Patrick Cégielski

Decidability of the theory of the natural integers with the cantor pairing function and the successor

Abstract The binary Cantor pairing function C from N × N into N is defined by C(x,y) = ( 1 2 )(x+y)(x+y+1)+y . We consider the theory of natural integers equipped with the Cantor pairing function and an extra relation or function X on N . When X is equal either to multiplication, or coprimeness, or divisibility, or addition or natural ordering, it can be proved that the theory Th( N ,C,X) is undecidable. Let S be the successor function. We provide an algorithm solving the decision problem for Th( N ,C,S) .

Window subsequence problems for compressed texts

Given two strings (a text t of length n and a pattern p) and a natural number w, window subsequence problems consist in deciding whether p occurs as a subsequence of t and/or finding the number of size (at most) w windows of text t which contain pattern p as a subsequence, i.e. the letters of pattern p occur in the text window, in the same order as in p, but not necessarily consecutively (they may be interleaved with other letters). We are searching for subsequences in a text which is compressed using Lempel-Ziv-like compression algorithms, without decompressing the text, and we would like our algorithms to be almost optimal, in the sense that they run in time O(m) where m is the size of the compressed text. The pattern is uncompressed (because the compression algorithms are evolutive: various occurrences of a same pattern look different in the text).

On arithmetical first-order theories allowing encoding and decoding of lists

Abstract In Computer Science, n -tuples and lists are usual tools; we investigate both notions in the framework of first-order logic within the set of nonnegative integers. Godel had firstly shown that the objects which can be defined by primitive recursion schema, can also be defined at first-order, using natural order and some coding devices for lists. Second he had proved that this encoding can be defined from addition and multiplication. We show this can be also done with addition and a weaker predicate, namely the coprimeness predicate. The theory of integers equipped with a pairing function can be decidable or not. The theory of decoding of lists (under some natural condition) is always undecidable. We distinguish the notions encoding of n -tuples and encoding of lists via some properties of decidability-undecidability. At last, we prove it is possible in some structure to encode lists although neither addition nor multiplication are definable in this structure.

Window-accumulated subsequence matching problem is linear

Abstract Given two strings, text t of length n , and pattern p = p 1 …p k of length k , and given a natural number w , the subsequence matching problem consists in finding the number of size w windows of text t which contain pattern p as a subsequence, i.e. the letters p 1 ,…,p k occur in the window, in the same order as in p , but not necessarily consecutively (they may be interleaved with other letters). Subsequence matching is used for finding frequent patterns and association rules in databases. We generalize the Knuth–Morris–Pratt (KMP) pattern matching algorithm; we define a non-conventional kind of RAM, the MP-RAMs which model more closely the microprocessor operations; we design an O (n) on-line algorithm for solving the subsequence matching problem on MP-RAMs.

Definability, decidability, complexity

We survey how the definability problem in first-order logic was born and the relations between this problem and the question of decidability of logical theories. We also show present connections between definability and the important theoretical problems of computational complexity.

Weakly maximal decidable structures

We prove that there exists a structure M whose monadic second order theory is decidable, and such that the first-order theory of every expansion of M by a constant is undecidable.

Newton representation of functions over natural integers having integral difference ratios

Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying f(a) - f(b) ≡ 0 (mod (a - b)) for all a > b. We characterize this class of functions via their representations as Newton series. This class, which obviously contains all polynomials with integral coefficients, also contains unexpected functions, for instance, all functions x ↦ ⌊e1/a ax x!⌋, with a ∈ ℤ\{0, 1}, and a function equal to ⌊e x!⌋ except on 0. Finally, to study the complement class, we look at functions ℕ → ℝ which are not uniformly close to any function having integral difference ratios.

On lattices of regular sets of natural integers closed under decrementation

We consider lattices of regular sets of non negative integers, i.e. of sets definable in Presburger arithmetic. We prove that if such a lattice is closed under decrement then it is also closed under many other functions: quotients by an integer, roots, etc. We characterize the family of such functions. We study lattices of regular subsets of N closed under decrement.Lattices of regular subsets of N closed under decrement are closed under quotient.We characterize functions such that closure under decrement yields closure under f - 1 .

Modeles Recursivement Satures De L'addition Et De La Multiplication Des Entiers Naturels

IΣo denotes the subtheory of first–order Peano arithmetic obtained by restricting the induction schema to formulae with only bounded quantifiers. Let EXP denote the corresponding theory obtained by adding to the language a function symbol to denote exponentiation. Let PT denote the naturally axiomatized theory (“Peano with top”) corresponding to the structures (n,+,.) for n a natural number. We show that the restriction to addition of a non-standard model of IΣo or of PT and the restriction to multiplication of a segment of a model of EXP closed under xlog x are both recursively saturated. Certain other results concerning PT are included in section III.

The Elementary Theory of the Natural Lattice Is Finitely Axiomatizable

Caracterisation du treillis naturel (ensemble des entiers positifs avec la relation de divisibilite)