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Proof-theoretic methods are developed for subsystems of Johansson's logic obtained by extending the positive fragment of intuitionistic logic with weak negations. These methods are exploited to establish properties of the logical systems.…
One-parameter interpolations between any two unitary matrices (e.g., quantum gates) $U_1$ and $U_2$ along efficient paths contained in the unitary group are constructed. Motivated by applications, we propose the continuous unitary path…
Large language models (LLMs) have showcased remarkable capabilities in complex reasoning through chain of thought (CoT) prompting. Recently, there has been a growing interest in transferring these reasoning abilities from LLMs to smaller…
This paper presents experiments on common knowledge logic, conducted with the help of the proof assistant Coq. The main feature of common knowledge logic is the eponymous modality that says that a group of agents shares a knowledge about a…
Most interesting proofs in mathematics contain an inductive argument which requires an extension of the LK-calculus to formalize. The most commonly used calculi for induction contain a separate rule or axiom which reduces the valid proof…
The solution of Shareshian-Wachs conjecture by Brosnan-Chow and Guay-Paquet tied the graded chromatic symmetric functions on indifference graphs (or unit interval graphs) and the cohomology of regular semisimple Hessenberg varieties with…
We develop foundations for computing Craig interpolants and similar intermediates of two given formulas with first-order theorem provers that construct clausal tableaux. Provers that can be understood in this way include efficient…
Multipoint secant and interpolation methods are effective tools for solving systems of nonlinear equations. They use quasi-Newton updates for approximating the Jacobian matrix. Owing to their ability to more completely utilize the…
In this paper we study the functions that can be learned through the polynomial interpolation quantum algorithm designed by Childs et al. This algorithm was initially intended to find the coefficients of a multivariate polynomial function…
We present a sequent-based deductive system for automatically proving entailments in separation logic by using mathematical induction. Our technique, called mutual explicit induction proof, is an instance of Noetherian induction.…
Using polyadic MV algebras, we show that many predicate many valued logics have the interpolation property.
The bisimulation proof method can be enhanced by employing `bisimulations up-to' techniques. A comprehensive theory of such enhancements has been developed for first-order (i.e., CCS-like) labelled transition systems (LTSs) and…
We present a uniform characterisation of three-valued logics by means of the bisequent calculus (BSC). It is a generalised form of a sequent calculus (SC) where rules operate on the ordered pairs of ordinary sequents. BSC may be treated as…
This paper argues that the operations of a 'Universal Turing Machine' (UTM) and equivalent mechanisms such as the 'Post Canonical System' (PCS) - which are widely accepted as definitions of the concept of `computing' - may be interpreted as…
We devise three strategies for recognizing admissibility of non-standard inference rules via interpolation, uniform interpolation, and model completions. We apply our machinery to the case of symmetric implication calculus $\mathsf{S^2IC}$,…
We introduce the calculus of Classical Transitions (CT), which extends the research line on the relationship between linear logic and processes to labelled transitions. The key twist from previous work is registering parallelism in typing…
Prompt tuning has become a popular strategy for adapting Vision-Language Models (VLMs) to zero/few-shot visual recognition tasks. Some prompting techniques introduce prior knowledge due to its richness, but when learnable tokens are…
In this paper, we consider methods to compute the coefficients of interpolants relative to a basis of polynomials satisfying a three-term recurrence relation. Two new algorithms are presented: the first constructs the coefficients of the…
We propose a new approach to computing global minimizers of singular value functions in two real variables. Specifically, we present new algorithms to compute the Kreiss constant of a matrix and the distance to uncontrollability of a linear…
Training a unified language model that adapts between intuitive System 1 and deliberative System 2 remains challenging due to interference between their cognitive modes. Recent studies have thus pursued making System 2 models more…