72 research outputs found
Non-termination using Regular Languages
We describe a method for proving non-looping non-termination, that is, of
term rewriting systems that do not admit looping reductions. As certificates of
non-termination, we employ regular (tree) automata.Comment: Published at International Workshop on Termination 201
Proving Looping and Non-Looping Non-Termination by Finite Automata
A new technique is presented to prove non-termination of term rewriting. The
basic idea is to find a non-empty regular language of terms that is closed
under rewriting and does not contain normal forms. It is automated by
representing the language by a tree automaton with a fixed number of states,
and expressing the mentioned requirements in a SAT formula. Satisfiability of
this formula implies non-termination. Our approach succeeds for many examples
where all earlier techniques fail, for instance for the S-rule from combinatory
logic
Automating the mean-field method for large dynamic gossip networks
We investigate an abstraction method, called mean- field method, for the performance evaluation of dynamic net- works with pairwise communication between nodes. It allows us to evaluate systems with very large numbers of nodes, that is, systems of a size where traditional performance evaluation methods fall short.\ud
While the mean-field analysis is well-established in epidemics and for chemical reaction systems, it is rarely used for commu- nication networks because a mean-field model tends to abstract away the underlying topology.\ud
To represent topological information, however, we extend the mean-field analysis with the concept of classes of states. At the abstraction level of classes we define the network topology by means of connectivity between nodes. This enables us to encode physical node positions and model dynamic networks by allowing nodes to change their class membership whenever they make a local state transition. Based on these extensions, we derive and implement algorithms for automating a mean-field based performance evaluation
Undecidability and Finite Automata
Using a novel rewriting problem, we show that several natural decision
problems about finite automata are undecidable (i.e., recursively unsolvable).
In contrast, we also prove three related problems are decidable. We apply one
result to prove the undecidability of a related problem about k-automatic sets
of rational numbers
Regularity Preserving but not Reflecting Encodings
Encodings, that is, injective functions from words to words, have been
studied extensively in several settings. In computability theory the notion of
encoding is crucial for defining computability on arbitrary domains, as well as
for comparing the power of models of computation. In language theory much
attention has been devoted to regularity preserving functions.
A natural question arising in these contexts is: Is there a bijective
encoding such that its image function preserves regularity of languages, but
its pre-image function does not? Our main result answers this question in the
affirmative: For every countable class C of languages there exists a bijective
encoding f such that for every language L in C its image f[L] is regular.
Our construction of such encodings has several noteworthy consequences.
Firstly, anomalies arise when models of computation are compared with respect
to a known concept of implementation that is based on encodings which are not
required to be computable: Every countable decision model can be implemented,
in this sense, by finite-state automata, even via bijective encodings. Hence
deterministic finite-state automata would be equally powerful as Turing machine
deciders.
A second consequence concerns the recognizability of sets of natural numbers
via number representations and finite automata. A set of numbers is said to be
recognizable with respect to a representation if an automaton accepts the
language of representations. Our result entails that there is one number
representation with respect to which every recursive set is recognizable
Decreasing Diagrams for Confluence and Commutation
Like termination, confluence is a central property of rewrite systems. Unlike
for termination, however, there exists no known complexity hierarchy for
confluence. In this paper we investigate whether the decreasing diagrams
technique can be used to obtain such a hierarchy. The decreasing diagrams
technique is one of the strongest and most versatile methods for proving
confluence of abstract rewrite systems. It is complete for countable systems,
and it has many well-known confluence criteria as corollaries.
So what makes decreasing diagrams so powerful? In contrast to other
confluence techniques, decreasing diagrams employ a labelling of the steps with
labels from a well-founded order in order to conclude confluence of the
underlying unlabelled relation. Hence it is natural to ask how the size of the
label set influences the strength of the technique. In particular, what class
of abstract rewrite systems can be proven confluent using decreasing diagrams
restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find
that two labels suffice for proving confluence for every abstract rewrite
system having the cofinality property, thus in particular for every confluent,
countable system.
Secondly, we show that this result stands in sharp contrast to the situation
for commutation of rewrite relations, where the hierarchy does not collapse.
Thirdly, investigating the possibility of a confluence hierarchy, we
determine the first-order (non-)definability of the notion of confluence and
related properties, using techniques from finite model theory. We find that in
particular Hanf's theorem is fruitful for elegant proofs of undefinability of
properties of abstract rewrite systems
Termination of Graph Transformation Systems Using Weighted Subgraph Counting
We introduce a termination method for the algebraic graph transformation
framework PBPO+, in which we weigh objects by summing a class of weighted
morphisms targeting them. The method is well-defined in rm-adhesive
quasitoposes (which include toposes and therefore many graph categories of
interest), and is applicable to non-linear rules. The method is also defined
for other frameworks, including SqPO and left-linear DPO, because we have
previously shown that they are naturally encodable into PBPO+ in the quasitopos
setting. We have implemented our method, and the implementation includes a REPL
that can be used for guiding relative termination proofs.Comment: 36 pages. Preprint submitted to LMCS. Extends the conference version
published at the 16th International Conference on Graph Transformation (ICGT
2023
A PBPO+ Graph Rewriting Tutorial
We provide a tutorial introduction to the algebraic graph rewriting formalism
PBPO+. We show how PBPO+ can be obtained by composing a few simple building
blocks, and model the reduction rules for binary decision diagrams as an
example. Along the way, we comment on how alternative design decisions lead to
related formalisms in the literature, such as DPO. We close with a detailed
comparison with Bauderon's double pullback approach.Comment: In Proceedings TERMGRAPH 2022, arXiv:2303.1421
- …