139 research outputs found

    Computability Theory

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    Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science

    Lattice nonembeddings and intervals of the recursively enumerable degrees

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    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]

    Multiple Permitting and Bounded Turing Reducibilities

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    We look at various properties of the computably enumerable (c.e.) not totally ω-c.e. Turing degrees. In particular, we are interested in the variant of multiple permitting given by those degrees. We define a property of left-c.e. sets called universal similarity property which can be viewed as a universal or uniform version of the property of array noncomputable c.e. sets of agreeing with any c.e. set on some component of a very strong array. Using a multiple permitting argument, we prove that the Turing degrees of the left-c.e. sets with the universal similarity property coincide with the c.e. not totally ω-c.e. degrees. We further introduce and look at various notions of socalled universal array noncomputability and show that c.e. sets with those properties can be found exactly in the c.e. not totally ω-c.e. Turing degrees and that they guarantee a special type of multiple permitting called uniform multiple permitting. We apply these properties of the c.e. not totally ω-c.e. degrees to give alternative proofs of well-known results on those degrees as well as to prove new results. E.g., we show that a c.e. Turing degree contains a left-c.e. set which is not cl-reducible to any complex left-c.e. set if and only if it is not totally ω-c.e. Furthermore, we prove that the nondistributive finite lattice S7 can be embedded into the c.e. Turing degrees precisely below any c.e. not totally ω-c.e. degree. We further look at the question of join preservation for bounded Turing reducibilities r and r′ such that r is stronger than r′. We say that join preservation holds for two reducibilities r and r′ if every join in the c.e. r-degrees is also a join in the c.e. r′-degrees. We consider the class of monotone admissible (uniformly) bounded Turing reducibilities, i.e., the reflexive and transitive Turing reducibilities with use bounded by a function that is contained in a (uniformly computable) family of strictly increasing computable functions. This class contains for example identity bounded Turing (ibT-) and computable Lipschitz (cl-) reducibility. Our main result of Chapter 3 is that join preservation fails for cl and any strictly weaker monotone admissible uniformly bounded Turing reducibility. We also look at the dual question of meet preservation and show that for all monotone admissible bounded Turing reducibilities r and r′ such that r is stronger than r′, meet preservation holds. Finally, we completely solve the question of join and meet preservation in the classical reducibilities 1, m, tt, wtt and T

    Preface

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

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    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

    Author index volume 87 (1997)

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    Communication protocols and quantum error-correcting codes from the perspective of topological quantum field theory

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    Topological quantum field theories (TQFTs) provide a general, minimal-assumption language for describing quantum-state preparation and measurement. They therefore provide a general language in which to express multi-agent communication protocols, e.g. local operations, classical communication (LOCC) protocols. Here we construct LOCC protocols using TQFT, and show that LOCC protocols induce quantum error-correcting codes (QECCs) on the agent-environment boundary. Such QECCs can be regarded as implementing, or inducing the emergence of, spacetimes on such boundaries. We investigate this connection between inter-agent communication and spacetime using BF and Chern-Simons theories, and then using topological M-theory.Comment: 52 page

    On Array Noncomputable Degrees, Maximal Pairs and Simplicity Properties

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    In this thesis, we give contributions to topics which are related to array noncomputable (a.n.c.) Turing degrees, maximal pairs and to simplicity properties. The outline is as follows. In Chapter 2, we introduce a subclass of the a.n.c. Turing degrees, the so called completely array noncomputable (c.a.n.c. for short) Turing degrees. Here, a computably enumerable (c.e.) Turing degree a is c.a.n.c. if any c.e. set A ∈ a is weak truth-table (wtt) equivalent to an a.n.c. set. We show in Section 2.3 that these degrees exist (indeed, there exist infinitely many low c.a.n.c. degrees) and that they cannot be high. Moreover, we apply some of the ideas used to show the existence of c.a.n.c. Turing degrees to show the stronger result that there exists a c.e. Turing degree whose c.e. members are halves of maximal pairs in the c.e. computably Lipschitz (cl) degrees, thereby solving the first part of the first open problem given in the paper by Ambos-Spies, Ding, Fan and Merkle [ASDFM13]. In Chapter 3, we present an approach to extending the notion of array noncomputability to the setting of almost-c.e. sets (these are the sets which correspond to binary representations of left-c.e. reals). This approach is initiated by the Heidelberg Logic Group and it is worked out in detail in an upcoming paper by Ambos-Spies, Losert and Monath [ASLM18], in the thesis of Losert [Los18] and in [ASFL+]. In [ASLM18], the authors introduce the class of sets with the universal similarity property (u.s.p. for short; throughout this thesis, sets with the u.s.p. will shortly be called u.s.p. sets) which is a strong form of array noncomputability in the setting of almost-c.e. sets and they show that sets with this property exist precisely in the c.e. not totally ω-c.e. degrees. Then it is shown that, using u.s.p. sets, one obtains a simplified method for showing the existence of almost-c.e. sets with a property P (for certain properties P) that are contained in c.e. not totally ω-c.e. degrees, namely by showing that u.s.p. sets have property P. This is demonstrated by showing that u.s.p. sets are computably bounded random (CB-random), thereby extending a result from Brodhead, Downey and Ng [BDN12]. Moreover, it is shown that the c.e. not totally ω-c.e. degrees can be characterized as those c.e. degrees which contain an almost-c.e. set which is not cl-reducible to any complex almost-c.e. set. This affirmatively answers a conjecture by Greenberg. For the if-direction of the latter result, we prove a new result on maximal pairs in the almost-c.e. sets by showing the existence of locally almost-c.e. sets which are halves of maximal pairs in the almost-c.e. sets such that the second half can be chosen to be c.e. and arbitrarily sparse. This extends Yun Fan’s result on maximal pairs [Fan09]. By our result, we also get a new proof of one of the main results in Barmpalias, Downey and Greenberg [BDG10], namely that in any c.e. a.n.c. degree there is a left-c.e. real which is not cl-reducible to any ML-random left-c.e. real. In this thesis, we give an overview of some of the results from [ASLM18] and sketch some of the proofs to illustrate this new methodology and, subsequently, we give a detailed proof of the above maximal pair result. In Chapter 4, we look at the interaction between a.n.c. wtt-degrees and the most commonly known simplicity properties by showing that there exists an a.n.c. wtt-degree which contains an r-maximal set. By this result together with the result by Ambos-Spies [AS18] that no a.n.c. wtt-degree contains a dense simple set, we obtain a complete characterization which of the classical simplicity properties may hold for a.n.c. wtt-degrees. The guiding theme for Chapter 5 is a theorem by Barmpalias, Downey and Greenberg [BDG10] in which they characterize the c.e. not totally ω-c.e. degrees as the c.e. degrees which contain a c.e. set which is not wtt-reducible to any hypersimple set. So Ambos-Spies asked what the above characterization would look like if we replaced hypersimple sets by maximal sets in the above theorem. In other words, what are the c.e. Turing degrees that contain c.e. sets which are not wtt-reducible to any maximal set. We completely solve this question on the set level by introducing the new class of eventually uniformly wtt-array computable (e.u.wtt-a.c.) sets and by showing that the c.e. sets with this property are precisely those c.e. sets which are wtt-reducible to maximal sets. Indeed, this characterization can be extended in that we can replace wtt-reducible by ibT-reducible and maximal sets by dense simple sets. By showing that the c.e. e.u.wtt-a.c. sets are closed downwards under wtt-reductions and under the join operation, it follows that the c.e. wtt-degrees containing e.u.wtt-a.c. sets form an ideal in the upper semilattice of the c.e. wtt-degrees and, further, we obtain a characterization of the c.e. wtt-degrees which contain c.e. sets that are not wtt-reducible to any maximal set. Moreover, we give upper and lower bounds (with respect to ⊆) for the class of the c.e. e.u.wtt-a.c. sets. For the upper bound, we show that any c.e. e.u.wtt-a.c. set has array computable wtt-degree. For the lower bound, we introduce the notion of a wtt-superlow set and show that any wtt-superlow c.e. set is e.u.wtt-a.c. Besides, we show that the wtt-superlow c.e. sets can be characterized as the c.e. sets whose bounded jump is ω-computably approximable (ω-c.a. for short); hence, they are precisely the bounded low sets as introduced in the paper by Anderson, Csima and Lange [ACL17]. Furthermore, we prove a hierarchy theorem for the wtt-superlow c.e. sets and we show that there exists a Turing complete set which lies in the intersection of that hierarchy. Finally, it is shown that the above bounds are strict, i.e., there exist c.e. e.u.wtta. c. sets which are not wtt-superlow and that there exist c.e. sets whose wtt-degree is array computable and which are not e.u.wtt-a.c. (where here, we obtain the separation even on the level of Turing degrees). The results from Chapter 5 will be included in a paper which is in preparation by Ambos-Spies, Downey and Monath [ASDM19]
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