arXiv subjects

Yuki Nishida

Publications and source records attributed to Yuki Nishida.

12 recordsLinked to original sources

arXiv2026

Tropical linearization and stability analysis of discrete dynamical systems at the tropical origin }

The tropical semiring is a semiring of extended real numbers, where the operations of `max' and `+' replace the usual addition and multiplication, respectively. Difference equations obtained from the ultradiscrete limit of discrete dynamical systems are described in terms of the tropical semiring. We propose a tropical linearization approach for the stability analysis of difference equations, including those describing ulradiscrete dynamical systems. We show that the fixed point at the tropical origin is asymptotically stable if the maximum eigenvalue of the tropical Jacobian matrix is negative. On the other hand, it is unstable if the maximum eigenvalue of the tropical Jacobian matrix is positive. Since $0$ is the tropical multiplicative identity, these results are analogous to those in the usual linearization process.

math.DS↗

arXiv2026

Combinatorial Algorithm for Tropical Linearly Factorized Programming

The tropical semiring is an algebraic system with addition ``$\max$'' and multiplication ``$+$''. As well as in conventional algebra, linear programming in the tropical semiring has been developed. In this study, we introduce a new type of tropical optimization problem, namely, tropical linearly factorized programming. This problem involves minimizing the objective function given by a product of tropical linear forms divided by a tropical monomial, subject to tropical linear inequality constraints. As the objective function is equivalent to the dual of the transportation problem, it is convex in the conventional sense but not in the tropical sense, while the feasible set is convex in the tropical sense but not in the conventional sense. Our algorithm for tropical linearly factorized programming is based on the descent method. We first show that a feasible descent direction can be characterized in terms of a specific digraph, called a tangent digraph. Especially in non-degenerate cases, we present a simplex-like algorithm that updates the tree structure of tangent digraphs iteratively. Each iteration can be executed in $O(r_A+r_C)$ time, where $r_A$ and $r_C$ are the numbers of finite coefficients in the constraints and objective function, respectively. For integer instances, our algorithm finds a local optimum in pseudo-polynomial time.

math.OC↗

arXiv2025

Transformation from integral operator with separable kernel to matrix in eigenvalue problem

This paper investigates the eigenvalue problem of integral operators whose kernels can be expressed as a finite sum of pairwise products of single-variable functions, making them separable. By consdiering the matrix form of the separable kernel in the integral operator, we establish the relationship between the eigenvalues and eigenfunctions of the integral operator and the eigenpairs of a matrix. We next generalize the eigenfunction of an integral operator based on the concept of generalized eigenvectors of matrices, and show that solving the Fredholm integral equation of the second kind reduces to computing matrix eigenpairs and generalized eigenvectors. We also provide several examples to validate our results.

math.FA↗

arXiv2024

Law and Order for Typestate with Borrowing

Typestate systems are notoriously complex as they require sophisticated machinery for tracking aliasing. We propose a new, transition-oriented foundation for typestate in the setting of impure functional programming. Our approach relies on ordered types for simple alias tracking and its formalization draws on work on bunched implications. Yet, we support a flexible notion of borrowing in the presence of typestate. Our core calculus comes with a notion of resource types indexed by an ordered partial monoid that models abstract state transitions. We prove syntactic type soundness with respect to a resource-instrumented semantics. We give an algorithmic version of our type system and prove its soundness. Algorithmic typing facilitates a simple surface language that does not expose tedious details of ordered types. We implemented a typechecker for the surface language along with an interpreter for the core language.

cs.PL↗

arXiv2024

On max-plus two-sided linear systems whose solution sets are min-plus linear

The max-plus algebra $\mathbb{R}\cup \{-\infty \}$ is defined in terms of a combination of the following two operations: addition, $a \oplus b := \max(a,b)$, and multiplication, $a \otimes b := a + b$. In this study, we propose a new method to characterize the set of all solutions of a max-plus two-sided linear system $A \otimes x = B \otimes x$. We demonstrate that the minimum ``min-plus'' linear subspace containing the ``max-plus'' solution space can be computed by applying the alternating method algorithm, which is a well-known method to compute single solutions of two-sided systems. Further, we derive a sufficient condition for the ``min-plus'' and ``max-plus'' subspaces to be identical. The computational complexity of the method presented in this study is pseudo-polynomial.

math.CO↗

arXiv2023

Algorithm for the CSR expansion of max-plus matrices using the characteristic polynomial

Max-plus algebra is a semiring with addition $a\oplus b = \max(a,b)$ and multiplication $a\otimes b = a+b$. It is applied in cases, such as combinatorial optimization and discrete event systems. We consider the power of max-plus square matrices, which is equivalent to obtaining the all-pair maximum weight paths with a fixed length in the corresponding weighted digraph. Each $n$-by-$n$ matrix admits the CSR expansion that decomposes the matrix into a sum of at most $n$ periodic terms after $O(n^{2})$ times of powers. In this study, we propose an $O(n(m+n \log n))$ time algorithm for the CSR expansion, where $m$ is the number of nonzero entries in the matrix, which improves the $O(n^{4} \log n)$ algorithm known for this problem. Our algorithm is based on finding the roots of the characteristic polynomial of the max-plus matrix. These roots play a similar role to the eigenvalues of the matrix, and become the growth rates of the terms in the CSR expansion.

math.OC↗

arXiv2021

Independence and orthogonality of algebraic eigenvectors over the max-plus algebra

The max-plus algebra $\mathbb{R}\cup \{-\infty \}$ is a semiring with the two operations: addition $a \oplus b := \max(a,b)$ and multiplication $a \otimes b := a + b$. Roots of the characteristic polynomial of a max-plus matrix are called algebraic eigenvalues. Recently, algebraic eigenvectors with respect to algebraic eigenvalues were introduced as a generalized concept of eigenvectors. In this paper, we present properties of algebraic eigenvectors analogous to those of eigenvectors in the conventional linear algebra. First, we prove that for generic matrices algebraic eigenvectors with respect to distinct algebraic eigenvalues are linearly independent. We further prove that for symmetric matrices algebraic eigenvectors with respect to distinct algebraic eigenvalues are orthogonal to each other.

math.CO↗

arXiv2021

HELMHOLTZ: A Verifier for Tezos Smart Contracts Based on Refinement Types

A smart contract is a program executed on a blockchain, based on which many cryptocurrencies are implemented, and is being used for automating transactions. Due to the large amount of money that smart contracts deal with, there is a surging demand for a method that can statically and formally verify them. This article describes our type-based static verification tool HELMHOLTZ for Michelson, which is a statically typed stack-based language for writing smart contracts that are executed on the blockchain platform Tezos. HELMHOLTZ is designed on top of our extension of Michelson's type system with refinement types. HELMHOLTZ takes a Michelson program annotated with a user-defined specification written in the form of a refinement type as input; it then typechecks the program against the specification based on the refinement type system, discharging the generated verification conditions with the SMT solver Z3. We briefly introduce our refinement type system for the core calculus Mini-Michelson of Michelson, which incorporates the characteristic features such as compound datatypes (e.g., lists and pairs), higher-order functions, and invocation of another contract. \HELMHOLTZ{} successfully verifies several practical Michelson programs, including one that transfers money to an account and that checks a digital signature.

cs.PL↗

arXiv2020

Compilation of Coordinated Choice

Recently, we have proposed coordinated choices, which are nondeterministic choices equipped with names. The main characteristic of coordinated choices is that they synchronize nondeterministic decision among choices of the same name. The motivation of the synchronization mechanism is to solve a theoretical problem. So, as a practical programming language, we still want to use coordinated choices like standard ones. In other words, we want to avoid synchronization. Now, there are two problems: (i) practically, it is a bit complicated work to write a program using coordinated choices in which execution synchronization never happens; and (ii) theoretically, it is unknown whether any programs using standard choices can be written by using only coordinated ones. In this paper, we define two simply typed lambda calculi called $λ^\parallel$ equipped with standard choices and $λ^{\parallelω}$ equipped with coordinated choices, and give compilation rules from the former into the latter. The challenge is to show the correctness of the compilation because behavioral correspondence between expressions before and after compiling cannot be defined directly by the compilation rules. For the challenge, we give an effect system for $λ^{\parallelω}$ that characterizes expressions in which execution synchronization never happens. Then, we show that all compiled expressions can be typed by the effect system. As a result, we can easily show the correctness because the main concern of the correctness is whether synchronization happens or not.

cs.PL↗

arXiv2020

$q$-VFCA: $q$-state Vector-valued Fuzzy Cellular Automata

Elementary fuzzy Cellular Automata (CA) are known as continuous counterpart of elementary CA, which are 2-state CA, via the polynomial representation of local rules. In this paper, we first develop a new fuzzification methodology for $q$-state CA. It is based on the vector representation of $q$-state CA, that is, the $q$-states are assigned to the standard basis vectors of the $q$-dimensional real space and the local rule can be expressed by a tuple of $q$ polynomials. Then, the $q$-state vector-valued fuzzy CA are defined by expanding the set of the states to the convex hull of the standard basis vectors in the $q$-dimensional real space. The vector representation of states enables us to enumerate the number-conserving rules of 3-state vector-valued fuzzy CA in a systematic way.

nlin.CG↗

arXiv2019

Manifest Contracts with Intersection Types

We present a manifest contract system PCFv$Δ$H with intersection types. A manifest contract system is a typed functional calculus in which software contracts are integrated into a refinement type system and consistency of contracts is checked by combination of compile- and run-time type checking. Intersection types naturally arise when a contract is expressed by a conjunction of smaller contracts. Run-time contract checking for conjunctive higher-order contracts in an untyped language has been studied but our typed setting poses an additional challenge due to the fact that an expression of an intersection type $τ_1 \wedge τ_2$ may have to perform different run-time checking whether it is used as $τ_1$ or $τ_2$. We build PCFv$Δ$H on top of the $Δ$-calculus, a Church-style intersection type system by Liquori and Stolze. In the $Δ$-calculus, a canonical expression of an intersection type is a strong pair, whose elements are the same expressions except for type annotations. To address the challenge above, we relax strong pairs so that expressions in a pair are the same except for type annotations and casts, which are a construct for run-time checking. We give a formal definition of PCFv$Δ$H and show its basic properties as a manifest contract system: preservation, progress, and value inversion. Furthermore, we show that run-time checking does not affect essential computation.

cs.PL↗

arXiv2017

Sharper and Simpler Nonlinear Interpolants for Program Verification

Interpolation of jointly infeasible predicates plays important roles in various program verification techniques such as invariant synthesis and CEGAR. Intrigued by the recent result by Dai et al.\ that combines real algebraic geometry and SDP optimization in synthesis of polynomial interpolants, the current paper contributes its enhancement that yields sharper and simpler interpolants. The enhancement is made possible by: theoretical observations in real algebraic geometry; and our continued fraction-based algorithm that rounds off (potentially erroneous) numerical solutions of SDP solvers. Experiment results support our tool's effectiveness; we also demonstrate the benefit of sharp and simple interpolants in program verification examples.

cs.LO↗