Marco Voigt

Lower bounds for the runtime of a global multi-objective evolutionary algorithm

While for single-objective evolutionary algorithms many sharp run-time analyses exist, there are only few for multiobjective evolutionary algorithms (MOEAs), and even fewer for global MOEAs, that is, MOEAs using standard bit mutation (instead of 1-bit mutation, which is easier to analyze, but less common in practice). For example, there is not a single lower bound result for the runtime of the classic “global simple evolutionary multiobjective optimizer” (GSEMO) on the biobjective test function LeadingOnesTrailingZeros (LOTZ). An upper bound of O(n2/p), where p ≤ 1/n is the mutation probability, for this runtime was proven ten years ago by Giel (CEC 2003). In this work, we show that this bound is sharp for small values of p, namely p <; n-7/4.

On the Combination of the Bernays–Schönfinkel–Ramsey Fragment with Simple Linear Integer Arithmetic

In general, first-order predicate logic extended with linear integer arithmetic is undecidable. We show that the Bernays-Schonfinkel-Ramsey fragment (\(\exists ^* \forall ^*\)-sentences) extended with a restricted form of linear integer arithmetic is decidable via finite ground instantiation. The identified ground instances can be employed to restrict the search space of existing automated reasoning procedures considerably, e.g., when reasoning about quantified properties of array data structures formalized in Bradley, Manna, and Sipma’s array property fragment. Typically, decision procedures for the array property fragment are based on an exhaustive instantiation of universally quantified array indices with all the ground index terms that occur in the formula at hand. Our results reveal that one can get along with significantly fewer instances.

A fine-grained hierarchy of hard problems in the separated fragment

Recently, the separated fragment (SF) has been introduced and proved to be decidable. Its defining principle is that universally and existentially quantified variables may not occur together in atoms. The known upper bound on the time required to decide SFs satisfiability problem is formulated in terms of quantifier alternations: Given an SF sentence ∃z⃗∀x⃗<inf>1</inf>∃y⃗<inf>1</inf>…∀x⃗<inf>n</inf>∃y⃗<inf>n</inf>.ψ in which ψ is quantifier free, satisfiability can be decided in non-deterministic n-fold exponential time. In the present paper, we conduct a more fine-grained analysis of the complexity of SF-satisfiability. We derive an upper and a lower bound in terms of the degree ∂ of interaction of existential variables (short: degree)—a novel measure of how many separate existential quantifier blocks in a sentence are connected via joint occurrences of variables in atoms. Our main result is the k-NEXPTIME-completeness of the satisfiability problem for the set SF<inf>∂≤k</inf> of all SF sentences that have degree k or smaller. Consequently, we show that SF-satisfiability is non-elementary in general, since SF is defined without restrictions on the degree. Beyond trivial lower bounds, nothing has been known about the hardness of SF-satisfiability so far.

Deciding First-Order Satisfiability when Universal and Existential Variables are Separated

We introduce a new decidable fragment of first-order logic with equality, which strictly generalizes two already well-known ones—the Bernays–Schönfinkel–Ramsey (BSR) Fragment and the Monadic Fragment. The defining principle is the syntactic separation of universally quantified variables from existentially quantified ones at the level of atoms. Thus, our classification neither rests on restrictions on quantifier prefixes (as in the BSR case) nor on restrictions on the arity of predicate symbols (as in the monadic case). We demonstrate that the new fragment exhibits the finite model property and derive a non-elementary upper bound on the computing time required for deciding satisfiability in the new fragment. For the subfragment of prenex sentences with the quantifier prefix ∃*∀*∃* the satisfiability problem is shown to be complete for NEXPTIME. Finally, we discuss how automated reasoning procedures can take advantage of our results.

The Bernays–Schönfinkel–Ramsey Fragment with Bounded Difference Constraints over the Reals Is Decidable

First-order linear real arithmetic enriched with uninterpreted predicate symbols yields an interesting modeling language. However, satisfiability of such formulas is undecidable, even if we restrict the uninterpreted predicate symbols to arity one. In order to find decidable fragments of this language, it is necessary to restrict the expressiveness of the arithmetic part. One possible path is to confine arithmetic expressions to difference constraints of the form \(x - y \mathrel {\triangleleft }c\), where \(\mathrel {\triangleleft }\) ranges over the standard relations \( \) and x, y are universally quantified. However, it is known that combining difference constraints with uninterpreted predicate symbols yields an undecidable satisfiability problem again. In this paper, it is shown that satisfiability becomes decidable if we in addition bound the ranges of universally quantified variables. As bounded intervals over the reals still comprise infinitely many values, a trivial instantiation procedure is not sufficient to solve the problem.