Many-one reductions and the category of multivalued functions

Multi-valued functions are common in computable analysis (built upon the Type 2 Theory of Effectivity), and have made an appearance in complexity theory under the moniker search problems leading to complexity classes such as PPAD and PLS being studied. However, a systematic investigation of the resulting degree structures has only been initiated in the former situation so far (the Weihrauch-degrees). A more general understanding is possible, if the category-theoretic properties of multi-valued functions are taken into account. In the present paper, the category-theoretic framework is established, and it is demonstrated that many-one degrees of multi-valued functions form a distributive lattice under very general conditions, regardless of the actual reducibility notions used (e.g. Cook, Karp, Weihrauch). Beyond this, an abundance of open questions arises. Some classic results for reductions between functions carry over to multi-valued functions, but others do not. The basic theme here again depends on category-theoretic differences between functions and multi-valued functions.Comment: an earlier version was titled "Many-one reductions between search problems". in Mathematical Structures in Computer Science, 201

Lattice nonembeddings and intervals of the recursively enumerable degrees

AbstractLet b and c be r.e. Turing degrees such that b>c. We show that there is an r.e. degree a such that b>a>c and all lattices containing a critical triple, including the lattice M5, cannot be embedded into the interval [c, a]

Matroids, Complexity and Computation

The node deletion problem on graphs is: given a graph and integer k, can we delete no more than k vertices to obtain a graph that satisfies some property π. Yannakakis showed that this problem is NP-complete for an infinite family of well- defined properties. The edge deletion problem and matroid deletion problem are similar problems where given a graph or matroid respectively, we are asked if we can delete no more than k edges/elements to obtain a graph/matroid that satisfies a property π. We show that these problems are NP-hard for similar well-defined infinite families of properties. In 1991 Vertigan showed that it is #P-complete to count the number of bases of a representable matroid over any fixed field. However no publication has been produced. We consider this problem and show that it is #P-complete to count the number of bases of matroids representable over any infinite fixed field or finite fields of a fixed characteristic. There are many different ways of describing a matroid. Not all of these are polynomially equivalent. That is, given one description of a matroid, we cannot create another description for the same matroid in time polynomial in the size of the first description. Due to this, the complexity of matroid problems can vary greatly depending on the method of description used. Given one description a problem might be in P while another description gives an NP-complete problem. Based on these interactions between descriptions, we create and study the hierarchy of all matroid descriptions and generalize this to all descriptions of countable objects

Joins and Meets in the Partial Orders of the Computably Enumerable ibT- and cl-Degrees

A bounded reducibility is a preorder on the power set of the integers which is obtained from Turing reducibility by the additional requirement that, for a reduction of A to B, for every input x the oracle B is only asked oracle queries y < f(x)+1, where f is from some given set F of total computable functions. The most general example of a bounded reducibility is weak-truth-table reducibility, where F is just the set of all computable functions. In this thesis we study the so-called strongly bounded reducibilites, which are obtained by choosing F={id} and F={id+c: c constant}, respectively (where id is the identity function). We start by giving a machine-independent characterisation of these reducibilities, define the degree structures of the computably enumerable ibT- and cl-degrees and review some important properties of ibT- and cl-reducibility concerning strictly increasing computable functions (called shifts) and the permitting method. Then we turn to the degree structures mentioned above, and in particular to existence and nonexistence of joins and meets of a finite set of degrees. As Barmpalias and independently Fan and Lu have shown, these structures are not upper semi-lattices; it is also known that they are not lower semi-lattices. We extend these results by showing that the existence of a join or meet of n degrees does in general not imply the existence of a join or meet, respectively, of any subset containining more than one element of these degrees. We also show that even if deg(A) and deg(B) have a join, there is no uniform way to compute a member of this join from A and B, contrasting the join in the Turing degrees. We conclude this part by looking at the substructure which consists of the degrees of simple sets and show that this structure is not closed with respect to the join operation. This is the dual of a theorem of Ambos-Spies stating that the simple degrees are not closed with respect to meets. Next, we investigate lattice embeddings into the c.e. r-degrees. Due to an observation of Ambos-Spies, the proof that every finite distributive lattice can be embedded into the computably enumerable Turing degrees carries over to the c.e. r-degrees. We show that the smallest nondistributive lattices N5 and M3 can also be embedded, but only the N5 can be embedded preserving the least element. Since every nondistributive lattice contains at least one of these two lattices as a sublattice, this motivates the conjecture that every finite lattice can be embedded. We show this for two other nondistributive lattices, the S7 und S8. Finally, we compare the c.e. ibT- and c.e. cl-degrees and prove that these are not elementarily equivalent. To show this, we study under which conditions on two degrees a and c with a<c it holds that there exists a degree b<c such that c is the join of a and b. In this context we also show that, while shifts provide a simple method to produce a lesser r-degree a to some given noncomputable r-degree c, there is no computable shift which uniformly produces such an a with the additional property that no degree b as above exists