twistors and Riemann spheres assemble the picture of twistor space. The meaning of this work is discussed in the concluding remarks.

Abstract: This paper examines classical diagonal-based results (Cantor's uncountability, G\"odel's incompleteness, Turing's halting problem, and computational universality) through a finite-resource lens. We analyze the diagonal pattern and its dependence on completed enumerations and on unbounded time, space, and precision, then formalize a finite framework $S(T_{\max}, S_{\max}, P_{\max}, L_{\max})$ with integer bounds on time, memory, numerical precision, and symbolic length, and analyze each result within this framework. Within this setting: (i) finite-decimal reals admit explicit enumeration via constant-time bijections; (ii) for formal systems, when bounds are chosen adequate for the system under study, formulas and proofs are finitely enumerable and provability is decidable (complete within bounds); (iii) for the halting problem, adequacy (time beyond the finite-configuration threshold) yields a definitive HALTS/LOOP decision for every machine-input pair, whereas without adequacy the same procedure provides a sound bounded classification (HALTS/TIMEOUT); and (iv) no machine operating under fixed finite bounds is universal in the classical sense. These results show how classical results depend on infinite idealizations and exhibit different behavior under explicit finite resource constraints.

Abstract: Divisible residuated lattices and MTL-algebras are algebraic structures connected with algebras in t-norm based fuzzy logics, being examples of BLalgebras. They are an important significance in the study of fuzzy logic. The purpose of this paper is to investigate and give classifications of these types of algebras. From computational considerations, we analyze the structure of these residuated lattices of small size n (2 ≤ n ≤ 5) and we give summarizing statistics. To extend these results for higer size, we used computer and a constructive algorithm for generating all residuated lattices.

Abstract: This paper introduces Coordination Logic, a formal system designed to model lawful co-variation between domains of description without presupposing causal dependence. The logic is motivated by situations where distinct vocabularies (e.g., physiological and experiential descriptions, or clinical symptoms and behavioural reports) converge on the same underlying event, but where interpreting the relation in causal terms would be inappropriate or misleading. To capture these cases, we define a new conditional operator (⇒c), interpreted as conditional coordination. Unlike material implication, ⇒c is non-vacuous, symmetric and field-dependent: it holds only when both relata are instantiated and coordinated. Semantics is three-valued, with truth tables incorporating a coordination predicate C(p,q) that determines lawful pairing. We further define a biconditional (↔c), establish its properties and develop a sequent calculus for the system. Coordination Logic departs from classical reasoning in rejecting Modus Ponens and Explosion for ⇒c, thereby preserving the non-reductive character of coordination. Applications include the formalization of non-causal dependencies in philosophy of mind, epistemology of science, psychology and psychiatry, where mistaken causal attributions are common. Our framework provides a rigorous alternative to causal or reductive logics, enriching the landscape of non-classical logics with a system grounded in dual-aspect description.

Abstract: The present work studies the Riemann hypothesis from metalogical perspectives. It argues that Riemann hypothesis is independent of the current Riemann analytic continuation. Consequently, as a corollary, if the Riemann hypothesis held, its predicting power on the prime density would be incomplete. This argument is based on the modifications of Gödel’s independent result (1931). This paper shows integrations of Riemann hypothesis and the Gödel structure. On one hand, Riemann hypothesis is construed into the Gödel structure by making a number of modifications. On the other hand, the Gödel structure is applied to disclose the metalogic behind the Riemann hypothesis.

Abstract: Here, we we prove that there is a strictly increasing countable chain of finitary relatively finitely-axiomatizable extensions of ({the} truth-singular {version/extension of})[{the} bounded {expansion of}] first-degree entailments - (TS)[B]FDE, for short - /``relatively axiomatized by the Modus Ponens rule for material implication'', in which case the chain does not contain its join,and so this, being a finitary extension of (TS)[B]FDE, is not {relatively} finitely-axiomatizable. ([As a consequence, applying one of our previous works, we immediately get a strictly decreasing chain of finitely-axiomatizable quasi-varieties of bounded De Morgan lattices including the variety of bounded Kleene lattices with non-finitely-axiomatizable intersection.])

Abstract: The Λ-Invariance Convergence Theorem provides a universal logical framework for understanding the emergence, persistence, and decay of invariance across all domains of intelligibility, including physics, biology, and information systems. It demonstrates that every nontrivial invariant property within a system is a projection of a deeper, substrate-level invariance rooted in the generative substrate Λ, which functions as the foundational source of coherence, stability, and conservation from which all domain-specific laws and structures arise. The theorem rigorously formalizes the mechanisms by which invariance is projected from Λ into concrete system instances and introduces invariance density as a quantitative measure of system health, defining precise laws governing its preservation, regeneration, and decay under degrading transformations. These laws enable predictive modeling of system resilience, vulnerability, and collapse, offering tools to assess the lifecycle of coherent phenomena. By unifying diverse scientific disciplines under a single substrate-level principle, the Λ-Invariance framework reveals that stability and conservation are not isolated domain-specific features but are anchored in the structure of Λ itself, reframing invariance as a substrate-derived property whose manifestation in any system depends on the fidelity of projection from Λ. The framework’s mathematical formalism establishes criteria for determining when invariance can be sustained, when it can be regenerated, and when its decay is irreversible, enabling a cross-domain theory of systemic integrity applicable to the persistence of physical laws, the hereditary stability of biological systems, and the preservation of information in computational and social networks. Ultimately, the Λ-Invariance Convergence Theorem shows that the fate of any intelligible system is determined by its ongoing connection to the substrate of invariance, and that systems degrade not merely through external perturbation but through the erosion of the projection pathway linking them to Λ. This principle offers a comprehensive lens for analyzing the origin, maintenance, and loss of invariance, providing a unified approach to understanding resilience and collapse in complex systems.