Trichotomy for Integer Linear Systems Based on Their Sign Patterns

In this paper, we consider solving the integer linear systems, i.e., given a matrix A in R^{m*n}, a vector b in R^m, and a positive integer d, to compute an integer vector x in D^n such that Ax <= b, where m and n denote positive integers, R denotes the set of reals, and D={0,1,..., d-1}. The problem is one of the most fundamental NP-hard problems in computer science. For the problem, we propose a complexity index h which is based only on the sign pattern of A. For a real r, let ILS_=(r) denote the family of the problem instances I with h(I)=r. We then show the following trichotomy: - ILS_=(r) is linearly solvable, if r < 1, - ILS_=(r) is weakly NP-hard and pseudo-polynomially solvable, if r = 1, and - ILS_=(r) is strongly NP-hard, if r > 1. This, for example, includes the existing results that quadratic systems and Horn systems can be solved in pseudo-polynomial time

Spatial externality and indeterminacy

International audienc

Shared Integer Dichotomy

The Integer Dichotomy Diagram IDD(n) represents a natural number n ∈ N by a Directed Acyclic Graph in which equal nodes are shared to reduce the size s(n). That IDD also represents some finite set of integers by a Digital Search DAG where equal subsets are shared. The same IDD also represents representing Boolean Functions, IDDs are equivalent to (Zero-suppressed) ZDD or to (Binary Moment) BMD Decision Diagrams. The IDD data-structure and algorithms combines three standard software packages into one: arithmetics, sets and Boolean functions. Unlike the binary length l(n), the IDD size s(n) < l(n) is not monotone in n. Most integers are dense, and s(n) is near l(n). Yet, the IDD size of sparse integers can be arbitrarily smaller. We show that a single IDD software package combines many features from the best known specialized packages for operating on integers, sets, Boolean functions, and more. Over dense structures, the time/space complexity of IDD operations is proportional to that of its specialized competitors. Yet equality testing is performed in unit time with IDDs, and the complexity of some integer operations (e.g. n < m, n ± 2 m , 2 2 n ,. . .) is exponentially lower than with bit-arrays. In general, the IDD is best in class over sparse structures, where both the space and time complexities can be arbitrarily lower than those of un-shared representations. We show that sparseness is preserved by most integer operations, including arithmetic and logic operations, but excluding multiplication and division. Keywords: computer arithmetic, integer dichotomy & trichotomy, sparse & dense structures , dictionary package, digital search tree, minimal acyclic automata, binary Trie, boolean function, decision diagram, store/compute/code once.

Deductive Mathematics: an Introduction to Proof and Discovery for Mathematics Education

Mathematics has two fundamental aspects: (1) discovery/logical deduction and (2) description/ computation. Discovery/deductive mathematics asks the questions: 1. What is true about this thing being studied? 2. How do we know it is true? On the other hand, descriptive/computational mathematics asks questions of the type: 3. What is the particular number, function, and so on, that satisfies ... ? 4. How can we find the number, function, and so on? In descriptive/computational mathematics, some pictorial, physical, or business situation is described mathematically, and then computational techniques are applied to the mathematical description, in order to find values of interest. The foregoing is frequently called “problem solving”. Examples of the third question such as “How many feet of fence will be needed by a farmer to enclose ...” are familiar. The fourth type of question is answered by techniques such as solving equations, multiplying whole numbers, finding antiderivatives, substituting in formulas, and so on. The first two questions, however, are unfamiliar to most. The teaching of computational techniques continues to be the overwhelming focus of mathematics education. For most people, the techniques, and their application to real world or business problems, are mathematics. Mathematics is understood only in its descriptive role in providing a language for scientific, technical, and business areas. Mathematics, however, is really a deductive science. Mathematical knowledge comes from people looking at examples, and getting an idea of what may be true in general. Their idea is put down formally as a statement—a conjecture. The statement is then shown to be a logical consequence of what we already know. The way this is done is by logical deduction. The mathematician Jean Dieudonne has called logical deduction “the one and only true powerhouse of mathematical thinking”. Finding proofs for conjectures is also called “problem solving”. The “Problems” sections of several mathematics journals for students and teachers involve primarily problems of this type. The deductive and descriptive aspects of mathematics are complementary—not antagonistic—they motivate and enrich each other. The relation between the two aspects has been a source of wonder to thoughtful people

Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)

Oxford, UK, 26 August 200