12 research outputs found
On Functions Weakly Computable by Pushdown Petri Nets and Related Systems
We consider numerical functions weakly computable by grammar-controlled
vector addition systems (GVASes, a variant of pushdown Petri nets). GVASes can
weakly compute all fast growing functions for
, hence they are computationally more powerful than
standard vector addition systems. On the other hand they cannot weakly compute
the inverses or indeed any sublinear function. The proof relies
on a pumping lemma for runs of GVASes that is of independent interest
Reasoning about Data Repetitions with Counter Systems
We study linear-time temporal logics interpreted over data words with
multiple attributes. We restrict the atomic formulas to equalities of attribute
values in successive positions and to repetitions of attribute values in the
future or past. We demonstrate correspondences between satisfiability problems
for logics and reachability-like decision problems for counter systems. We show
that allowing/disallowing atomic formulas expressing repetitions of values in
the past corresponds to the reachability/coverability problem in Petri nets.
This gives us 2EXPSPACE upper bounds for several satisfiability problems. We
prove matching lower bounds by reduction from a reachability problem for a
newly introduced class of counter systems. This new class is a succinct version
of vector addition systems with states in which counters are accessed via
pointers, a potentially useful feature in other contexts. We strengthen further
the correspondences between data logics and counter systems by characterizing
the complexity of fragments, extensions and variants of the logic. For
instance, we precisely characterize the relationship between the number of
attributes allowed in the logic and the number of counters needed in the
counter system.Comment: 54 page
Alternating Vector Addition Systems with States
International audienceAlternating vector addition systems are obtained by equipping vector addition systems with states (VASS) with 'fork' rules, and provide a natural setting for infinite-arena games played over a VASS. Initially introduced in the study of propositional linear logic, they have more recently gathered attention in the guise of multi-dimensional energy games for quantitative verification and synthesis. We show that establishing who is the winner in such a game with a state reachability objective is 2-ExpTime-complete. As a further application, we show that the same complexity result applies to the problem of whether a VASS is simulated by a finite-state system
Constructive Decision via Redundancy-Free Proof-Search
International audienceWe give a constructive account of Kripke-Curry's method which was used to establish the decidability of Implicational Relevance Logic (R →). To sustain our approach, we mechanize this method in axiom-free Coq, abstracting away from the specific features of R → to keep only the essential ingredients of the technique. In particular we show how to replace Kripke/Dickson's lemma by a constructive form of Ramsey's theorem based on the notion of almost full relation. We also explain how to replace König's lemma with an inductive form of Brouwer's Fan theorem. We instantiate our abstract proof to get a constructive decision procedure for R → and discuss potential applications to other logical decidability problems
On the complexity of resource-bounded logics
We revisit decidability results for resource-bounded logics and use decision problems on vector addition systems with states (VASS) in order to establish complexity characterisations of (decidable) model checking problems. We show that the model checking problem for the logic RB+-ATL is 2EXPTIME-complete by using recent results on alternating VASS (and in EXPTIME when the number of resources is bounded). Moreover, we establish that the model checking problem for RBTL is EXPSPACE-complete. The problem is decidable and of the same complexity for RBTL*, proving a new decidability result as a by-product of the approach. When the number of resources is bounded, the problem is in PSPACE. We also establish that the model checking problem for RB+-ATL*, the extension of RB+-ATL with arbitrary path formulae, is decidable by a reduction to parity games for single-sided VASS (a variant of alternating VASS). Furthermore, we are able to synthesise values for resource parameters. Hence, the paper establishes formal correspondences between model checking problems for resource bounded logics advocated in the AI literature and decision problems on alternating VASS, paving the way for more applications and cross-fertilizations
Decidable Models of Recursive Asynchronous Concurrency
Asynchronously communicating pushdown systems (ACPS) that satisfy the
empty-stack constraint (a pushdown process may receive only when its stack is
empty) are a popular decidable model for recursive programs with asynchronous
atomic procedure calls. We study a relaxation of the empty-stack constraint for
ACPS that permits concurrency and communication actions at any stack height,
called the shaped stack constraint, thus enabling a larger class of concurrent
programs to be modelled. We establish a close connection between ACPS with
shaped stacks and a novel extension of Petri nets: Nets with Nested Coloured
Tokens (NNCTs). Tokens in NNCTs are of two types: simple and complex. Complex
tokens carry an arbitrary number of coloured tokens. The rules of NNCT can
synchronise complex and simple tokens, inject coloured tokens into a complex
token, and eject all tokens of a specified set of colours to predefined places.
We show that the coverability problem for NNCTs is Tower-complete. To our
knowledge, NNCT is the first extension of Petri nets, in the class of nets with
an infinite set of token types, that has primitive recursive coverability. This
result implies Tower-completeness of coverability for ACPS with shaped stacks
The covering and boundedness problems for branching vector addition systems
The covering and boundedness problems for branching vector addition systems are shown complete for doubly-exponential time