scholarly journals Some computational problems related to normal forms

2016 ◽  
Vol 13 (1) ◽  
pp. 53-65
Author(s):  
Vũ Đức Thi
Keyword(s):  
Normal Form ◽  
Time Complexity ◽  
Normal Forms ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Effective Algorithm ◽  
Database Theory ◽  
Relation Scheme ◽  
Sperner System

In the relational database theory the most desirable  normal form is the Boyce-Codd normal form (BCNF). This paper investigates some computational problems concerning BCNF relation scheme and BCNF relations. We give an effective algorithm finding a BCNF relation r such that r represents a given BCNF relation scheme s  (i.e., Kr=Ks, where Kr and Ks are  sets of all minimal keys of  r and s). This paper also gives an effective algorithm which  from a given  BCNF relation finds a BCNF relation scheme such that Kr=Ks. Based on these algorithms we prove that  the time  complexity of the  problem that  finds a BCNF relation r  representing a given BCNF relation scheme s is exponential in the size of s and conversely, the complexity of finding a BCNF relation scheme s from a given BCNF relation r such that r represents s also is exponential in the number of attributes. We give a new characterization of the relations and the relation scheme that are uniquely determined by their minimal keys. It is known that these relations and the relation schemes are in the BCNF class. From this characterization we give a polynomial time algorithm deciding whether an arbitrary relation is uniquely determined by its set of all  minimal keys. In the rest if this paper some new bounds of the  size of minimal Armstrong relations for  BCNF relation scheme are given. We show that given a Sperner system K and BCNF relation scheme s a set of minimal keys of which is K, the number of antikeys (maximal nonkeys) of K is polynomial in the number of attributes iff so is the size of minimal Armstrong relation of s.

ORDINAL AUTOMATA AND CANTOR NORMAL FORM

2012 ◽  
Vol 23 (01) ◽  
pp. 87-98
Author(s):  
ZOLTÁN ÉSIK
Keyword(s):  
Normal Form ◽  
Polynomial Time ◽  
Regular Language ◽  
Finite Automata ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Order Type ◽  
Deterministic Finite Automaton ◽  
Lexicographic Ordering ◽  
Language L

It is known that an ordinal is the order type of the lexicographic ordering of a regular language if and only if it is less than ωω. We design a polynomial time algorithm that constructs, for each well-ordered regular language L with respect to the lexicographic ordering, given by a deterministic finite automaton, the Cantor Normal Form of its order type. It follows that there is a polynomial time algorithm to decide whether two deterministic finite automata accepting well-ordered regular languages accept isomorphic languages. We also give estimates on the state complexity of the smallest "ordinal automaton" representing an ordinal less than ωω, together with an algorithm that translates each such ordinal to an automaton.

scholarly journals Fair Allocation based on Diminishing Differences

2017 ◽  
Author(s):  
Erel Segal-Halevi ◽  
Haris Aziz ◽  
Avinatan Hassidim
Keyword(s):  
School Choice ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Conservative Extension ◽  
Score Functions ◽  
Full Characterization ◽  
Proportional Allocation ◽  
The Difference ◽  
Natural Way

Ranking alternatives is a natural way for humans to explain their preferences. It is being used in many settings, such as school choice (NY, Boston), Course allocations, and the Israeli medical lottery. In some cases (such as the latter two), several ``items'' are given to each participant. Without having any information on the underlying cardinal utilities, arguing about fairness of allocation requires extending the ordinal item ranking to ordinal bundle ranking. The most commonly used such extension is stochastic dominance (SD), where a bundle X is preferred over a bundle Y if its score is better according to all additive score functions. SD is a very conservative extension, by which few allocations are necessarily fair while many allocations are possibly fair. We propose to make a natural assumption on the underlying cardinal utilities of the players, namely that the difference between two items at the top is larger than the difference between two items at the bottom. This assumption implies a preference extension which we call diminishing differences (DD), where a X is preferred over Y if its score is better according to all additive score functions satisfying the DD assumption. We give a full characterization of allocations that are necessarily-proportional or possibly-proportional according to this assumption. Based on this characterization, we present a polynomial-time algorithm for finding a necessarily-DD-proportional allocation if it exists. Using simulations, we show that with high probability, a necessarily-proportional allocation does not exist but a necessarily-DD-proportional allocation exists, and moreover, that allocation is proportional according to the underlying cardinal utilities.

scholarly journals Complete Characterization of Incorrect Orthology Assignments in Best Match Graphs

2021 ◽  
Vol 82 (3) ◽  
Author(s):  
David Schaller ◽  
Manuela Geiß ◽  
Peter F. Stadler ◽  
Marc Hellmuth
Keyword(s):  
Gene Transfer ◽  
Horizontal Gene Transfer ◽  
A Priori ◽  
Gene Tree ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Complete Characterization ◽  
Species Trees ◽  
Genome Scale

AbstractGenome-scale orthology assignments are usually based on reciprocal best matches. In the absence of horizontal gene transfer (HGT), every pair of orthologs forms a reciprocal best match. Incorrect orthology assignments therefore are always false positives in the reciprocal best match graph. We consider duplication/loss scenarios and characterize unambiguous false-positive (u-fp) orthology assignments, that is, edges in the best match graphs (BMGs) that cannot correspond to orthologs for any gene tree that explains the BMG. Moreover, we provide a polynomial-time algorithm to identify all u-fp orthology assignments in a BMG. Simulations show that at least $$75\%$$ 75 % of all incorrect orthology assignments can be detected in this manner. All results rely only on the structure of the BMGs and not on any a priori knowledge about underlying gene or species trees.

scholarly journals Liveness in broadcast networks

Computing ◽  
2021 ◽  
Author(s):  
Peter Chini ◽  
Roland Meyer ◽  
Prakash Saivasan
Keyword(s):  
Model Checking ◽  
Polynomial Time ◽  
Message Passing ◽  
Linear Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Final State ◽  
Broadcast Network ◽  
Broadcast Networks

AbstractWe study liveness and model checking problems for broadcast networks, a system model of identical clients communicating via message passing. The first problem that we consider is Liveness Verification. It asks whether there is a computation such that one clients visits a final state infinitely often. The complexity of the problem has been open. It was shown to be $$\texttt {P}$$ P -hard but in $$\texttt {EXPSPACE}$$ EXPSPACE . We close the gap by a polynomial-time algorithm. The latter relies on a characterization of live computations in terms of paths in a suitable graph, combined with a fixed-point iteration to efficiently check the existence of such paths. The second problem is Fair Liveness Verification. It asks for a computation where all participating clients visit a final state infinitely often. We adjust the algorithm to also solve fair liveness in polynomial time. Both problems can be instrumented to answer model checking questions for broadcast networks against linear time temporal logic specifications. The first problem in this context is Fair Model Checking. It demands that for all computations of a broadcast network, all participating clients satisfy the specification. We solve the problem via the Vardi–Wolper construction and a reduction to Liveness Verification. The second problem is Sparse Model Checking. It asks whether each computation has a participating client that satisfies the specification. We reduce the problem to Fair Liveness Verification.

scholarly journals POLYNOMIAL-TIME COMPLEXITY FOR INSTANCES OF THE ENDOMORPHISM PROBLEM IN FREE GROUPS

2007 ◽  
Vol 17 (02) ◽  
pp. 289-328 ◽  
Author(s):  
LAURA CIOBANU
Keyword(s):  
Free Group ◽  
Polynomial Time ◽  
Efficient Algorithm ◽  
Time Complexity ◽  
Free Groups ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Polynomial Time Complexity

We say the endomorphism problem is solvable for an element W in a free group F if it can be decided effectively whether, given U in F, there is an endomorphism ϕ of F sending W to U. This work analyzes an approach due to Edmunds and improved by Sims. Here we prove that the approach provides an efficient algorithm for solving the endomorphism problem when W is a two-generator word. We show that when W is a two-generator word this algorithm solves the problem in time polynomial in the length of U. This result gives a polynomial-time algorithm for solving, in free groups, two-variable equations in which all the variables occur on one side of the equality and all the constants on the other side.

scholarly journals On the nonkeys

2016 ◽  
Vol 13 (1) ◽  
pp. 11-15
Author(s):  
Vũ Đức Thi
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Relation Scheme ◽  
Np Complete

In this paper we give some results about nonkeys. We show that for relation scheme the problem decide whether there is a nonkey having cardinality greater than or equal to a give integer m is NP-complete. However, for relation this problem can be solved by a polynomial time algorithm.

Maximal Monoids of Normal Form

1982 ◽  
Vol 5 (2) ◽  
pp. 129-141
Author(s):  
Mariangola Dezani-Ciancaglini ◽  
Françoise Ermine
Keyword(s):  
Normal Form ◽  
Type Theory ◽  
Normal Forms

In this paper, a characterization of the sets of normal forms which are monoids with respect to the composition combinator is obtained as application of the type theory to λ-calculus developed in [5]. The main result is that there is a monoid of normal form which is maximal in the sense that all extensions lead to terms without normal forms.

scholarly journals Arc-Completion of 2-Colored Best Match Graphs to Binary-Explainable Best Match Graphs

Algorithms ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 110
Author(s):  
David Schaller ◽  
Manuela Geiß ◽  
Marc Hellmuth ◽  
Peter F. Stadler
Keyword(s):  
Phylogenetic Tree ◽  
Polynomial Time ◽  
Binary Tree ◽  
A Priori ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Cardinality ◽  
Mathematical Phylogenetics ◽  
Special Case

Best match graphs (BMGs) are vertex-colored digraphs that naturally arise in mathematical phylogenetics to formalize the notion of evolutionary closest genes w.r.t. an a priori unknown phylogenetic tree. BMGs are explained by unique least resolved trees. We prove that the property of a rooted, leaf-colored tree to be least resolved for some BMG is preserved by the contraction of inner edges. For the special case of two-colored BMGs, this leads to a characterization of the least resolved trees (LRTs) of binary-explainable trees and a simple, polynomial-time algorithm for the minimum cardinality completion of the arc set of a BMG to reach a BMG that can be explained by a binary tree.

scholarly journals A characterization of terms of the λI-calculus having a normal form

1973 ◽  
Vol 38 (3) ◽  
pp. 441-445 ◽  
Author(s):  
Henk Barendregt
Keyword(s):  
Normal Form ◽  
Normal Forms

The theorem proved in this paper answers some transitivity questions (in the geometric sense) for the type free λ-calculus: Which objects can be mapped on all other objects? How much can an object do by applying it to other objects (see footnote 2)?The main result is that, for closed terms of the λI-calculus, the following conditions are equivalent:(a) M has a normal form.(b) FM = I for some λI-term F.(c) MN1 … Nn = I for some λI-terms N1 …, Nn.By the same method it follows that if M is a closed term of the λK-calculus having a normal form, then for some λI-terms (sic) N1, …, Nn, MN1… Nn = I is provable in the λK-calculus.The theorem of Böhm [2] states that if M1, M2 are terms of the λK-calculus having different βη-normal forms, then ∀A1, A2 ∃N1, …, NnMiN1 … Nn = Ai is provable in the λK-βη-calculus for i = 1, 2. As a consequence of this it was shown (implicitly) in [1, 3.2.20 1/2 (1)] that if M has a normal form, then for some λK-terms N1, …, Nn, MN1 … Nn =I is provable in the λK-calculus.It was not clear that this also could be proved for the λK-calculus since the proof of the theorem of Böhm essentially made use of λK-terms.We conjecture that, using the results of this paper, the full theorem of Böhm can be proved for the λI-calculus.

Normal forms for elementary patterns

2012 ◽  
Vol 77 (1) ◽  
pp. 174-194 ◽  
Author(s):  
Timothy J. Carlson ◽  
Gunnar Wilken
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Simple Characterization ◽  
The Given

AbstractA notation for an ordinal using patterns of resemblance is based on choosing an isominimal set of ordinals containing the given ordinal. There are many choices for this set meaning that notations are far from unique. We establish that among all such isominimal sets there is one which is smallest under inclusion thus providing an appropriate notion of normal form notation in this context. In addition, we calculate the elements of this isominimal set using standard notations based on collapsing functions. This provides a capstone to the results in [2, 6, 8, 9, 7], using further refinement of ordinal arithmetic developed in [8] which then both allows for a simple characterization of normal forms for patterns of order one and will play a key role in the arithmetical analysis of pure patterns of order two, [5].

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