Minimal Counterexamples Research Articles

The structure of digraphs with excess one

AbstractA digraph is ‐geodetic if for any (not necessarily distinct) vertices there is at most one directed walk from to with length not exceeding . The order of a ‐geodetic digraph with minimum out‐degree is bounded below by the directed Moore bound . The Moore bound can be met only in the trivial cases and , so it is of interest to look for ‐geodetic digraphs with out‐degree and smallest possible order , where is the excess of the digraph. Miller, Miret and Sillasen recently ruled out the existence of digraphs with excess one for and and for and . We conjecture that there are no digraphs with excess one for and in this paper we investigate the structure of minimal counterexamples to this conjecture. We severely constrain the possible structures of the outlier function and prove the nonexistence of certain digraphs with degree three and excess one, as well closing the open cases and left by the analysis of Miller et al. We further show that there are no involutary digraphs with excess one, that is, the outlier function of any such digraph must contain a cycle of length .

On minimal flows and definable amenability in some distal NIP theories

We study the definable topological dynamics (G(M),SG(M)) of a definable group acting on its type space, where M is either an o-minimal structure or a p-adically closed field, and G a definably amenable group. We focus on the problem raised in [13] of whether weakly generic types coincide with almost periodic types, showing that the answer is positive when G has boundedly many global weakly generic types. We also give two “minimal counterexamples” where G has unboundedly many global weakly generic types, extending the main results in [21] to a more general context.

On the union-closed sets conjecture

We give a new proof of a theorem of Lo Faro concerning minimal counterexamples to the union-closed conjecture, and we prove some new results concerning the “ε-version” of the union-closed conjecture, and concerning the minimal maximal frequency in union-closed families.

Semigenerated Carnot algebras and applications to sub-Riemannian perimeter

This paper contributes to the study of sets of finite intrinsic perimeter in Carnot groups. Our intent is to characterize in which groups the only sets with constant intrinsic normal are the vertical half-spaces. Our viewpoint is algebraic: such a phenomenon happens if and only if the semigroup generated by each horizontal half-space is a vertical half-space. We call semigenerated those Carnot groups with this property. For Carnot groups of nilpotency step 3 we provide a complete characterization of semigeneration in terms of whether such groups do not have any Engel-type quotients. Engel-type groups, which are introduced here, are the minimal (in terms of quotients) counterexamples. In addition, we give some sufficient criteria for semigeneration of Carnot groups of arbitrary step. For doing this, we define a new class of Carnot groups, which we call type (Diamond ) and which generalizes the previous notion of type (star ) defined by M. Marchi. As an application, we get that in type (Diamond ) groups and in step 3 groups that do not have any Engel-type algebra as a quotient, one achieves a strong rectifiability result for sets of finite perimeter in the sense of Franchi, Serapioni, and Serra-Cassano.

Approximation ratio of LD algorithm for multi-processor scheduling and the Coffman–Sethi conjecture

Coffman and Sethi proposed a heuristic algorithm, called LD (Longest Decreasing), for multi-processor scheduling, to minimize makespan over flowtime-optimal schedules. The LD algorithm is an extension of a very well-known list scheduling algorithm, Longest Processing Time (LPT) list scheduling, to this bicriteria scheduling problem. Coffman and Sethi conjectured (in 1976) that the LD algorithm has the following precise worst-case performance bound: 54−34(4m−1), where m is the number of machines. In this paper, utilizing some recent work by the authors and Huang (2016), which exposed some very strong combinatorial properties of various presumed minimal counterexamples to the conjecture, we provide a proof of this conjecture. The problem and the LD algorithm have connections to some other fundamental problems (such as the assembly line-balancing problem) and algorithms.

PBW Degenerate Schubert Varieties: Cartan Components and Counterexamples

In recent years PBW degenerations of Demazure modules and Schubert varieties were defined and studied in several papers. Various interesting properties (such as these PBW degenerations embedding naturally into the corresponding degenerate representations and flag varieties) were obtained in type A but only with restrictions on the Weyl group element or the highest weight. We show that these properties cannot hold in full generality due to the following issue with the definition. The degenerate variety depends on the highest weight used to define it and not only on its Weyl group stabilizer (as is the case for PBW degenerate flag varieties as well as classical Schubert varieties). Perhaps surprisingly, the minimal counterexamples appear only for \(\mathfrak {sl}_{6}\). The counterexamples are constructed with the help of a study of the Cartan components appearing in this context.

On a minimal counterexample to Brauer’s k(B)-conjecture

We study Brauer’s long-standing k(B)-conjecture on the number of characters in p-blocks for finite quasi-simple groups and show that their blocks do not occur as a minimal counterexample for p ≥ 5 nor in the case of abelian defect. For p = 3 we obtain that the principal 3-blocks do not provide minimal counterexamples. We also determine the precise number of irreducible characters in unipotent blocks of classical groups for odd primes.

Derivative-Based Diagnosis of Regular Expression Ambiguity

Regular expressions are often ambiguous. We present a novel method based on Brzozowski’s derivatives to aid the user in diagnosing ambiguous regular expressions. We introduce a derivative-based finite state transducer to generate parse trees and minimal counter-examples. The transducer can be easily customized to either follow the POSIX or Greedy disambiguation policy and based on a finite set of examples it is possible to examine if there are any differences between POSIX and Greedy.

Worst-case performance analysis of some approximation algorithms for minimizing makespan and flowtime

In 1976, Coffman and Sethi conjectured that a natural extension of LPT list scheduling to the bicriteria scheduling problem of minimizing makespan over flowtime-optimal schedules, called the LD algorithm, has a simple worst-case performance bound: $$\frac{5m-2}{4m-1}$$5m-24m-1, where m is the number of machines. We study the structure of potential minimal counterexamples to this conjecture, provide some new tools and techniques for the analysis of such algorithms, and prove that to verify the conjecture, it suffices to analyze the following case: for every $$m \ge 4$$mź4, $$n \in \{4m, 5m\}$$nź{4m,5m}, where n is the number of jobs.

On stable equivalences with endopermutation source

We show that a bimodule between block algebras which has a fusion stable endopermutation module as a source and which induces Morita equivalences between centralisers of nontrivial subgroups of a defect group induces a stable equivalence of Morita type; this is the converse to a theorem of Puig. The special case where the source is trivial has long been known by many authors. The earliest instance of a result deducing a stable equivalence of Morita type from local Morita equivalences with possibly nontrivial endopermutation source is due to Puig, in the context of blocks with abelian defect groups with a Frobenius inertial quotient. The present note is motivated by an application, due to Biland, to blocks of finite groups with structural properties known to hold for hypothetical minimal counterexamples to the Zp⁎-Theorem.

Are There Good Mistakes? A Theoretical Analysis of CEGIS

Counterexample-guided inductive synthesis CEGIS is used to synthesize programs from a candidate space of programs. The technique is guaranteed to terminate and synthesize the correct program if the space of candidate programs is finite. But the technique may or may not terminate with the correct program if the candidate space of programs is infinite. In this paper, we perform a theoretical analysis of counterexample-guided inductive synthesis technique. We investigate whether the set of candidate spaces for which the correct program can be synthesized using CEGIS depends on the counterexamples used in inductive synthesis, that is, whether there are good mistakes which would increase the synthesis power. We investigate whether the use of minimal counterexamples instead of arbitrary counterexamples expands the set of candidate spaces of programs for which inductive synthesis can successfully synthesize a correct program. We consider two kinds of counterexamples: minimal counterexamples and history bounded counterexamples. The history bounded counterexample used in any iteration of CEGIS is bounded by the examples used in previous iterations of inductive synthesis. We examine the relative change in power of inductive synthesis in both cases. We show that the synthesis technique using minimal counterexamples MinCEGIS has the same synthesis power as CEGIS but the synthesis technique using history bounded counterexamples HCEGIS has different power than that of CEGIS, but none dominates the other.

Minimal counterexamples for linear-time probabilistic verification

Counterexamples for property violations have a number of important applications like supporting the debugging of erroneous systems and verifying large systems via counterexample-guided abstraction refinement. In this paper, we propose the usage of minimal critical subsystems of discrete-time Markov chains and Markov decision processes as counterexamples for violated ω-regular properties. Minimality can thereby be defined in terms of the number of states or transitions. This problem is known to be NP-complete for Markov decision processes. We show how to compute such subsystems using mixed integer linear programming and evaluate the practical applicability in a number of experiments. They show that our method yields substantially smaller counterexample than using existing techniques.

The 1,2,3-Conjecture and 1,2-Conjecture for sparse graphs

The 1, 2, 3-Conjecture states that the edges of a graph without isolated edges can be labeled from {1, 2, 3} so that the sums of labels at adjacent vertices are distinct. The 1, 2-Conjecture states that if vertices also receive labels and the vertex label is added to the sum of its incident edges, then adjacent vertices can be distinguished using only {1, 2}. We show that various configurations cannot occur in minimal counterexamples to these conjectures. Discharging then confirms the conjectures for graphs with maximum average degree less than 8/3. The conjectures are already confirmed for larger families, but the structure theorems and reducibility results are of independent interest.

Robustness Analysis of Embedded Control Systems with Respect to Signal Perturbations: Finding Minimal Counterexamples Using Fault Injection

Fault-tolerance of embedded control systems is of great importance, given their wide usage in various domains such as aeronautics, automotive, medical, and so on. Signal perturbations such as small amounts of noise, shift, and spikes, can sometimes severely hamper the performance of the system, apart from complete failure of components and links. Finding minimal counterexamples (perturbations on the system leading to violation of fault-tolerance requirements) can be of great assistance to control system designers in understanding and adjusting the fault-tolerance behavior of the system. Fault injection is an effective method for dependability analysis of such systems. In this paper, we introduce the concept of dominating sets of perturbations, and define a minimal set of counterexamples called the basis. We propose effective methods using a simulation-based fault injection technique on Simulink models for finding the basis set at an early stage of design, given the fault specification and fault-tolerance requirements. Experimental results on two different control system examples from the Simulink automotive library demonstrate the efficacy of the proposed framework.

Connectivities for k-knitted graphs and for minimal counterexamples to Hadwigerʼs Conjecture

For a given subset S⊆V(G) of a graph G, the pair (G,S) is knitted if for every partition of S into non-empty subsets S1,S2,…,St, there are disjoint connected subgraphs C1,C2,…,Ct in G so that Si⊆Ci. A graph G is ℓ-knitted if (G,S) is knitted for all S⊆V(G) with |S|=ℓ. In this paper, we prove that every 9ℓ-connected graph is ℓ-knitted.Hadwigerʼs Conjecture states that every k-chromatic graph contains a Kk-minor. We use the above result to prove that the connectivity of minimal counterexamples to Hadwigerʼs Conjecture is at least k/9, which was proved to be at least 2k/27 in Kawarabayashi (2007) [4].

Metabelian groups that admit triality

Abstract.A problem proposed by Grishkov asks if every metabelian triality group has a corresponding Moufang loop which is a group. Here we show that if minimal counter-examples exist, then such triality groups have to be

Unfaithful minimal Heilbronn characters of finite groups

We obtain necessary and sufficient conditions for a finite group G to possess an “unfaithful minimal Heilbronn character”—a virtual character but not a character of G whose inner product with every monomial character is nonnegative, whose restriction to every proper subgroup and quotient is a character, and whose restriction to some proper subgroup is unfaithful. We give an application constraining hypothetical minimal counterexamples to Artin's Conjecture on the holomorphy of L-series.

Uniqueness cases in odd-type groups of finite Morley rank

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. One of the major theorems in the area is Borovik's trichotomy theorem. The ‘trichotomy’ here is a case division of the generic minimal counterexamples within odd type, that is, groups with a large and divisible Sylow° 2-subgroup. The so-called ‘uniqueness case’ in the trichotomy theorem is the existence of a proper 2-generated core. It is our aim to drive the presence of a proper 2-generated core to a contradiction, and hence bind the complexity of the Sylow° 2-subgroup of a minimal counterexample to the Cherlin–Zilber conjecture. This paper shows that the group in question is a minimal connected simple group and has a strongly embedded subgroup, a far stronger uniqueness case. As a corollary, a tame counterexample to the Cherlin–Zilber conjecture has Prüfer rank at most two.

A game-based framework for CTL counterexamples and 3-valued abstraction-refinement

This work exploits and extends the game-based framework of CTL model checking for counterexample and incremental abstraction-refinement. We define a game-based CTL model checking for abstract models over the 3-valued semantics, which can be used for verification as well as refutation. The model checking process of an abstract model may end with an indefinite result, in which case we suggest a new notion of refinement, which eliminates indefinite results of the model checking. This provides an iterative abstraction-refinement framework. This framework is enhanced by an incremental algorithm, where refinement is applied only where indefinite results exist and definite results from prior iterations are used within the model checking algorithm. We also define the notion of annotated counterexamples , which are sufficient and minimal counterexamples for full CTL. We present an algorithm that uses the game board of the model checking game to derive an annotated counterexample in case the examined system model refutes the checked formula.

On the connectivity of minimum and minimal counterexamples to Hadwiger's Conjecture

The main result of this paper is the following: Any minimal counterexample to Hadwiger's Conjecture for the k-chromatic case is ⌈ 2 k 27 ⌉ -connected. This improves the previous known bound due to Mader [W. Mader, Über trennende Eckenmengen in homomorphiekritischen Graphen, Math. Ann. 175 (1968) 243–252], which says that any minimal counterexample to Hadwiger's Conjecture for the k-chromatic case is 7-connected for k ⩾ 7 . This is the first result on the vertex connectivity of minimal counterexamples to Hadwiger's Conjecture for general k. Consider the following problem: There exists a constant c such that any ck-chromatic graph has a K k -minor. This problem is still open, but together with the recent result in [T. Böhme, K. Kawarabayashi, J. Maharry, B. Mohar, Linear connectivity forces large complete bipartite graph minors, preprint], our main result implies that there are only finitely many minimal counterexamples to the above problem when c ⩾ 27 . This would be the first step to attach the above problem. We also prove that the vertex connectivity of minimum counterexamples to Hadwiger's Conjecture is at least ⌈ k 3 ⌉ -connected. This is also the first result on the vertex connectivity of minimum counterexamples to Hadwiger's Conjecture for general k.