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Page 9 of Mathematical Reviews Vol. 40, Issue 2 [page]

1970 Mathematical Reviews  
Lerman, Manuel 1276 Some nondistributive lattices as initial segments of the degrees of unsolvability. J. Symbolic Logic 34 (1969), 85-98. G. E. Sacks [Degrees of unsolvability, Ann. of Math.  ...  The case of nondistributive lattices remained open until Shoenfield (unpublished) proved that the lattice 1-3-1 is order-isomorphic to an initial segment of degrees.  ... 

Degrees of Unsolvability [chapter]

Klaus Ambos-Spies, Peter A. Fejer
2014 Handbook of the History of Logic  
We finish our history by quoting Gerald Sacks, who through his own work and the work of his students (including, Harrington, Robinson, Shore, Simpson, Slaman, and Thomason) shaped degree theory throughout  ...  a good part of its existence: "Farewell to higher recursion theory, but not to recursion theory; there is no way to say good-bye to recursion theory" ( [Sacks, 1999] ).  ...  a finite initial segment of the degrees if and only if S is order-isomorphic to a finite initial segment of some upper semi-lattice with a least element.  ... 
doi:10.1016/b978-0-444-51624-4.50010-1 fatcat:clf7varewrh6lc4jq2c2sp6pgy

Page 4722 of Mathematical Reviews Vol. , Issue 87i [page]

1987 Mathematical Reviews  
T- degrees, where all countable distributive and some but not all finite nondistributive lattices are embeddable, and the r.e. wtt- and m- degrees, where the finite lattices which are embeddable are exactly  ...  New results: (1) Every generating set of R has nonempty intersection with each initial segment of R\ {0}.  ... 

Conjectures and questions from Gerald Sacks's Degrees of Unsolvability

Richard A. Shore
1997 Archive for Mathematical Logic  
We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the  ...  -the conjectures and questions posed at the end of the two editions of Sacks's first book, the classic monograph Degrees of Unsolvability (Annals of Mathematics Studies, Number 55, Princeton University  ...  (C6) S is a finite initial segment of degrees if and only if S is orderisomorphic to a finite, initial segment of some upper semi-lattice with a least member.  ... 
doi:10.1007/s001530050063 fatcat:wmp32iusbvffnbkv6rsosghtoq

Not every finite lattice is embeddable in the recursively enumerable degrees

A.H Lachlan, R.I Soare
1980 Advances in Mathematics  
Lachlan [4] proved undecidability of Th(g) by proving that all finite lattices could be embedded as initial segments of degrees.  ...  Using a much more complex technique of enumeration Lachlan [5] showed how the two 5-element nondistributive lattices can be embedded in 9? as lattices.  ... 
doi:10.1016/0001-8708(80)90027-4 fatcat:2vwdewhs2zf5hjsm4dfcasrhqa

The Recursively Enumerable Degrees [chapter]

Richard A. Shore
1999 Studies in Logic and the Foundations of Mathematics  
The idea here is that the standard models are the models M such that each initial segment of M can be mapped into an initial segment of every model.  ...  As Church's Thesis became widely accepted the word "recursive" was dropped and they became simply the degrees of unsolvability.  ... 
doi:10.1016/s0049-237x(99)80022-6 fatcat:yviyyg7uhfbolegitvgof4ga7m

Strong reducibilities

Piergiorgio Odifreddi
1981 Bulletin of the American Mathematical Society  
Some necessary conditions are known (see [La2] ), e.g. every finite initial segment is a lattice of a certain kind, but the situation seems complicated since some nondistributive finite lattices are not  ...  Every finite distributive lattice is embeddable as an initial segment, and the sets of first order sentences true of no distributive lattice and true of some finite distributive lattice are recursively  ... 
doi:10.1090/s0273-0979-1981-14863-1 fatcat:qbyl6mci6rd7jbxfwqjfmkit34

Recursively enumerable sets and degrees

Robert I. Soare
1978 Bulletin of the American Mathematical Society  
The relation of the structure of an r.e. set to its degree. 1. Post's program and simple sets. 2. Dominating functions and quotient lattices. 3. Maximal sets and high degrees. 4.  ...  The minimal pair method and lattice embeddings in R. 15. Cupping and splitting r.e. degrees. 16. Automorphisms and decidability of R. Introduction. G. E.  ...  Lachlan proved undecidability of the elementary theory of degrees by showing that every finite distributive lattice can be embedded as an initial segment of the degrees.  ... 
doi:10.1090/s0002-9904-1978-14552-2 fatcat:arxp4btvhzfzjeoufybmnemt2u

The Book Review Column

William Gasarch
2000 ACM SIGACT News  
In this column we review the following books.  ...  After introducing the concept of equivalent queries, the proof that nonequivalent queries form a nondistribution lattice (called a query lattice) is given.  ...  Equipped with a large selection of solved examples and even a larger set of unsolved exercises, this book renders itself a useful tool for an algorithm student (as well as the instructor).  ... 
doi:10.1145/348210.1042071 fatcat:3fnw5huw6vguhbdzac7sqzsqxi

The annual meeting of the society

W. M. Whyburn
1951 Bulletin of the American Mathematical Society  
If F(x) is of the aperiodic lattice type, then lim ^*P(Sn -x) is equal to 1/m or 0 according as #-•+ oo or x->-<*> on the lattice.  ...  The &th segment/w of the reciprocal of (J+zI) m k plays the role of mth approximant.  ...  of the Proceedings.  ... 
doi:10.1090/s0002-9904-1951-09479-3 fatcat:gv6o2ywuyjgtvoukqa4advbtd4

Categorical Ontology of Complex Spacetime Structures: The Emergence of Life and Human Consciousness

I. C. Baianu, R. Brown, J. F. Glazebrook
2007 Axiomathes  
biological evolution in terms of colimits of variable biogroupoids, or as chains and compositions of local procedures (COLPs) of locally Lie groupoids in the evolution and co-evolution of species Á What  ...  The claim is defended in this essay that human consciousness is unique and should be viewed as an ultra-complex, global process of processes.  ...  There still remains the unsolved question why humans-as well as parrots-have the inherited inclination to talk whereas the apes do not.  ... 
doi:10.1007/s10516-007-9011-2 fatcat:m3kuram2fjcwze4mkmfxk2akwu

Computability theory, algorithmic randomness and Turing's anticipation [chapter]

Rod Downey, Rod Downey
Turing's Legacy  
This article gives a brief history of some of the main lines of investigation in computability theory, a major part of Turing's Legacy.  ...  Turing's theory has substantially developed under the names of recursion theory and computability theory. Turing's work can be seen as perhaps the high point in the confluence of ideas in 1936.  ...  Also, many lattices can be embedded densely, such as all distributive lattices (Slaman [132] ), and some nondistributive lattices (Ambos-Spies, Hirschfeldt and Shore [2] ) but not all embeddable lattices  ... 
doi:10.1017/cbo9781107338579.005 fatcat:yg2zjyeobng4diuwiqh2uj5qcm

Algorithmic Information Theoretic Issues in Quantum Mechanics [article]

Gavriel Segre
2004 arXiv   pre-print
taking aside the review part, a finite-cardinality's set of new ideas concerning algorithmic information issues in Quantum Mechanics is introduced and analyzed  ...  Theorem 2.2.4 KRAFT-CHAITIN'S THEOREM HP: φ ∈ C M −C Φ −∆ 0 0 − • M AP (N, N) : HALT IN G(φ) is an initial segment of N + TH: The following statements are equivalent: 1.  ...  can be seen as the starting point of the of the open intellectual challenge summarized by the path: Quantum Mechanics as Nondistributive Logic → Quantum Mechanics as Noncommutative Probability → Quantum  ... 
arXiv:quant-ph/0110018v7 fatcat:3a5beei555g4vhgwge7v5ykqee

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

Thorsten Kräling
We say that α is an initial segment of a set A if α(i) = A(i) for i < |α|, and we write α < A. The unique initial segment of A of length n is denoted by A n.  ...  < m, we call α a proper initial segment of β and write α < β.  ... 
doi:10.11588/heidok.00016720 fatcat:sjljp2dhm5dslazkftifzvlsha

Multivariable regulation in geometric terms: Old and new results [chapter]

Giovanni Marro
Colloquium on Automatic Control  
Property 2.3 The unobservability subspace of the pair (C, A), i.e., the set of all the initial states that cannot be recognized from the output function, is Q = maxL(A, C).  ...  The aim of this lecture is to present some well-known and some new results in multivariable regulation in a strictly geometric framework, each compared with the corresponding singlevariable result approached  ...  D ⊆ E, and a linear map A : X → X , it is well known that the set of all A-invariants L satisfying D ⊆ L ⊆ E is a nondistributive lattice with respect to ⊆, +, ∩.  ... 
doi:10.1007/bfb0043577 fatcat:3z7hfqmp25dqnkwhzcapk2iha4
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