108,909 research outputs found
Automatic Termination Analysis of Programs Containing Arithmetic Predicates
For logic programs with arithmetic predicates, showing termination is not
easy, since the usual order for the integers is not well-founded. A new method,
easily incorporated in the TermiLog system for automatic termination analysis,
is presented for showing termination in this case.
The method consists of the following steps: First, a finite abstract domain
for representing the range of integers is deduced automatically. Based on this
abstraction, abstract interpretation is applied to the program. The result is a
finite number of atoms abstracting answers to queries which are used to extend
the technique of query-mapping pairs. For each query-mapping pair that is
potentially non-terminating, a bounded (integer-valued) termination function is
guessed. If traversing the pair decreases the value of the termination
function, then termination is established. Simple functions often suffice for
each query-mapping pair, and that gives our approach an edge over the classical
approach of using a single termination function for all loops, which must
inevitably be more complicated and harder to guess automatically. It is worth
noting that the termination of McCarthy's 91 function can be shown
automatically using our method.
In summary, the proposed approach is based on combining a finite abstraction
of the integers with the technique of the query-mapping pairs, and is
essentially capable of dividing a termination proof into several cases, such
that a simple termination function suffices for each case. Consequently, the
whole process of proving termination can be done automatically in the framework
of TermiLog and similar systems.Comment: Appeared also in Electronic Notes in Computer Science vol. 3
A Simple and Scalable Static Analysis for Bound Analysis and Amortized Complexity Analysis
We present the first scalable bound analysis that achieves amortized
complexity analysis. In contrast to earlier work, our bound analysis is not
based on general purpose reasoners such as abstract interpreters, software
model checkers or computer algebra tools. Rather, we derive bounds directly
from abstract program models, which we obtain from programs by comparatively
simple invariant generation and symbolic execution techniques. As a result, we
obtain an analysis that is more predictable and more scalable than earlier
approaches. Our experiments demonstrate that our analysis is fast and at the
same time able to compute bounds for challenging loops in a large real-world
benchmark. Technically, our approach is based on lossy vector addition systems
(VASS). Our bound analysis first computes a lexicographic ranking function that
proves the termination of a VASS, and then derives a bound from this ranking
function. Our methodology achieves amortized analysis based on a new insight
how lexicographic ranking functions can be used for bound analysis
Fifty years of Hoare's Logic
We present a history of Hoare's logic.Comment: 79 pages. To appear in Formal Aspects of Computin
The Hardness of Finding Linear Ranking Functions for Lasso Programs
Finding whether a linear-constraint loop has a linear ranking function is an
important key to understanding the loop behavior, proving its termination and
establishing iteration bounds. If no preconditions are provided, the decision
problem is known to be in coNP when variables range over the integers and in
PTIME for the rational numbers, or real numbers. Here we show that deciding
whether a linear-constraint loop with a precondition, specifically with
partially-specified input, has a linear ranking function is EXPSPACE-hard over
the integers, and PSPACE-hard over the rationals. The precise complexity of
these decision problems is yet unknown. The EXPSPACE lower bound is derived
from the reachability problem for Petri nets (equivalently, Vector Addition
Systems), and possibly indicates an even stronger lower bound (subject to open
problems in VAS theory). The lower bound for the rationals follows from a novel
simulation of Boolean programs. Lower bounds are also given for the problem of
deciding if a linear ranking-function supported by a particular form of
inductive invariant exists. For loops over integers, the problem is PSPACE-hard
for convex polyhedral invariants and EXPSPACE-hard for downward-closed sets of
natural numbers as invariants.Comment: In Proceedings GandALF 2014, arXiv:1408.5560. I thank the organizers
of the Dagstuhl Seminar 14141, "Reachability Problems for Infinite-State
Systems", for the opportunity to present an early draft of this wor
A General Framework for Automatic Termination Analysis of Logic Programs
This paper describes a general framework for automatic termination analysis
of logic programs, where we understand by ``termination'' the finitenes s of
the LD-tree constructed for the program and a given query. A general property
of mappings from a certain subset of the branches of an infinite LD-tree into a
finite set is proved. From this result several termination theorems are
derived, by using different finite sets. The first two are formulated for the
predicate dependency and atom dependency graphs. Then a general result for the
case of the query-mapping pairs relevant to a program is proved (cf.
\cite{Sagiv,Lindenstrauss:Sagiv}). The correctness of the {\em TermiLog} system
described in \cite{Lindenstrauss:Sagiv:Serebrenik} follows from it. In this
system it is not possible to prove termination for programs involving
arithmetic predicates, since the usual order for the integers is not
well-founded. A new method, which can be easily incorporated in {\em TermiLog}
or similar systems, is presented, which makes it possible to prove termination
for programs involving arithmetic predicates. It is based on combining a finite
abstraction of the integers with the technique of the query-mapping pairs, and
is essentially capable of dividing a termination proof into several cases, such
that a simple termination function suffices for each case. Finally several
possible extensions are outlined
Homeomorphic Embedding for Online Termination of Symbolic Methods
Well-quasi orders in general, and homeomorphic embedding in particular, have gained popularity to ensure the termination of techniques for program analysis, specialisation, transformation, and verification. In this paper we survey and discuss this use of homeomorphic embedding and clarify the advantages of such an approach over one using well-founded orders. We also discuss various extensions of the homeomorphic embedding relation. We conclude with a study of homeomorphic embedding in the context of metaprogramming, presenting some new (positive and negative) results and open problems
Polytool: polynomial interpretations as a basis for termination analysis of Logic programs
Our goal is to study the feasibility of porting termination analysis
techniques developed for one programming paradigm to another paradigm. In this
paper, we show how to adapt termination analysis techniques based on polynomial
interpretations - very well known in the context of term rewrite systems (TRSs)
- to obtain new (non-transformational) ter- mination analysis techniques for
definite logic programs (LPs). This leads to an approach that can be seen as a
direct generalization of the traditional techniques in termination analysis of
LPs, where linear norms and level mappings are used. Our extension general-
izes these to arbitrary polynomials. We extend a number of standard concepts
and results on termination analysis to the context of polynomial
interpretations. We also propose a constraint-based approach for automatically
generating polynomial interpretations that satisfy the termination conditions.
Based on this approach, we implemented a new tool, called Polytool, for
automatic termination analysis of LPs
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