Related papers: Partial Reductions for Kleene Algebra with Linear …
We prove that the equational theory of Kleene algebra with commutativity conditions on primitives (or atomic terms) is undecidable, thereby settling a longstanding open question in the theory of Kleene algebra. While this question has also…
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our…
A plausible definition of "reasoning" could be "algebraically manipulating previously acquired knowledge in order to answer a new question". This definition covers first-order logical inference or probabilistic inference. It also includes…
One of my recent papers transforms an NP-Complete problem into the question of whether or not a feasible real solution exists to some Linear Program. The unique feature of this Linear Program is that though there is no explicit bound on the…
Inference problems with conjectured statistical-computational gaps are ubiquitous throughout modern statistics, computer science and statistical physics. While there has been success evidencing these gaps from the failure of restricted…
We exhibit a uniform method for obtaining (wellfounded and non-wellfounded) cut-free sequent-style proof systems that are sound and complete for various classes of action algebras, i.e., Kleene algebras enriched with meets and residuals.…
Let $f:V\times V\to F$ be a totally arbitrary bilinear form defined on a finite dimensional vector space $V$ over a a field $F$, and let $L(f)$ be the subalgebra of $\gl(V)$ of all skew-adjoint endomorphisms relative to $f$. Provided $F$ is…
We develop a fully diagrammatic approach to finite-state automata, based on reinterpreting their usual state-transition graphical representation as a two-dimensional syntax of string diagrams. In this setting, we are able to provide a…
The emergence of tools based on artificial intelligence has also led to the need of producing explanations which are understandable by a human being. In most approaches, the system is considered a black box, making it difficult to generate…
A Lie algebra $K$ over a field of characteristic zero $E$ is called a completion of a rational Lie algebra $L$, if it contains $L$ as $\mathbb{Q}$-subalgebra and the $E$-span of $L$ is equal to $K$. The class of all completions of a…
Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and…
Program synthesis is the task of automatically deriving a program that has been specified by a user in advance. Combining automated theorem proving with program synthesis enables the automated construction of proven-to-be-correct programs,…
For any finitely generated abelian group $Q$, we reduce the problem of classification of $Q$-graded simple Lie algebras over an algebraically closed field of "good" characteristic to the problem of classification of gradings on simple Lie…
We propose an algorithm for general nonlinear conic programming which does not require the knowledge of the full cone, but rather a simpler, more tractable, approximation of it. We prove that the algorithm satisfies a strong global…
Many engineering processes can be accurately modelled using partial differential equations (PDEs), but high dimensionality and non-convexity of the resulting systems pose limitations on their efficient optimisation. In this work, a model…
Regular algebraic $K$-theory for groups is a homology theory for discrete groups closely connected (but different from) group homology. It also gives a version of algebraic $K$-theory for rings by the simple functorial mapping assigning to…
Kleene's computability theory based on his S1-S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing's `machine model' which formalises computing with real numbers. A fundamental…
The work is devoted to Computability Logic (CoL) -- the philosophical/mathematical platform and long-term project for redeveloping classical logic after replacing truth} by computability in its underlying semantics (see…
Automated theorem proving, or more broadly automated reasoning, aims at using computer programs to automatically prove or disprove mathematical theorems and logical statements. It takes on an essential role across a vast array of…
Pre-trained language models (PLMs) have achieved remarkable success on various natural language understanding tasks. Simple fine-tuning of PLMs, on the other hand, might be suboptimal for domain-specific tasks because they cannot possibly…