Truly stateless, optimal dynamic partial order reduction

Dynamic partial order reduction (DPOR) verifies concurrent programs by exploring all their interleavings up to some equivalence relation, such as the Mazurkiewicz trace equivalence. Doing so involves a complex trade-off between space and time. Existing DPOR algorithms are either exploration-optimal (i.e., explore exactly only interleaving per equivalence class) but may use exponential memory in the size of the program, or maintain polynomial memory consumption but potentially explore exponentially many redundant interleavings. In this paper, we show that it is possible to have the best of both worlds: exploring exactly one interleaving per equivalence class with linear memory consumption. Our algorithm, TruSt, formalized in Coq, is applicable not only to sequential consistency, but also to any weak memory model that satisfies a few basic assumptions, including TSO, PSO, and RC11. In addition, TruSt is embarrassingly parallelizable: its different exploration options have no shared state, and can therefore be explored completely in parallel. Consequently, TruSt outperforms the state-of-the-art in terms of memory and/or time.

A Graph Theory of Rook Placements

Two boards are rook equivalent if they have the same number of non-attacking rook placements for any number of rooks. Define a rook equivalence graph on an equivalence class of Ferrers boards by specifying that two boards are connected by an edge if you can obtain one of the boards by moving squares in the other board out of one column and into a single other column. Given such a graph, we characterize which boards will yield connected graphs. We also provide some cases where common graphs will or will not be the graph for some set of rook equivalent Ferrers boards. Finally, we extend this graph definition to the m-level rook placement generalization developed by Briggs and Remmel. This yields a graph on the set of rook equivalent singleton boards, and we characterize which singleton boards give rise to a connected graph.

An Automorphic Distance Metric and Its Application to Node Embedding for Role Mining

Role is a fundamental concept in the analysis of the behavior and function of interacting entities in complex networks. Role discovery is the task of uncovering the hidden roles of nodes within a network. Node roles are commonly defined in terms of equivalence classes. Two nodes have the same role if they fall within the same equivalence class. Automorphic equivalence, where two nodes are equivalent when they can swap their labels to form an isomorphic graph, captures this notion of role. The binary concept of equivalence is too restrictive, and nodes in real-world networks rarely belong to the same equivalence class. Instead, a relaxed definition in terms of similarity or distance is commonly used to compute the degree to which two nodes are equivalent. In this paper, we propose a novel distance metric called automorphic distance, which measures how far two nodes are from being automorphically equivalent. We also study its application to node embedding, showing how our metric can be used to generate role-preserving vector representations of nodes. Our experiments confirm that the proposed automorphic distance metric outperforms a state-of-the-art automorphic equivalence-based metric and different state-of-the-art techniques for the generation of node embeddings in different role-related tasks.

ON SOME EQUIVALENCE RELATION ON NON-ABELIAN $\CA$-GROUPS

A non-abelian group $G$ is called a $\CA$-group ($\CC$-group) if $C_G(x)$ is abelian(cyclic) for all $x\in G\setminus Z(G)$. We say $x\sim y$ if and only if $C_G(x)=C_G(y)$.We denote the equivalence class including $x$ by$[x]_{\sim}$. In this paper, we prove thatif $G$ is a $\CA$-group and $[x]_{\sim}=xZ(G)$, for all $x\in G$, then $2^{r-1}\leq|G'|\leq 2^{r\choose 2}$.where $\frac {|G|}{|Z(G)|}=2^{r}, 2\leq r$ and characterize all groups whose $[x]_{\sim}=xZ(G)$for all $x\in G$ and $|G|\leq 100$. Also, we will show that if $G$ is a $\CC$-group and $[x]_{\sim}=xZ(G)$,for all $x \in G$, then $G\cong C_m\times Q_8$ where $C_m$ is a cyclic group of odd order $m$ andif $G$ is a $\CC$-group and $[x]_{\sim}=x^G$, for all $x\in G\setminus Z(G)$, then $G\cong Q_8$.

Integrable triples in semisimple Lie algebras

We classify all integrable triples in simple Lie algebras, up to equivalence. The importance of this problem stems from the fact that for each such equivalence class one can construct the corresponding integrable hierarchy of bi-Hamiltonian PDE. The simplest integrable triple (f, 0, e) in $${\mathfrak {sl}}_2$$ sl 2 corresponds to the KdV hierarchy, and the triple $$(f,0,e_\theta )$$ ( f , 0 , e θ ) , where f is the sum of negative simple root vectors and $$e_\theta $$ e θ is the highest root vector of a simple Lie algebra, corresponds to the Drinfeld–Sokolov hierarchy.

On the Epistemic Logic of Incomplete Argumentation Frameworks

We study the relation between two existing formalisms: incomplete argumentation frameworks (IAFs) and epistemic logic of visibility (ELV). We show that the set of completions of a given IAF naturally corresponds to a specific equivalence class of possible worlds within the model of visibility. This connection is further strengthened in two directions. First, we show how to reduce argument acceptance problems of IAFs to ELV model-checking problems. Second, we highlight the epistemic assumptions that underlie IAFs by providing a minimal epistemic logic for IAFs.

Polynomial-time Classification of Skew-symmetrizable Matrices with a Positive Definite Quasi-Cartan Companion

Skew-symmetrizable matrices play an essential role in the classification of cluster algebras. We prove that the problem of assigning a positive definite quasi-Cartan companion to a skew-symmetrizable matrix is in polynomial class P. We also present an algorithm to determine the finite type Δ ∈ {𝔸n; 𝔻n; 𝔹n; ℂn; 𝔼6; 𝔼7; 𝔼8; 𝔽4; 𝔾2} of a cluster algebra associated to the mutation-equivalence class of a connected skew-symmetrizable matrix B, if it has one.

Analisis Rekomendasi Produk Menggunakan Algoritma ECLAT Berdasarkan Riwayat Data Penjualan PT XYZ

PT XYZ merupakan perusahaan yang menyediakan sarana produksi peternakan. Tranksaksi penjualan dicatat sebagai arsip perusahaan, laporan penjualan, dan laporan laba rugi. Lebih dari 1.500 lembar faktur tercetak setiap bulan. Namun, dalam hal promosi produk, belum menggunakan hasil analisis dari riwayat transaksi penjualan. Penelitian ini bertujuan untuk memberikan rekomendasi produk menggunakan algoritma ECLAT. Algoritma ECLAT (Equivalence Class Transformation) menggunakan konsep pencarian depth-first untuk menemukan itemset yang sering muncul dalam transaksi. Langkah penelitian, yaitu wawancara untuk akuisisi data, pra-pemrosesan data, transformasi data, dan proses data mining dengan algoritma ECLAT untuk menemukan frequent itemset dan menggunakan hasil frequent itemset sebagai basis pembuatan pola aturan asosiasi. Hasil analisis menujukkan bahwa sistem dapat memberikan rekomendasi aturan asosiasi secara efektif dari 14.617 riwayat transaksi. Minimum support tertinggi yang dapat digunakan untuk menemukan kombinasi k-itemset adalah 1%. Hasil aturan asosiasi pertahun dari riwayat transaksi tahun 2018-2020 menunjukkan perbedaan hasil dengan ragam terbanyak terjadi tahun 2020, yaitu 5 aturan asosiasi. Setiap aturan asosiasi yang muncul memiliki nilai confidence yang kuat yakni di atas 50%.

Effects of Limited Hold on Equivalence Class Formation

In an attempt to limit the opportunity to engage in mediating behavior, two groups of adult participants received preliminary training in identity matching with limited hold levels (LH) for responding of 0.7 s for the sample and 1.2 s for the comparisons. The two groups were subsequently trained to form three 5-member classes, using the same LH levels, where the A, B, D, and E stimuli were abstract stimuli, and the C stimuli were meaningful pictures. In two tests for emergent relations, the LH for Group Short was unchanged, whereas 5 s were added to the LH for the comparisons for Group Long. None of the participants in Group Short responded in accordance with stimulus equivalence in either of the two tests. In Group Long, one participant responded in accordance with stimulus equivalence in the first test, and an additional eight participants formed equivalence classes in the second test.

Ordered Structures of Polynomials over Max-Plus Algebra

The ordered structures of polynomial idempotent algebras over max-plus algebra are investigated in this paper. Based on the antisymmetry, the partial orders on the sets of formal polynomials and polynomial functions are introduced to generate two partially ordered idempotent algebras (POIAs). Based on the symmetry, the quotient POIA of formal polynomials is then obtained. The order structure relationships among these three POIAs are described: the POIA of polynomial functions and the POIA of formal polynomials are orderly homomorphic; the POIA of polynomial functions and the quotient POIA of formal polynomials are orderly isomorphic. By using the partial order on formal polynomials, an algebraic method is provided to determine the upper and lower bounds of an equivalence class in the quotient POIA of formal polynomials. The criterion for a formal polynomial to be the minimal element of an equivalence class is derived. Furthermore, it is proven that any equivalence class is either an uncountable set with cardinality of the continuum or a finite set with a single element.