Minimal Counterexamples Research Articles

Using heuristic search for finding deadlocks in concurrent systems

Model checking is a formal technique for proving the correctness of a system with respect to a desired behavior. This is accomplished by checking whether a structure representing the system (typically a labeled transition system) satisfies a temporal logic formula describing the expected behavior. Model checking has a number of advantages over traditional approaches that are based on simulation and testing: it is completely automatic and when the verification fails it returns a counterexample that can be used to pinpoint the source of the error. Nevertheless, model checking techniques often fail because of the state explosion problem: transition systems grow exponentially with the number of components. The aim of this paper is to attack the state explosion problem that may arise when looking for deadlocks in concurrent systems described through the calculus of communicating systems. We propose to use heuristics-based techniques, namely the A* algorithm, both to guide the search without constructing the complete transition system, and to provide minimal counterexamples. We have realized a prototype tool to evaluate the methodology. Experiments we have conducted on processes of different size show the benefit from using our technique against building the whole state space, or applying some other methods.

Advances in packing directed joins

Several of the finest unclaimed prizes in directed graph theory involve the packing of directed joins. One difficulty in claiming these prizes is that the broad conjecture posed by Edmonds and Giles, whether the maximum number of disjoint directed joins equals the smallest weight of a directed cut in every weighted directed graph, is not true in general. This is despite the fact that the conjecture is true in several special cases, and is also true if the roles of directed joins and directed cuts are reversed. Another difficulty is that the known minimal counterexamples, one found by Schrijver and two found by Cornuejols and Guenin during a computer search, are mysterious in nature. We dispel some of this mystery by providing a framework for understanding the known counterexamples. We then use this framework to construct several new counterexamples, and to prove that every “smallest” minimal counterexample has now been found. Finally, we temper these advances by introducing an NP-completeness result for a more difficult problem.

Variations on a Theme of Desmond MacHale

This article is based on a talk that the author gave at Groups in Galway 2005. We discuss four conjectures, known to be false, and give a progress report on finding min- imal counterexamples for them. These conjectures are varia- tions on the 47 (false) conjectures Desmond MacHale gave in his 1981 paper Minimum Counterexamples in Group The- ory. My variations are on an Irish tune, the theme is in Desmond MacHale's 1981 paper Minimum Counterexamples in Group The- ory (7). In this paper, 47 conjectures are stated, all known to be false, and MacHale asks for minimal counterexamples. Conjecture 7 in (7) reads as follows. Conjecture 7. In any group G, the set of commutators forms a subgroup.

A signalizer functor theorem for 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. Towards this end, the development of the theory of groups of finite Morley rank has achieved a good theory of Sylow 2-subgroups. It is now common practice to divide the Cherlin–Zilber conjecture into different cases depending on the nature of the connected component of the Sylow 2-subgroup, known as the Sylow◦ 2-subgroup. We shall be working with groups whose Sylow◦ 2-subgroup is divisible, or odd type groups. To date, the main theorem in the area of odd type groups is Borovik’s trichotomy theorem. The “trichotomy” here is a case division of the minimal counterexamples within odd type. More technically, Borovik’s result represents a major success at transferring signalizer functors and their applications from finite group theory to the finite Morley rank setting. The major difference between the two settings is the absence of a solvable signalizer functor theorem. This forced Borovik to work only with nilpotent signalizer functors, and the trichotomy theorem ends up depending on the assumption of tameness to assure that the necessary signalizer functors are nilpotent. The present paper shows that one may obtain a connected nilpotent signalizer functor from any sufficiently non-trivial solvable signalizer functor. This result plugs seamlessly into Borovik’s work to eliminate the assumption of tameness from his trichotomy theorem. In the meantime, a new approach to the trichotomy theorem has been developed by Borovik [7], based on the “generic identification theorem” of Berkman and Borovik [5]. Borovik uses his original signalizer functor arguments, and incorporates the result of the present paper. The paper is organized as follows. The first section will develop a limited characteristic zero notion of unipotence to complement the usual p-unipotence theory. The section on centralizers and generation which follows will establish some background needed in the rest of the paper. In Section 4 we prove our main result on signalizer functors, and in Section 5 we discuss some applications. With Borovik’s kind permission, we include a proof of the nilpotent signalizer functor theorem as an appendix. The results of Section 3 are based in part on a section of an unpublished version of [3].

Elementary Bipartite Graphs and Unique Colourability

The Clique-Pair-Conjecture (CPC) states that a uniquely colourable perfect graph, different from a clique, contains two maximum size cliques having a two element symmetric difference. One can make an auxiliary graphB from a minimal counterexample for the CPC (if any exists), this B is bipartite. We prove thatB is elementary. Furthermore, we give a characterization of the minimal counterexamples, in terms of the constructed auxiliary graphs.

Minimal counterexamples to a conjecture of Hall and Paige

A complete map for a group G is a permutation \( \varphi\colon G\to G \) such that \( g\mapsto g\varphi(g) \) is still a permutation of G. A conjecture of M. Hall and L. J. Paige states that every finite group whose Sylow 2-subgroup is non-trivial and non-cyclic admits a complete map. In the present paper it is proved that a potential counterexample G of minimal order to this conjecture either is almost simple or G has only one involution, the Sylow 2-subgroups of G are quaternionic, \( |G/G'|\leqq 2$, $G'\cong \ SL(2,q) \) for some odd prime power \( q>5 \) and if G is not a perfect group then \( G/Z(G')\cong \rm{PGL}(2,\it{q}) \).

Sylow 2-subgroups of galois groups arising as minimal counterexamples to artin's conjecture

Let E/F be a Galois extension of number fields with Galois group G. The purpose of this paper is to place limitations on the structure of a Sylow 2-subgroup of G in the case when the extension E/F is a minimal counterexample to Artin’s Conjecture on the holomorphy of L-series. More specifically, assume for some s0 ∈ C − {1} and some irreducible character χ of G that the Artin L-series L(s, χ,E/F ) has a pole at s0. Assume further that this is a minimal configuration in the sense that: for no intermediate Galois extension E1/F1 of smaller degree, where F ⊆ F1 ⊆ E1 ⊆ E, and no irreducible character ψ of Gal(E1/F1) does L(s, ψ,E1/F1) have a pole at s0. Then we show that a Sylow 2-subgroup of G has a faithful complex representation of degree r, where r is the order of zero of the zeta function of E at s0. In particular, this implies that the 2-rank of G is at most r.

The topological Vaught's conjecture and minimal counterexamples

Let G be a Polish topological group, let X be a Polish space, let J: G × X → X be a Borel-measurable action of G on X, and let A ⊂ X be a Borel set which is invariant with respect to J, i.e., a Borel set of orbits. The following statement, or various equivalent versions of it, is known as the Topological Vaught's Conjecture.Let (G, X, J, A) be as above. Either A contains only countably many orbits, or else, A contains perfectly many orbits.We say that A contains perfectly many orbits if there is a perfect set P ⊂ A such that no two elements of P are in the same orbit. (Assuming ¬CH, A contains perfectly many orbits iff it contains 2ℵ0 orbits.) The Topological Vaught's Conjecture implies the usual, model theoretic, Vaught's Conjecture for Lω1ω, since the isomorphism classes are the orbits of an action of the group of permutations of ω; we give details in §0. The “Borel” assumption cannot be weakened for either A or J.Given a Borel-measurable Polish action (G,X,J) and an invariant Borel set B ⊂ X, we say that B is a minimal counterexample if (G,X,J,B) is a counterexample to the Topological Vaught's Conjecture and for every invariant Borel C ⊂ B, either C or B\C contains only countably many orbits. This paper is concerned with counterexamples to the Topological Vaught's Conjecture (of course, there may not be any), and in particular, with minimal counterexamples. First, there is a theorem on the existence of minimal counterexamples. This theorem was known for the model theoretic case (it is due to Harnik and Makkai), and is here generalized to arbitrary Borel-measurable Polish actions. Second, we study the properties of minimal counterexamples. We give two different necessary and sufficient conditions for a counterexample to be minimal, as well as some consequences of minimality. Some of these results are proved assuming determinacy axioms.This second part seems to be new even in the model theoretic case.

Irreducible subrepresentations of the conjugation representation of finitep-groups

For finitep-groupsG we study the conjugation representation γG which is defined by lettingG act on itself by conjugation. Roth conjectured that every irreducible representation ofG which is trivial on the center ofG, occurs in γG. However, this is not true in general. We construct minimal counterexamples and verify Roth's conjecture for various classes of finitep-groups, for instance those of maximal class.

Constructing a Minimal Counterexample in Group Theory

Desmond MacHale's intriguing article [2] neatly illustrates one mode of interplay between proof and example in group theory by describing the commonly used inductive technique of minimal counterexamples. In this Note, we use some ideas from group theory and linear algebra to lead the reader through a geometric and algebraic construction of one of the minimal counterexamples sought by MacHale. Recall that a group is called nilpotent if each of its Sylow subgroups is normal. One of MacHale's list of known-to-be-false conjectures is this: if G is a finite group having a fixed-pointfree automorphism 4, then G is nilpotent. The true theorem on which the conjecture is based assumes that 4 is of prime order; it was established by Thompson in his dissertation in 1959 [4]. His proof so excited the mathematical community that even the New York Times reported his result and the reaction to it [3]. MacHale indicates that [1, p. 336] contains details of a counterexample to the conjecture with IGI = 147 = 72 3 and 1j1 = 4. In our minimal counterexample to be constructed, I4 = 6 and IGI = 48 = 24 43. In what follows, we use some standard conventions of notation in group theory. In particular, we write operators on the right and denote both the action of a group homomorphism and conjugation using superscripts. Recall that if H and K are subgroups of a group G such that G = HK, H n K = 1, and H is normal in G, then G is called the semidirect product of H by K. In this situation, since H is normal in G, each element of K acts via conjugation as an automorphism of H. Of particular interest here is the situation in which each nontrivial element of K acts as a nontrivial (i.e., nonidentity) automorphism on H. In this case K is isomorphic to a subgroup of Aut H, the group of all automorphisms of H. A well-known geometric example leads directly to an important link between group theory and geometry. If A is the group of rotations of a regular tetrahedron, then A contains eight elements of order 3, three of order 2, and an identity element. Each element of order 3 is a 120?rotation of the tetrahedron about an axis passing through one of the 4 vertices and perpendicular to the opposite face. Each element of order 2 is a rotation of 1800 about one of three axes joining the midpoints of two nonadjacent edges of the tetrahedron. (See FIGURE 1.) If the vertices are labeled, and each rotation is identified with the permutation of the labels it produces, this identification provides a natural isomorphism between A and A4, the group of even permutations on 4 letters. Thus it is possible to use the algebra of permutations or geometry to analyze A.

Chords and disjoint paths in matroids

A chord of a circuit C of a matroid M on E is a cell e ϵ S\\ C such that C spans e. Menger's theorem gives necessary and sufficient conditions for a cell of a graphic matroid to be a chord of some circuit. We extend this result to a large class of matroids and find all minimal counterexamples. The theorem is used to obtain results on disjoint paths and to characterize a class of matroid sums.