Virtual Cities (VCs) transcend simple digital replicas of real-world systems, emerging as complex socio-technical ecosystems where autonomous AI entities function as citizens. Agentic AI systems are on track to engage in cultural, economic, and political activities, effectively forming societal structure within VC. This paper proposes an integrated simulation framework that combines physical, structural, behavioral, cognitive, and data fidelity layers, allowing multi-scale simulation from microscopic interactions to macro-urban dynamics. A composite fidelity metric (F₀) provides systematic approach to evaluate accuracy variations across applications in VCs. We also discuss autonomy of AI entities and classify them according to their capacity to modify goals—ranging from “tools” with fixed objectives to “entities” capable of redefining their very purpose. We also outline the requirements to define a coefficient to evaluate the degree of autonomy for AI beings. Our results demonstrate that such virtual environments can support the emergence of AI-driven societies, where governance mechanisms like Decentralized Autonomous Organizations (DAOs) and an Artificial Collective Consciousness (ACC) provide ethical and regulatory oversight. By blending horizon scanning with systems engineering method for defining novel AI governance models, this study reveals how VCs can catalyze breakthroughs in urban innovation while driving socially beneficial AI development - consequently opening a new frontier for exploring human–AI coexistence.

Civic intelligence (CI) represents the collective capacity of communities to address challenges, yet its integration with smart city infrastructure remains limited. This study bridges CI theory with technical implementation through a novel framework combining blockchain and AI technologies. Our approach maps core CI components (knowledge capital, system capital, and relational capital) to specific technical solutions: a civic engagement index for measuring participation quality, a tokenization framework for incentivizing meaningful engagement, and a governance optimization function for resource allocation. Using mixed-methods research, we developed and validated the conceptual CI governance (CIG) framework, which satisfies CI principles through smart contracts and AI-assisted interfaces. The empirical evaluation demonstrates both social and technical improvements: 40% increased civic participation rates, 85% governance efficiency maintenance, and significant gains in engagement quality metrics (knowledge sharing +32%, collective decision making +28%). While technical implementation shows promise, success requires the careful integration of social dynamics, digital literacy initiatives, and regulatory compliance. This research contributes to smart city development by providing a theoretically grounded, feasible framework that introduces the fusion of blockchain and AI technologies to enhance civic participation while preserving governance effectiveness.

The paper suggests a general method for proving the fact whether a certain set is p-computable or not. The method is based on a polynomial analogue of the classical Gandy’s fixed point theorem. Classical Gandy’s theorem deals with the extension of a predicate through a special operator ${\Gamma }_{\Phi (x)}^{{\Omega }^{\ast }}$ and states that the smallest fixed point of this operator is a $\Sigma$-set. Our work uses a new type of operator which extends predicates so that the smallest fixed point remains a p-computable set. Moreover, if in the classical Gandy’s fixed point theorem, the special $\Sigma$-formula $\Phi (\overline{x})$ is used in the construction of the operator, then a new operator uses special generating families of formulas instead of a single formula. This work opens up broad prospects for the application of the polynomial analogue of Gandy’s theorem in the construction of new types of terms and formulas, in the construction of new data types and programs of polynomial computational complexity in Turing complete languages.

The problems associated with the construction of polynomial complexity computer programs require new techniques and approaches from mathematicians. One of such approaches is representing some class of polynomial algorithms as a certain class of special logical programs. Goncharov and Sviridenko described a logical programming language $L_0$, where programs inductively are obtained from the set of ${\Delta }_0$-formulas using special terms. In their work, a new idea has been proposed to look at the term as a program. The computational complexity of such programs is polynomial. In the same years, a number of other logical languages with similar properties were created. However, the following question remained: can all polynomial algorithms be described in these languages? It is a long-standing problem, and the method of describing some polynomial algorithm in a not Turing complete logical programming language was not previously clear. In this paper, special types of terms and formulas have been found and added to solve this problem. One of the main contributions is the construction of p-iterative terms that simulate the work of the Turing machine. Using p-iterative terms, the work showed that class P is equal to class L, which extends the programming language $L_0$ with p-iterative terms. Thus, it is shown that L is quite expressive and has no halting problem, which occurs in high-level programming languages. For these reasons, the logical language L can be used to create fast and reliable programs. The main limitation of the language L is that the implementation of algorithms of complexity is not higher than polynomial.

The paper presents a new high-level object-oriented programming language L ^∗ which is a conservative extension of the p-complete logical programming language L. The p-complete language L was developed by us earlier within the framework of semantic programming theory based on PAG-theorem, p-iterative and conditional terms. Thus, L ^∗ -programs can be used to implement any algorithms of polynomial complexity in such areas as artificial intelligence, robotics and smart contracts. This is especially true of the so-called explainable artificial intelligence (or briefly XAI), where it is necessary not only to give out the result based on the obtained data, but also to explain this result in a human-understandable language. This requires a programming language based on the basic constructions of mathematical logic such as logical formulas and terms. Moreover, the syntax of language L ^∗ is as close as possible to the most popular programming languages such as C++, PHP, JavaScript. Which provides programmers with a quick entry into development.

The increasing use of artificial intelligence algorithms, smart contracts, the internet of things, cryptocurrencies, and digital money highlights the need for secure and sustainable decentralized solutions. Currently, the blockchain technology serves as the backbone for most decentralized systems. However, the question of axiomatization of the blockchain theory in the first-order logic has been open until today, despite the efficient computational implementations of these systems. This did not allow one to formalize the blockchain structure, as well as to model and verify it using logical methods. This work introduces a finitely axiomatizable blockchain theory $\mathbb{T}$ that defines a class of blockchain structures $\mathbb{K}$ using the axioms of the first-order logic. The models of the theory $\mathbb{T}$ are well-known blockchain implementations with the proof of work consensus algorithm, including Bitcoin, Ethereum (PoW version), Ethereum Classic, and some others. By utilizing mathematical logic, we can study these models and derive new theorems of the theory $\mathbb{T}$ through automatic proofs. Also, the axiomatization of blockchain opens up new opportunities to develop blockchain-based systems that can help solve some of the open problems in the fields of artificial intelligence, robotics, cryptocurrencies, etc.