arXiv subjects

Zoran Majkic

Publications and source records attributed to Zoran Majkic.

At least 19 recordsLinked to original sources

arXiv2026

Probabilistic Extension of Neuro-Symbolic AGI Robots based on Belnap's Typed Intensional FOL

Neuro-symbolic AI based on $IFOL_B$ is a way to combine neural learning and symbolic reasoning to overcome limitations of purely neural systems (like lack of interpretability and logical structure) with formal logical machinery for self-reference. In this paper we expand the cognitive power of $IFOL_B$ by using the probability computation for the currently unknown sentences, based on Nilsson's probability structure for the $IFOL_B$. We introduce the global symmetry transformation that preserves the current knowledge database and logical deduction, and the local one used for real-time decisions about concrete (sub)problems that involve only a very strict subset of $IFOL_B$ predicates. The computation of probability density function $KI$ in both cases, based on the Shannon's maximum information entropy, is provided by neural networks of this probabilistic neuro-symbolic AGI.

cs.AI↗

arXiv2026

General Categorial Geometry and Algebraic Topology

In Categorial Topology, given a category (as a "geometric object") we can consider its properties preserved under continuous action (a "deformation") of a comma-propagation operation. However, the Metacategory space, valid for all categories, cannot be defined by using well-know Grothendeick's approach with discrete ringed spaces. So, we can consider any category $\textbf{C}$ as an abstract geometric object, that is, a discrete space where the points are the objects of this category and morphisms between objects as the oriented paths. \\Based on this approach, we define the Cat-arrows space $V$ valid for all categories with commutative (and associative) partial addition operation $\oplus$ for the vectors, based on partial operation of categorial composition of morphisms, their inner and outer products in 3D Cat-arrows space, like in 3D Clifford algebra. We provide a general definition of the norm ("weight") of the vectors in $V$, cumulative under the composition of the category morphisms. This norm assigned to the morphisms, non existing in metacategory theory, enriches each individual category with new semantical interpretation of the morphisms: they are "weighted" with a real number. So, this general transformation of metacategory theory into 3D Cat-arrows space of vectors is not only another example of applicability of 3D Clifford algebra, but the new semantic (of their "weight") and topological enrichment of the Metacategory theory with its abstract undefined primitive concept of morphisms.

math.GM↗

arXiv2026

Neuro-Symbolic Strong-AI Robots with Closed Knowledge Assumption: Learning and Deductions

Knowledge representation formalisms are aimed to represent general conceptual information and are typically used in the construction of the knowledge base of reasoning agent. A knowledge base can be thought of as representing the beliefs of such an agent. Like a child, a strong-AI (AGI) robot would have to learn through input and experiences, constantly progressing and advancing its abilities over time. Both with statistical AI generated by neural networks we need also the concept of \textsl{causality} of events traduced into directionality of logic entailments and deductions in order to give to robots the emulation of human intelligence. Moreover, by using the axioms we can guarantee the \textsl{controlled security} about robot's actions based on logic inferences. For AGI robots we consider the 4-valued Belnap's bilattice of truth-values with knowledge ordering as well, where the value "unknown" is the bottom value, the sentences with this value are indeed unknown facts, that is, the missed knowledge in the AGI robots. Thus, these unknown facts are not part of the robot's knowledge database, and by learn through input and experiences, the robot's knowledge would be naturally expanded over time. Consequently, this phenomena can be represented by the Closed Knowledge Assumption and Logic Inference provided by this paper. Moreover, the truth-value "inconsistent", which is the top value in the knowledge ordering of Belnap's bilattice, is necessary for strong-AI robots to be able to support such inconsistent information and paradoxes, like Liar paradox, during deduction processes.

cs.LO↗

arXiv2026

Internal Symmetry Group in Categorial Topology

The interdefinability of the universal concepts of category theory has been introduced by Lawvere. The perfect interdefinability between the objects and arrows of some category, defines the class of Perfectly Symmetric Categories (PSC) where each category can be represented equivalently by its arrows or by its objects only. Such symmetry, differently from the global categorial symmetry ( categorial-symmetry group $CS(\mathbb{Z})$ of all comma-propagation transformations), ia a local internal symmetry inside a given PSC category. Given a PSC category (as a "geometric object") $\textbf{C}$ we can consider its properties (the categorial commutative diagrams) preserved under actions of a particular endofunctor $E$ which transforms any commutative diagram into an invariant "up to isomorphism" diagram. We show that this kind of internal categorial invariance is a phenomena of a local categorial symmetry under an Internal Catergorial Symmetry group $ICS(\mathbb{N})$ of all local enfdofunctorial transformations. Then we establish the relationships between this local internal symmetry and global general symmetry between n-dimensional levels (the comma categories obtained from a PSC category $\textbf{C}$) . We show that if a base category $\textbf{C}$ is a PSC, then all its ne-dimensional levels are PSC as well.

math.GM↗

arXiv2025

Intensional FOL over Belnap's Billatice for Strong-AI Robotics

AGI (Strong AI) aims to create intelligent robots that are quasi indistinguishable from the human mind. Like a child, the AGI robot would have to learn through input and experiences, constantly progressing and advancing its abilities over time. The AGI robot would require an intelligence more close to human's intelligence: it would have a self-aware consciousness that has the ability to solve problems, learn, and plan. Based on this approach an Intensional many-sorted First-order Logic (IFOL), as an extension of a standard FOL with Tarskian's semantics, is proposed in order to avoid the problems of standard 2-valued FOL with paradoxes (inconsistent formulae) and a necessity for robots to work with incomplete (unknown) knowledge as well. This is a more sophisticated version of IFOL with the same syntax but different semantics, able to deal with truth-ordering and knowledge-ordering as well, based on the well known Belnap's billatice with four truth-values that extend the set of classical two truth-values.

cs.LO↗

arXiv2024

Intensional FOL: Many-Sorted Extension

The concepts used in IFOL have associated to them a list of sorted attributes, and the sorts are the intensional concepts as well. The requirement to extend the unsorted IFOL (Intensional FOL) to many-sorted IFOL is mainly based on the fact that a natural language is implicitly many-sorted and that we intend to use IFOL to support applications that use natural languages. Thus, the proposed version of many-sorted IFOL is just the completion of this conceptual feature of the IFOL.

cs.AI↗

arXiv2024

Category Theory: Symmetry Group of Comma-propagation Transformations

In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. The arrow categories are more simple forms of the \emph{comma} categories and were introduced by Lawvere in the context of the interdefinability of the universal concepts of category theory. The basic idea is the elevation of arrows of one category $\textbf{C}$ to objects in another. Given a category (as a "geometric object") $\textbf{C}$ we can consider its properties (the universal categorial commutative diagrams) preserved under actions of a comma-propagation operation $\{\}$ in the infinite hierarchy of its arrow-categories (n-dimensional levels, such that for any $n\geq 1$, $\textbf{C}_{n+1} = {\textbf{C}_n}$, with $\textbf{C}_1 =\textbf{C}$) and on the functors (and their natural transformations) between such n-dimensional levels, which is a phenomena of a general categorial symmetry under a categorial-symmetry group $CS(\mathbb{Z})$ of all comma-propagation transformations.

math.GM↗

arXiv2023

Strong-AI Autoepistemic Robots Build on Intensional First Order Logic

Neuro-symbolic AI attempts to integrate neural and symbolic architectures in a manner that addresses strengths and weaknesses of each, in a complementary fashion, in order to support robust strong AI capable of reasoning, learning, and cognitive modeling. In this paper we consider the intensional First Order Logic (IFOL) as a symbolic architecture of modern robots, able to use natural languages to communicate with humans and to reason about their own knowledge with self-reference and abstraction language property. We intend to obtain the grounding of robot's language by experience of how it uses its neuronal architectures and hence by associating this experience with the mining (sense) of non-defined language concepts (particulars/individuals and universals) in PRP (Properties/Relations/Propositions) theory of IFOL.\\ We consider the robot's four-levels knowledge structure: The syntax level of particular natural language (Italian, French, etc..), two universal language levels: its semantic logic structure (based on virtual predicates of FOL and logic connectives), and its corresponding conceptual PRP structure level which universally represents the composite mining of FOL formulae grounded on the last robot's neuro-system level. Finally, we provide the general method how to implement in IFOL (by using the abstracted terms) different kinds of modal logic operators and their deductive axioms: we present a particular example of robots autoepistemic deduction capabilities by introduction of the special temporal $Konow$ predicate and deductive axioms for it: reflexive, positive introspection and distributive axiom.

cs.AI↗

arXiv2014

Saturation of the morphisms in the database category

In this paper we present the problem of saturation of a given morphism in the database category DB, which is the base category for the functiorial semantics of the database schema mapping systems used in Data Integration theory. This phenomena appears in the case when we are using the Second-Order tuple-generating dependencies (SOtgd) with existentially quantified non-built-in functions, for the database schema mappings. We provide the algorithm of the saturation for a given morphism, which represents a mapping between two relational databases, and show that the original morphism in DB can be equivalently substituted by its more powerful saturated version in any commutative diagram in DB.

cs.LO↗

arXiv2014

Intensional RDB Manifesto: a Unifying NewSQL Model for Flexible Big Data

In this paper we present a new family of Intensional RDBs (IRDBs) which extends the traditional RDBs with the Big Data and flexible and 'Open schema' features, able to preserve the user-defined relational database schemas and all preexisting user's applications containing the SQL statements for a deployment of such a relational data. The standard RDB data is parsed into an internal vector key/value relation, so that we obtain a column representation of data used in Big Data applications, covering the key/value and column-based Big Data applications as well, into a unifying RDB framework. We define a query rewriting algorithm, based on the GAV Data Integration methods, so that each user-defined SQL query is rewritten into a SQL query over this vector relation, and hence the user-defined standard RDB schema is maintained as an empty global schema for the RDB schema modeling of data and as the SQL interface to stored vector relation. Such an IRDB architecture is adequate for the massive migrations from the existing slow RDBMSs into this new family of fast IRDBMSs by offering a Big Data and new flexible schema features as well.

cs.DB↗

arXiv2014

Intensional RDB for Big Data Interoperability

A new family of Intensional RDBs (IRDBs), introduced in [1], extends the traditional RDBs with the Big Data and flexible and 'Open schema' features, able to preserve the user-defined relational database schemas and all preexisting user's applications containing the SQL statements for a deployment of such a relational data. The standard RDB data is parsed into an internal vector key/value relation, so that we obtain a column representation of data used in Big Data applications, covering the key/value and column-based Big Data applications as well, into a unifying RDB framework. Such an IRDB architecture is adequate for the massive migrations from the existing slow RDBMSs into this new family of fast IRDBMSs by offering a Big Data and new flexible schema features as well. Here we present the interoperability features of the IRDBs by permitting the queries also over the internal vector relations created by parsing of each federated database in a given Multidatabase system. We show that the SchemaLog with the second-order syntax and ad hoc Logic Programming and its querying fragment can be embedded into the standard SQL IRDBMSs, so that we obtain a full interoperabilty features of IRDBs by using only the standard relational SQL for querying both data and meta-data.

cs.DB↗

arXiv2011

Data Base Mappings and Theory of Sketches

In this paper we will present the two basic operations for database schemas used in database mapping systems (separation and Data Federation), and we will explain why the functorial semantics for database mappings needed a new base category instead of usual Set category. Successively, it is presented a definition of the graph G for a schema database mapping system, and the definition of its sketch category Sch(G). Based on this framework we presented functorial semantics for database mapping systems with the new base category DB.

cs.DB↗

arXiv2011

Binary Sequent Calculi for Truth-invariance Entailment of Finite Many-valued Logics

In this paper we consider the class of truth-functional many-valued logics with a finite set of truth-values. The main result of this paper is the development of a new \emph{binary} sequent calculi (each sequent is a pair of formulae) for many valued logic with a finite set of truth values, and of Kripke-like semantics for it that is both sound and complete. We did not use the logic entailment based on matrix with a strict subset of designated truth values, but a different new kind of semantics based on the generalization of the classic 2-valued truth-invariance entailment. In order to define this non-matrix based sequent calculi, we transform many-valued logic into positive 2-valued multi-modal logic with classic conjunction, disjunction and finite set of modal connectives. In this algebraic framework we define an uniquely determined axiom system, by extending the classic 2-valued distributive lattice logic (DLL) by a new set of sequent axioms for many-valued logic connectives. Dually, in an autoreferential Kripke-style framework we obtain a uniquely determined frame, where each possible world is an equivalence class of Lindenbaum algebra for a many-valued logic as well, represented by a truth value.

cs.LO↗

arXiv2011

Temporal Probabilistic Logic Programs: State and Revision

There are numerous applications where we have to deal with temporal uncertainty associated with events. The Temporal Probabilistic (TP) Logic Programs should provide support for valid-time indeterminacy of events, by proposing the concept of an indeterminate instant, that is, an interval of time-points (event's time-window) with an associated, lower and upper, probability distribution. In particular, we propose the new semantics, for the TP Logic Programs of Dekhtyar and Subrahmanian. Our semantics, based on the possible world semantics is a generalization of the possible world semantics for (non temporal) Probabilistic Logic Programming, and we define the new syntax for PT-programs, with time variable explicitly represented in all atoms, and show how the standard role of Herbrand interpretations used as possible worlds for probability distributions is coherently extended to Temporal Probabilistic Logic Programming.

cs.LO↗

arXiv2011

Reduction of Many-valued into Two-valued Modal Logics

In this paper we develop a 2-valued reduction of many-valued logics, into 2-valued multi-modal logics. Such an approach is based on the contextualization of many-valued logics with the introduction of higher-order Herbrand interpretation types, where we explicitly introduce the coexistence of a set of algebraic truth values of original many-valued logic, transformed as parameters (or possible worlds), and the set of classic two logic values. This approach is close to the approach used in annotated logics, but offers the possibility of using the standard semantics based on Herbrand interpretations. Moreover, it uses the properties of the higher-order Herbrand types, as their fundamental nature is based on autoreferential Kripke semantics where the possible worlds are algebraic truth-values of original many-valued logic. This autoreferential Kripke semantics, which has the possibility of flattening higher-order Herbrand interpretations into ordinary 2-valued Herbrand interpretations, gives us a clearer insight into the relationship between many-valued and 2-valued multi-modal logics. This methodology is applied to the class of many-valued Logic Programs, where reduction is done in a structural way, based on the logic structure (logic connectives) of original many-valued logics. Following this, we generalize the reduction to general structural many-valued logics, in an abstract way, based on Suszko's informal non-constructive idea. In all cases, by using developed 2-valued reductions we obtain a kind of non truth-valued modal meta-logics, where two-valued formulae are modal sentences obtained by application of particular modal operators to original many-valued formulae.

cs.LO↗

arXiv2011

Intensionality and Two-steps Interpretations

In this paper we considered the extension of the First-order Logic (FOL) by Bealer's intensional abstraction operator. Contemporary use of the term 'intension' derives from the traditional logical Frege-Russell's doctrine that an idea (logic formula) has both an extension and an intension. Although there is divergence in formulation, it is accepted that the extension of an idea consists of the subjects to which the idea applies, and the intension consists of the attributes implied by the idea. From the Montague's point of view, the meaning of an idea can be considered as particular extensions in different possible worlds. In the case of the pure FOL we obtain commutative homomorphic diagram that holds in each given possible world of the intensional FOL, from the free algebra of the FOL syntax, toward its intensional algebra of concepts, and, successively, to the new extensional relational algebra (different from Cylindric algebras). Then we show that it corresponds to the Tarski's interpretation of the standard extensional FOL in this possible world.

cs.LO↗

arXiv2011

Probabilistic Logic: Many-valuedness and Intensionality

The probability theory is a well-studied branch of mathematics, in order to carry out formal reasoning about probability. Thus, it is important to have a logic, both for computation of probabilities and for reasoning about probabilities, with a well-defined syntax and semantics. Both current approaches, based on Nilsson's probability structures/logics, and on linear inequalities in order to reason about probabilities, have some weak points. In this paper we have presented the complete revision of both approaches. We have shown that the full embedding of Nilsson'probabilistic structure into propositional logic results in a truth-functional many-valued logic, differently from Nilsson's intuition and current considerations about propositional probabilistic logic. Than we have shown that the logic for reasoning about probabilities can be naturally embedded into a 2-valued intensional FOL with intensional abstraction, by avoiding current ad-hoc system composed of two different 2-valued logics: one for the classical propositional logic at lower-level, and a new one at higher-level for probabilistic constraints with probabilistic variables. The obtained theoretical results are applied to Probabilistic Logic Programming.

cs.LO↗

arXiv2011

First-order Logic: Modality and Intensionality

Contemporary use of the term 'intension' derives from the traditional logical Frege-Russell's doctrine that an idea (logic formula) has both an extension and an intension. From the Montague's point of view, the meaning of an idea can be considered as particular extensions in different possible worlds. In this paper we analyze the minimal intensional semantic enrichment of the syntax of the FOL language, by unification of different views: Tarskian extensional semantics of the FOL, modal interpretation of quantifiers, and a derivation of the Tarskian theory of truth from unified semantic theory based on a single meaning relation. We show that not all modal predicate logics are intensional, and that an equivalent modal Kripke's interpretation of logic quantifiers in FOL results in a particular pure extensional modal predicate logic (as is the standard Tarskian semantics of the FOL). This minimal intensional enrichment is obtained by adopting the theory of properties, relations and propositions (PRP) as the universe or domain of the FOL, composed by particulars and universals (or concepts), with the two-step interpretation of the FOL that eliminates the weak points of the Montague's intensional semantics. Differently from the Bealer's intensional FOL, we show that it is not necessary the introduction of the intensional abstraction in order to obtain the full intensional properties of the FOL. Final result of this paper is represented by the commutative homomorphic diagram that holds in each given possible world of this new intensional FOL, from the free algebra of the FOL syntax, toward its intensional algebra of concepts, and, successively, to the new extensional relational algebra (different from Cylindric algebras), and we show that it corresponds to the Tarski's interpretation of the standard extensional FOL in this possible world.

cs.LO↗