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Page 1844 of Mathematical Reviews Vol. , Issue 91D [page]

1991 Mathematical Reviews  
(English and Russian summaries) [A new proof of E. L. Post’s results on closed classes of Boolean functions] J. Inform. Process. Cybernet. 25 (1989), no. 7, 361-380.  ...  More recently, there have appeared several new proofs of Post’s result based on universal algebra results nonexistent 50 years ago.  ... 

Structure identification of Boolean relations and plain bases for co-clones

Nadia Creignou, Phokion Kolaitis, Bruno Zanuttini
2008 Journal of computer and system sciences (Print)  
Our algorithm is derived by first introducing the concept of a plain basis for a co-clone and then identifying natural plain bases for every co-clone in Post's lattice.  ...  We give a quadratic algorithm for the following structure identification problem: given a Boolean relation R and a finite set S of Boolean relations, can the relation R be expressed as a conjunctive query  ...  A (Boolean) clone is a set of Boolean functions closed under composition and containing all projections functions.  ... 
doi:10.1016/j.jcss.2008.02.005 fatcat:byizubbfjff7fc5yartl7wx5ha

A Trichotomy in the Complexity of Propositional Circumscription [chapter]

Gustav Nordh
2005 Lecture Notes in Computer Science  
We prove that there exists a trichotomy for the complexity of the inference problem in propositional variable circumscription.  ...  Circumscription is one of the most important and well studied formalisms in the realm of nonmonotonic reasoning.  ...  Acknowledgments The author thanks Steffen Reith for providing the visualization of Post's lattice in Figure 1 .  ... 
doi:10.1007/978-3-540-32275-7_18 fatcat:wr4bfzscdzhtrl2j26jto2zadq

On Boolean primitive positive clones

Miki Hermann
2008 Discrete Mathematics  
This paper presents a new proof of his result, relating it to Post's lattice and exhibiting finite bases for those clones.  ...  Kuznecov introduced the concept of primitive positive clones and proved in 1977 that there are 25 Boolean primitive positive clones in a notoriously unavailable article.  ...  I also thank the referees for valuable comments and suggestions on the previous version of this paper.  ... 
doi:10.1016/j.disc.2007.06.018 fatcat:rbpwvqlrxbgm5oplhhlzoslwu4

Emil Post [chapter]

UrquharT. Alasdair
2009 Handbook of the History of Logic  
The biographical essay [Gleiser, 1980] is a very informative source on Post's life. The description of Post's papers was obtained from the web site of the American Philosophical Society, which is  ...  ACKNOWLEDGEMENTS My greatest debt is to Martin Davis, whose editorial work and writings on Post were indispensable in preparing this chapter.  ...  A self-contained proof of Post's theorem on functional completeness can be found in [Pelletier and Martin, 1990] . Similar criteria can be given for all the other classes in Post's diagram.  ... 
doi:10.1016/s1874-5857(09)70016-0 fatcat:erwzbkejovc7vfrx2xbqqmtybe

Page 605 of Mathematical Reviews Vol. , Issue 87b [page]

1987 Mathematical Reviews  
The proof uses a direct coding of the theory of irreflexive binary relations on finite sets. (Aside: The interpolation theorem is itself a Il, sentence in the language of Prop’.  ...  Author summary: “We show that the minimum degree of Turing complexity of a recursively representable choice function {on a re- cursive space of alternatives derived from a recursive metric space} is 0”  ... 

Recursively enumerable sets and degrees

Robert I. Soare
1978 Bulletin of the American Mathematical Society  
A subset A of <o (the set of nonnegative integers) is recursive (also called decidable or computable) if there is an algorithm for determining whether a  ...  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.  ...  (All sets will be subsets of co and all functions will be on co.) A partial recursive (p.r.) function (from co to co) is a partial function whose graph is r.e.  ... 
doi:10.1090/s0002-9904-1978-14552-2 fatcat:arxp4btvhzfzjeoufybmnemt2u

Notions of semicomputability in topological algebras over the reals

Mark Armstrong, Jeffery Zucker
2018 Computability - The Journal of the Assosiation  
of semicomputable and projectively (semi)computable sets (Chapter 5 Theorem 7, p.45), and Post's Theorem (i.e. a set is computable iff both it and its complement are semicomputable) (Appendix B Theorem  ...  We will consider a "continuity principle", which states that computability should imply continuity; however, equality, order, and other total boolean-valued functions on the reals are clearly discontinuous  ...  Appendix B A counterexample to Post's Theorem for partial algebras Theorem 9 ( 9 Post's theorem for While(R) semicomputability).  ... 
doi:10.3233/com-180087 fatcat:ra6vmzvqtnbetfn5piszhmhwiq

Page 6596 of Mathematical Reviews Vol. , Issue 99j [page]

1999 Mathematical Reviews  
Using Boolean models for first-order logic, the author gives a constructive proof (in a constructive metalanguage) for Herbrand’s theorem and a conservative extension theorem.  ...  The authors develop a notion of minimality similar to o- minimality, called P-minimality, for p-adically closed fields.  ... 

Page 1613 of Mathematical Reviews Vol. , Issue 2003C [page]

2003 Mathematical Reviews  
Summary: “We define the .2/-ideals of a poset—or equally of a quasi-ordered set—for various collections of subsets % and corresponding .%-ideal continuity for functions.  ...  Summary: “A list of four terms is given such that a subset of a Brouwerian semilattice S containing | is a kernel (i.e., 1-class) of some congruence on S if and only if it is closed with respect to these  ... 

Boolean Constraint Satisfaction Problems: When Does Post's Lattice Help? [chapter]

Nadia Creignou, Heribert Vollmer
2008 Lecture Notes in Computer Science  
Pol(Γ) is a clone, i.e., a set of Boolean functions that contains all projections and is closed under composition.  ...  Post's lattice [Emil Post, 1921 /1941 One-one correspondence between clones and co-clones; obtain complete list of co-clones from Post's lattice. Determine easy bases for relational clones!  ...  Post's lattice: L 2 ⊆ R 2 , hence Inv(R 2 ) ⊆ Inv(L 2 ). Hence: Undirected graph accessibility is in ⊕L, in other words: [Karchmer, Wigderson, 1993] (Today we even know SL ⊆ L.)  ... 
doi:10.1007/978-3-540-92800-3_2 fatcat:5ahjgjte5vfalii7u7rre6kdkq

Fixed-parameter Approximability of Boolean MinCSPs [article]

Édouard Bonnet, László Egri, Bingkai Lin, Dániel Marx
2018 arXiv   pre-print
The minimum unsatisfiability version of a constraint satisfaction problem (MinCSP) asks for an assignment where the number of unsatisfied constraints is minimum possible, or equivalently, asks for a minimum-size  ...  Here we study the fixed-parameter (FP-) approximability of the problem: given an instance and an integer k, one has to find a solution of size at most g(k) in time f(k)n^O(1) if a solution of size at most  ...  Post's lattice, co-clone lattice, and a simple reduction A clone is a set of Boolean functions that contains all projections (that is, the functions f (a 1 , . . . , a n ) = a k for 1 ≤ k ≤ n) and is closed  ... 
arXiv:1601.04935v2 fatcat:w2lpubpnc5ccfndq67gjfezdmm

The Complexity of Orbits of Computably Enumerable Sets

Peter A. Cholak, Rodney Downey, Leo A. Harrington
2008 Bulletin of Symbolic Logic  
This result and proof have a number of nice corollaries: the Scott rank of ε is + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of ε; for all finite α ≥  ...  9, there is a properly orbit (from the proof).  ...  (Additionally they use the algebraic methods for new results such as new orbits.)  ... 
doi:10.2178/bsl/1208358844 fatcat:frqyk375tfbepbii74zfrqwski

Weak Bases of Boolean Co-Clones [article]

Victor Lagerkvist
2013 arXiv   pre-print
In this article we consider strong partial clones which can be seen as a more fine-grained framework than Post's lattice where each clone splits into an interval of strong partial clones.  ...  Universal algebra and clone theory have proven to be a useful tool in the study of constraint satisfaction problems since the complexity, up to logspace reductions, is determined by the set of polymorphisms  ...  Acknowledgements The author is grateful towards Peter Jonsson, Gustav Nordh and Bruno Zanuttini for helpful comments and suggestions.  ... 
arXiv:1310.3674v1 fatcat:lm4rlpfzn5crncqegeoat7oky4

An Algebraic Approach to the Complexity of Generalized Conjunctive Queries [chapter]

Michael Bauland, Philippe Chapdelaine, Nadia Creignou, Miki Hermann, Heribert Vollmer
2005 Lecture Notes in Computer Science  
We derive a dichotomy theorem for this audit problem that sheds more light on audit applicability within database systems.  ...  We study the Boolean conjunctive queries under a more detailed scope, where we investigate their counting problem by means of the algebraic approach through Galois theory, taking advantage of Post's lattice  ...  We thank Elmar Böhler, Matthias Galota, and Steffen Reith for helpful discussions.  ... 
doi:10.1007/11527695_3 fatcat:pc7kel2r45ef3ketqmeq7dlkhi
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