On properties not inherited by monoids from their Schützenberger groups

We give an example of a monoid with finitely many left and right ideals, all of whose Schützenberger groups are presentable by finite complete rewriting systems, and so each have finite derivation type, but such that the monoid itself does not have finite derivation type, and therefore does not admit a presentation by a finite complete rewriting system. The example also serves as a counterexample to several other natural questions regarding complete rewriting systems and finite derivation type. Specifically it allows us to construct two finitely generated monoids M and N with isometric Cayley graphs, where N has finite derivation type (respectively, admits a presentation by a finite complete rewriting system) but M does not. This contrasts with the case of finitely generated groups for which finite derivation type is known to be a quasi-isometry invariant. The same example is also used to show that neither of these two properties is preserved under finite Green index extensions.     

An extension of the Lyndon-Schützenberger result to pseudoperiodic words

One of the particularities of information encoded as DNA strands is that a string u contains basically the same information as its Watson-Crick complement, denoted here as θ(u). Thus, any expression consisting of repetitions of u and θ(u) can be considered in some sense periodic. In this paper, we give a generalization of Lyndon and Schützenberger’s classical result about equations of the form u^l=vⁿw^m, to cases where both sides involve repetitions of words as well as their complements. Our main results show that, for such extended equations, if l?5,n,m?3, then all three words involved can be expressed in terms of a common word t and its complement θ(t). Moreover, if l?5, then n=m=3 is an optimal bound. These results are established based on a complete characterization of all possible overlaps between two expressions that involve only some word u and its complement θ(u), which is also obtained in this paper.     

Poitín: Distilling Theorems From Conjectures

In this paper, we describe a new fully automatic theorem prover called Poitín which makes use of a novel transformation algorithm called distillation to prove input conjectures. The input conjectures are defined in a functional language and are transformed using the distillation algorithm. The result of this transformation can be easily inspected to see whether the original conjecture is true. Possible divergence of the transformation algorithm is detected, and this information is used to perform generalizations to ensure termination. We give several examples of the application of the theorem prover, and compare it to related work.     

关于煎饼网络及层次环煎饼网络的几个猜想

煎饼网络是由互连网络的群论模型设计出来的一类典型的超级计算机互连网络。关于煎饼网络师海忠提出了一个猜想-猜想1,但煎饼网络有一个弱点即结点度随着规模的增大而迅速增大,为了改进这一缺点师海忠提出了互连网络的层次环群论模型。在这篇文章中,首先,汪生龙给出了煎饼网络当n=5时的两种圈分解,其次师海忠提出了关于该网络的一个猜想-猜想2,当Cayley图层次环网络中的Cayley图取煎饼网络时得到煎饼层次环网络的猜想-猜想2/,进而汪生龙证明了猜想2/在低维度情形下是正确的     

Conjectures on Liapunov qualitative game

In this paper the stability of a class of linear and non-linear stochastic transformations arising from control system analysis is considered. Explicit criteria for stability are derived. An algorithm for computing the moments of linear stochastic transformations of a specific type is presented.     

Un theoreme de Chomsky-Schützenberger pour les forets algebriques

Résumé On démontre un théorème de Chomsky-Schützenberger dans le cas des forêts algébriques. On définit les forêts de semi-Dyck et les forêts de semi-Dyck simples et on montre qu“une forêt est algébrique si et seulement si elle est l“image par un homomorphisme linéaire de l“intersection d“une forêt de semi-Dyck [ou de semi-Dyck simple] et d“une forêt locale [ou reconnaissable] ou équivalemment si elle est la transductée linéaire complète ascendante ou descendante d“une forêt de semi-Dyck ou de semi-Dyck simple.     

A generalization of the Schützenberger product of finite monoids

For each n?1, an n-ary product ? on finite monoids is constructed. This product has the following property: Let Σ be a finite alphabet and Σ¹ the free monoid generated by Σ. For i = 1, …,n, let A_i be a recognizable subset of Σ¹, M(A_i) the syntactic monoid of A_n and M(A₁?A_n) the syntactic monoid of the concatenation product A₁?A_n. Then M(A₁?A_n)< ? (M(A₁),…,M(A_n)). The case n = 2 was studied by Schützenberger. As an application of the generalized product, I prove the theorem of Brzozowski and Knast that the dot-depth hierarchy of star-free sets is infinite.     

关于修正冒泡排序网络的一簇猜想

修正冒泡排序网络是互连网络设计中的一个重要的Cayley图模型,关于修正冒泡排序网络的一簇猜想如下:对于任意的自然数n≥3,修正冒泡排序网络Yn是i个边不交的哈密尔顿圈以及n-2i个完美对集的并,其中1≤i≤︱n/2︱。证明了当i=1,2时,这个猜想是正确的。     

Attacking Group Protocols by Refuting Incorrect Inductive Conjectures

Automated tools for finding attacks on flawed security protocols often fail to deal adequately with group protocols. The reason is that the abstractions made to improve performance on fixed two- or three-party protocols either preclude the modeling of group protocols altogether or permit modeling only in a fixed scenario, which can prevent attacks from being discovered. This paper describes Coral, a tool for finding counterexamples to incorrect inductive conjectures, which we have used to model protocols for both group key agreement and group key management, without any restrictions on the scenario. We show how we used Coral to discover six previously unknown attacks on three group protocols.     

关于互连网络群论模型的一簇猜想

连通图生成的Cayley图是作为互连网络的群论模型提出来的概念。猜想:设G=(V,E)是具有顶点集{1,2,…,n}(n>2)和m条边的连通图。如果m=2r,则由G生成的Cayley图是边不交的k(0≤k≤r)个Hamilton图和m－2k个完美对集的并;如果m=2r+1,则由G生成的Cayley图是边不交的k(0≤k≤r)个Hamilton图和m－2k个完美对集的并。特别地,对于k=r和星网络,这个猜想的特殊情形是1998年由师海忠提出来的。     

Semantic Generalizations for Proving and Disproving Conjectures by Analogy

Taking an extension of resolution as a base calculus (though the same principles are applicable to other calculi) for searching proofs (refutations) and counterexamples (models), we introduce a new method able to find refutations and also models by analogy with refutations and models in a knowledge base. The source objects for the analogy process are generalizations of the refutations (models). They are included in the knowledge base, and then higher-order matching techniques for the choice of the relevant source objects as well as the building of a new proof or a model by analogy are used. Some comparisons with existing methods as well as two detailed running examples on generalization show evidence of the interest of our approach.     

半图的计数

顶点标记半图是用n个不同符号标记的半图G.通过分析得出只包含不相邻边的顶点标记半图的计数和包含两个相邻S边的顶点标记半图的数目的计算的多种结果.同时也计算了包含1到8个顶点的顶点标记半图的数目.     

Predicate Synthesis for Correcting Faulty Conjectures: The Proof Planning Paradigm

The synthesis of programs as well as other synthetic tasks often end up with an unprovable, partially false conjecture. A successful subsequent synthesis attempt depends on determining why the conjecture is faulty and how it can be corrected. Hence, it is highly desirable to have an automated means for detecting and correcting faulty conjectures.We introduce a method for patching faulty conjectures. The method is based on abduction and performs its task during an attempt to prove a given conjecture. On input X. G(X), the method builds a definition for a corrective predicate, P(X), such that X. P(X) G(X) is a theorem. The synthesis of a corrective predicate is guided by the constructive principle of formulae as types, relating inference with computation.We take the construction of a corrective predicate as a program transformation task. The method consists of a collection of construction commands. A construction command is a small program that makes use of one or more program editing commands, geared towards building recursive, equational procedures.A synthesised corrective predicate is guaranteed to be correct, turning a faulty conjecture into a theorem. Whether conditional or not, it will be well-defined. If recursive, it will also be terminating.Our method is amenable to mechanisation, but careful search guidance is required for making a productive use of the failure of a proof. A failed proof attempt quickly yields a huge, possibly infinite, deduction tree, giving rise to exponentially many abductive explanations. We suggest that a proof planning approach can structure the task of correcting a formula in such a way as to allow significant automation, while dramatically restricting the search space.     

Combining SAT Solvers with Computer Algebra Systems to Verify Combinatorial Conjectures

We present a method and an associated system, called MathCheck, that embeds the functionality of a computer algebra system (CAS) within the inner loop of a conflict-driven clause-learning SAT solver. SAT+CAS systems, à la MathCheck, can be used as an assistant by mathematicians to either find counterexamples or finitely verify open universal conjectures on any mathematical topic (e.g., graph and number theory, algebra, geometry, etc.) supported by the underlying CAS. Such a SAT+CAS system combines the efficient search routines of modern SAT solvers, with the expressive power of CAS, thus complementing both. The key insight behind the power of the SAT+CAS combination is that the CAS system can help cut down the search-space of the SAT solver, by providing learned clauses that encode theory-specific lemmas, as it searches for a counterexample to the input conjecture (just like the T in DPLL (T)). In addition, the combination enables a more efficient encoding of problems than a pure Boolean representation. In this paper, we leverage the capabilities of several different CAS, namely the SAGE, Maple, and Magma systems. As case studies, we study three long-standing open mathematical conjectures, two from graph theory regarding properties of hypercubes, and one from combinatorics about Hadamard matrices. The first conjecture states that any matching of any d-dimensional hypercube can be extended to a Hamiltonian cycle; the second states that given an edge-antipodal coloring of a hypercube there always exists a monochromatic path between two antipodal vertices; the third states that Hadamard matrices exist for all orders divisible by 4. Previous results on the graph theory conjectures have shown the conjectures true up to certain low-dimensional hypercubes, and attempts to extend them have failed until now. Using our SAT+CAS system, MathCheck, we extend these two conjectures to higher-dimensional hypercubes. Regarding Hadamard matrices, we demonstrate the advantages of SAT+CAS by constructing Williamson matrices up to order 42 (equivalently, Hadamard up to order \(4\times 42=168\)), improving the bounds up to which Williamson matrices of even order have been constructed. Prior state-of-the-art construction was only feasibly performed for odd numbers, where possible.     

关于企业应用集成EAI若干核心技术

随着各种应用软件的迅速增加以及向电子商务转变,越来越多的企业开始采用EAI解决方案将企业内部的应用软件与外部客户和供应商的应用软件进行集成。本文介绍了围绕着EAI的几项核心技术,并对以后的EAI发展趋势作了一些探讨。