The theory of lattices, partially ordered sets all subsets of which have suprema and infima inside the set.
Questions tagged [lattices]
40 questions
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What are lattices used for?
Wikipedia says:
Complete lattices appear in many applications in mathematics and computer science
Is it just referring to the fact that the standard Boolean algebra used in computation is a complete lattice? Is there anything we gain by working at…
Xodarap
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Is there a data-structure for semilattices similar to a tree data-structure?
If we regard a tree as a partial ordered set, it becomes a special case of a join-semilattice. For a join-semilattice, we want to be able to compute the (unique) least upper bound of two elements (more or less) efficiently. In the case of a tree, a…
Thomas Klimpel
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Algorithm to generate self-avoiding random walk on a lattice
Where can I find some code to generate random self-avoiding walks on 2 and 3-dimensional lattices whose side-lengths are powers of two? The walk should pass through every point on the lattice More specifically, how can I find a random hamiltonian…
Alecto
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Bisimulation and the Knaster–Tarski theorem: What does the least fixed point mean?
Given a suitable lattice and a monotonic function $F$, we can compute the bisimilarity of a labeled transition system (its greatest bisimulation) by computing the greatest fixed point of $F$ using Knaster–Tarski. I was wondering if the least fixed…
Karla
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Testing if a given DAG is a lattice
I am given a directed acyclic graph (DAG) with a unique source and sink. Is there an efficient way to test whether the partial order represented by this graph is a lattice?
In other words, I need to test whether any two vertices have a unique least…
Mangara
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Correctness of a zigzag algorithm to find the most similar vector in a bounded integer lattice
I am currently working on an integer lattice problem, called the "most similar vector problem," and wondering if can be solved correctly by a simple "zig-zagging" algorithm.
Given a real vector $u \in \mathbb{R}^n$, the most similar vector problem…
Berk U.
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What does Tarski's Fixed-Point theorem give us that that Y-Combinator does't
I'm taking a graduate course on the theory of functional programming, based on Paul Taylor's "Practical Foundations of Mathematics." I understand the statement of Tarski's theorem about how for any $\omega$-compelte poset $X$, and any…
Joey Eremondi
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Closure operator and set of fixpoint
In chapter 2.2 of
Giacobazzi, Roberto; Ranzato, Francesco, Uniform closures: Order-theoretically reconstructing logic program semantics and abstract domain refinements, Inf. Comput. 145, No.2, 153-190 (1998). ZBL0921.68057.
it's said:
An (upper)…
incud
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Finding optimal set for a sum of a product function over a 2D lattice
Given a 2D lattice with coordinates $1 \leq x \leq c$ and $1 \leq y \leq d$, we define $f(x, y) = xy$.
We wish to find a boolean function $I(x,y)$ that determines in $O(1)$ time whether or not $(x,y)$ belongs to the set of points $S$ of size $k$…
Alexandre
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Relation between Lattice and Boolean Algebra
In discrete math, I have read that lattice is a generalized form of boolean lattice. But those places where boolean algebra is mentioned, they don't tell about lattices (digital logic, binary,...). Whether the meet and join is same as and and or in…
user5507
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"Most Similar Vector Problem" on an Integer Lattice
I am currently working on problem that I think could be expressed as an integer lattice problem, and hoping to find some guidance on this forum.
Given $u \in \mathbb{R}^n$ and a bounded integer lattice $L = \mathbb{Z}^n \cap [-M,M]^n$, I would like…
Berk U.
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Halting Problem and Turing Degree and Reduction?
This is a Local Olympiad question on computation and computer science on 2013. How can explain it and says some hint for understanding such an example question.
for $ A \subseteq \mathbb{N}$ we have $a=deg_T(A)=\{B | B \equiv_T A \} $ and…
Maryam Panahi
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Unranking paths in a graph/lattice
A ranking algorithm determines the position (or rank) of a combinatorial object among all the objects (with respect to a given order); an unranking algorithm finds the object having a specified rank. Thus, ranking and unranking can be considered as…
Jean-Baptiste
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Finding lattice points with fewest non-zero terms
Finding a short point on a lattice is hard, but I was wondering if finding a vector on the lattice which the fewest non-zero terms was easier?
Are there any good approximations like LLL, but which produce lattice points with lots of zeros?
yberman
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Greatest fixpoint of the type of lists
I'm working through Samuel Mimram's book Program = Proof. In the first chapter, he discusses recursive types in OCaml, and inductive types. An exercise he provides on the topic has me a little bit confused. In the exercise, he asks the reader to…
Finley
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