349 research outputs found
Interpretations of Presburger Arithmetic in Itself
Presburger arithmetic PrA is the true theory of natural numbers with
addition. We study interpretations of PrA in itself. We prove that all
one-dimensional self-interpretations are definably isomorphic to the identity
self-interpretation. In order to prove the results we show that all linear
orders that are interpretable in (N,+) are scattered orders with the finite
Hausdorff rank and that the ranks are bounded in terms of the dimension of the
respective interpretations. From our result about self-interpretations of PrA
it follows that PrA isn't one-dimensionally interpretable in any of its finite
subtheories. We note that the latter was conjectured by A. Visser.Comment: Published in proceedings of LFCS 201
Multi-Dimensional Interpretations of Presburger Arithmetic in Itself
Presburger Arithmetic is the true theory of natural numbers with addition. We
study interpretations of Presburger Arithmetic in itself. The main result of
this paper is that all self-interpretations are definably isomorphic to the
trivial one. Here we consider interpretations that might be multi-dimensional.
We note that this resolves a conjecture by A. Visser. In order to prove the
result we show that all linear orderings that are interpretable in
are scattered orderings with the finite Hausdorff rank and
that the ranks are bounded in the terms of the dimensions of the respective
interpretations.Comment: Submitted to the JLC. arXiv admin note: text overlap with
arXiv:1709.0734
The First-Order Theory of Sets with Cardinality Constraints is Decidable
We show that the decidability of the first-order theory of the language that
combines Boolean algebras of sets of uninterpreted elements with Presburger
arithmetic operations. We thereby disprove a recent conjecture that this theory
is undecidable. Our language allows relating the cardinalities of sets to the
values of integer variables, and can distinguish finite and infinite sets. We
use quantifier elimination to show the decidability and obtain an elementary
upper bound on the complexity.
Precise program analyses can use our decidability result to verify
representation invariants of data structures that use an integer field to
represent the number of stored elements.Comment: 18 page
Foundations of Declarative Data Analysis Using Limit Datalog Programs
Motivated by applications in declarative data analysis, we study
---an extension of positive Datalog with
arithmetic functions over integers. This language is known to be undecidable,
so we propose two fragments. In
predicates are axiomatised to keep minimal/maximal numeric values, allowing us
to show that fact entailment is coNExpTime-complete in combined, and
coNP-complete in data complexity. Moreover, an additional
requirement causes the complexity to drop to ExpTime and PTime, respectively.
Finally, we show that stable can express many
useful data analysis tasks, and so our results provide a sound foundation for
the development of advanced information systems.Comment: 23 pages; full version of a paper accepted at IJCAI-17; v2 fixes some
typos and improves the acknowledgment
Tarski's influence on computer science
The influence of Alfred Tarski on computer science was indirect but
significant in a number of directions and was in certain respects fundamental.
Here surveyed is the work of Tarski on the decision procedure for algebra and
geometry, the method of elimination of quantifiers, the semantics of formal
languages, modeltheoretic preservation theorems, and algebraic logic; various
connections of each with computer science are taken up
Stratified Negation in Limit Datalog Programs
There has recently been an increasing interest in declarative data analysis,
where analytic tasks are specified using a logical language, and their
implementation and optimisation are delegated to a general-purpose query
engine. Existing declarative languages for data analysis can be formalised as
variants of logic programming equipped with arithmetic function symbols and/or
aggregation, and are typically undecidable. In prior work, the language of
was proposed, which is sufficiently powerful to
capture many analysis tasks and has decidable entailment problem. Rules in this
language, however, do not allow for negation. In this paper, we study an
extension of limit programs with stratified negation-as-failure. We show that
the additional expressive power makes reasoning computationally more demanding,
and provide tight data complexity bounds. We also identify a fragment with
tractable data complexity and sufficient expressivity to capture many relevant
tasks.Comment: 14 pages; full version of a paper accepted at IJCAI-1
Linear Temporal Logic and Propositional Schemata, Back and Forth (extended version)
This paper relates the well-known Linear Temporal Logic with the logic of
propositional schemata introduced by the authors. We prove that LTL is
equivalent to a class of schemata in the sense that polynomial-time reductions
exist from one logic to the other. Some consequences about complexity are
given. We report about first experiments and the consequences about possible
improvements in existing implementations are analyzed.Comment: Extended version of a paper submitted at TIME 2011: contains proofs,
additional examples & figures, additional comparison between classical
LTL/schemata algorithms up to the provided translations, and an example of
how to do model checking with schemata; 36 pages, 8 figure
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