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A novel family of peptide antimycotics, termed ecomycins, is described from Pseudomonas viridiflava, a plant-associated bacterium. Ecomycins B and C have molecular masses of 1153 and 1181. They contain equimolar amounts of a b hydroxyaspartic acid, homoserine, threonine, serine, alanine, glycine and one unknown amino acid. Fatty acids were detectable after hydrolysis, methylation and gas chromatography and mass spectroscopy. The ecomycins have significant bioactivities against a wide range of human and plant pathogenic fungi. The minimum inhibitory concentration values for ecomycin B were 4·0 mg ml −1 against Cryptococcus neoformans and 31 mg ml −1 against Candida albicans. Pseudomonas viridiflava also produces what appears to be syringotoxin, an antifungal lipopeptide previously described from Ps. syringae.

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The notion of computable reducibility between equivalence relations on the natural numbers provides a natural computable analogue of Borel reducibility. We investigate the computable reducibility hierarchy, comparing and contrasting it with the Borel reducibility hierarchy from descriptive set theory. Meanwhile, the notion of computable reducibility appears well suited for an analysis of equivalence relations on the c.e. sets, and more specifically, on various classes of c.e. structures. This is a rich context with many natural examples, such as the isomorphism relation on c.e. graphs or on computably presented groups. Here, our exposition extends earlier work in the literature concerning the classification of computable structures. An abundance of open questions remains.

We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations R, S, a componentwise reducibility is defined byHere, f is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and f must be computable. We show that there is a Π 0 1 -complete equivalence relation, but no Π 0 k -complete for k ≥ 2. We show that Σ 0 k preorders arising naturally in the above-mentioned areas are Σ 0 k -complete. This includes polynomial time m-reducibility on exponential time sets, which is Σ 0 2 , almost inclusion on r.e. sets, which is Σ 0 3 , and Turing reducibility on r.e. sets, which is Σ 0 4 . 859 860 EGOR IANOVSKI, RUSSELL MILLER, KENG MENG NG, AND ANDRÉ NIES so that we can actually learn something from the reduction. In our example, one should ask how hard it is to compute the dimension of a rational vector space. It is natural to restrict the question to computable vector spaces over Q (i.e., those where the vector addition is given as a Turing-computable function on the domain of the space). Yet even when its domain D E such vector spaces, computing the function which maps each one to its dimension requires a 0 -oracle, hence is not as simple as one might have hoped. (The reasons why 0 is required can be gleaned from [7] or [8].) 1.1. Effective reductions. Reductions are normally ranked by the ease of computing them. In the context of Borel theory, for instance, a large body of research is devoted to the study of Borel reductions (the standard book reference is [17]). Here, the domains D E and D F are the set 2 or some other standard Borel space, and a Borel reduction f is a reduction (from E to F , these being equivalence relations on 2 ) which, viewed as a function from 2 to 2 , is Borel. If such a reduction exists, one says that E is Borel reducible to F , and writes E ≤ B F . A stronger possible requirement is that f be continuous, in which case we have (of course) a continuous reduction. In case the reduction is given by a Turing functional from reals to reals, it is a (type-2) computable reduction.A further body of research is devoted to the study of the same question for equivalence relations E and F on , and reductions f : → between them which are computable. If such a reduction from E to F exists, we say that E is computably reducible to F , and write E ≤ c F , or often just E ≤ F . These reductions will be the focus of this paper. Computable reducibility on equivalence relations was perhaps first studied by Ershov [12] in a category theoretic setting.The main purpose of this paper is to investigate the complexity of equivalence relations under these reducibilities. In certain cases we will generalize from equivalence relations to preorders on . We restrict most of our discussion to relatively low levels of the hierarchy, usually to Π 0 n and Σ 0 n with n ≤ 4. One can focus more closely on very low levels: Such articles as [3,5,18], for instance, have dealt exclusively with Σ 0 1 equivalen...

We exploit properties of certain directed graphs, obtained from the families of sets with special effective enumeration properties, to generalize several results in computable model theory to higher levels of the hyperarithmetical hierarchy. Families of sets with such enumeration features were previously built by Selivanov, Goncharov, and Wehner. For a computable successor ordinal α, we transform a countable directed graph G into a structure G * such that G has a ∆ 0 α isomorphic copy if and only if G * has a computable isomorphic copy.A computable structure A is ∆ 0 α categorical (relatively ∆ 0 α categorical, respectively) if for all computable (countable, respectively) isomorphic copies B of A, there is an isomorphism from A onto B, which is ∆ 0 α (∆ 0 α relative to the atomic diagram of B, respectively). We prove that for every computable successor ordinal α, there is a computable, ∆ 0 α categorical structure, which is not relatively ∆ 0 α categorical. This generalizes the result of Goncharov that there is a computable, computably categorical structure, which is not relatively computably categorical.

Both transduction of single chromosomal loci and cotransduction of closely linked loci were observed between lysogenic and nonlysogenic strains of Pseudomonas aeruginosa in a freshwater habitat. Transductants were recovered at frequencies of 10-6 to 10-5 transductants per CFU. Transductants of lysogenized strains were recovered 10-to 100-fold more frequently than were transductants of nonlysogenic parents. Lysogens are thus capable of introducing phages which mediate generalized transduction into the natural microbial community and serving as recipients of transduced DNA. It would appear that lysogeny has the potential of increasing the size and flexibility of the gene pool available to natural populations of bacteria. The ability to generate and select new genetic combinations through phage-mediated exchange can be significant in the face of a continually changing environment and may contribute to the apparent fitness of the lysogenic state in natural ecosystems.

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