Periodic behavior in families of numerical and affine semigroups via parametric Presburger arithmetic

Parametric Presburger Arithmetic: Logic, Combinatorics, and Quasi-polynomial Behavior

Parametric Presburger arithmetic concerns families of sets St ⊆Zd , for t ∈N, that are defined using addition, inequalities, constants in Z, Boolean operations, multiplication by t, and quantifiers on variables ranging over Z. That is, such families are defined using quantifiers and Boolean combinations of formulas of the form a(t) ·x≤ b(t), where a(t) ∈ Z[t]d ,b(t) ∈ Z[t]. A function g : N→ Z ...

Bounding Quantification in Parametric Expansions of Presburger Arithmetic

Generalizing Cooper’s method of quantifier elimination for Presburger arithmetic, we give a new proof that all parametric Presburger families {St : t ∈ N} (as defined by Woods in [8]) are definable by formulas with polynomially bounded quantifiers in an expanded language with predicates for divisibility by f(t) for every polynomial f ∈ Z[t]. In fact, this quantifier bounding method works more g...

Parametric Presburger Arithmetic: Complexity of Counting and Quantifier Elimination

We consider an expansion of Presburger arithmetic which allows multiplication by k parameters t1, . . . , tk. A formula in this language defines a parametric set St ⊆ Z d as t varies in Z, and we examine the counting function |St| as a function of t. For a single parameter, it is known that |St| can be expressed as an eventual quasi-polynomial (there is a period m such that, for sufficiently la...

Durations, Parametric Model-Checking in Timed Automata with Presburger Arithmetic

Invariants of Generalized Arithmetic Numerical Semigroups

A generalized arithemtic numerical monoid is of the form S = 〈a, ah+d, ah+2d, . . . , ah+ xd〉 where the gcd(a, d) = 1 and a > x. Much is known for the arithmetic numerical monoid, when h = 1, due to known information for that specific monoid’s length set. Therefore, this paper will explore various invariants of the generalized arithmetic numerical monoid.