Decidability Results Research Articles

The Bernays-Schönfinkel-Ramsey class for set theory: decidability

AbstractAs proved recently, the satisfaction problem for all prenex formulae in the set-theoretic Bernays-Shönfinkel-Ramsey class is semi-decidable over von Neumann's cumulative hierarchy. Here that semi-decidability result is strengthened into a decidability result for the same collection of formulae.

Interrogative Belief Revision Based on Epistemic Strategies

I develop a dynamic logic for reasoning about "interrogative belief revision", a new branch of belief revision theory that has been developed in a small number of papers, beginning with E. J. Olsson and D. Westlund's paper "On the role of the research agenda in epistemic change" [12]. In interrogative belief revision, epistemic states are taken to include a research agenda, consisting of questions the agent seeks to answer. I present a logic for revision of such epistemic states based on the notion of an epistemic strategy, a stable plan of action that determines changes in the agent's research agenda. This idea is a further development of an idea put forward in [6], that changes in the research agenda of an agent should be determined by stable, "long term" research interests. I provide complete axioms and a decidability result for the logic.

Regular set of representatives for time-constrained MSC graphs

Systems involving both time and concurrency are notoriously difficult to analyze. Existing decidability results apply when clocks on different processes cannot be compared or when the set of timed executions is regular. We prove new decidability results for timed concurrent systems, requiring neither restriction. We consider the formalism of time-constrained MSC graphs (TC-MSC graphs for short) introduced in Akshay et al. (2007) [1], and study if the set of timed executions generated by a TC-MSC graph is empty. This emptiness problem is known to be undecidable in general (Gastin et al., 2009) [11]. Our approach consists of finding a regular set R of representative timed executions, i.e., such that every timed execution of the system has an equivalent, up to commutation, timed execution in R. This allows us to solve the emptiness problem under the assumption that the TC-MSC graph is prohibited from (1) forcing any basic scenario to take an arbitrarily long time to complete and (2) enforcing unboundedly many events to occur within one unit of time.

Decidability Results for Soundness Criteria of Resource-Constrained Workflow Nets

This paper focuses on the decidability status of various forms of behavioral correctness criteria for resource-constrained workflow (RCWF) nets (Petri net models of RCWF systems). These behavioral correctness criteria, usually called soundness criteria, are natural extensions of similar correctness criteria for workflow nets (Petri net models of workflow systems). While all forms of soundness are known to be decidable for workflow nets, only soundness for RCWF nets with just one resource type is known to be decidable. In this paper, we show that if we limit the number of cases, then soundness for RCWF nets with arbitrarily many resource types is decidable. Moreover, we show that some “intermediate” forms of soundness, as well as a restrictive form of structural soundness for RCWF nets, are decidable too. The proof technique is based on instantiation nets as a general tool for dealing with arbitrarily many cases and arbitrarily large resources in workflow nets and RCWF nets. It is also shown why this technique cannot be extended to the most general form of soundness.

On the Computational Complexity of Matrix Semigroup Problems

Most computational problems for matrix semigroups and groups are inherently difficult to solve and even undecidable starting from dimension three. The questions about the decidability and complexity of problems for two-dimensional matrix semigroups remain open and are directly linked with other challenging problems in the field. In this paper we study the computational complexity of the problem of determining whether the identity matrix belongs to a matrix semigroup (the Identity Problem) generated by a finite set of 2 × 2 integral unimodular matrices. The Identity Problem for matrix semigroups is a well-known challenging problem, which has remained open in any dimension until recently. It is currently known that the problem is decidable in dimension two and undecidable starting from dimension four. In particular, we show that the Identity Problem for 2 × 2 integral unimodular matrices is NP-hard by a reduction of the Subset Sum Problem and several new encoding techniques. An upper bound for the nontrivial decidability result by C. Choffrut and J. Karhumaki is unknown. However, we derive a lower bound on the minimum length solution to the Identity Problem for a constructible set of instances, which is exponential in the number of matrices of the generator set and the maximal element of the matrices. This shows that the most obvious candidate for an NP algorithm, which is to guess the shortest sequence of matrices which multiply to give the identity matrix, does not work correctly since the certificate would have a length which is exponential in the size of the instance. Both results lead to a number of corollaries confirming the same bounds for vector reachability, scalar reachability and zero in the right upper corner problems.

Conservative Elementary Object Systems

This contribution presents decidability results for the formalism of Elementary Object Systems EOS. Object nets are Petri nets which have Petri nets as tokens --an approach known as the nets-within-nets paradigm. In this paper we study the relationship of the reachability and the liveness problem. We prove that both problems are undecidable for EOS even for the subclass of conservative EOS while it is well known that both are decidable for classical p/t nets. Despite these undecidability results, boundedness can be decided for conservative EOS using a monotonicity argument similar to that for p/t nets.

Reasoning with Forest Logic Programs and f-hybrid knowledge bases

AbstractOpen Answer Set Programming (OASP) is an undecidable framework for integrating ontologies and rules. Although several decidable fragments of OASP have been identified, few reasoning procedures exist. In this paper, we provide a sound, complete, and terminating algorithm for satisfiability checking w.r.t. Forest Logic Programs (FoLPs), a fragment of OASP where rules have a tree shape and allow for inequality atoms and constants. The algorithm establishes a decidability result for FoLPs. Although believed to be decidable, so far only the decidability for two small subsets of FoLPs, local FoLPs and acyclic FoLPs, has been shown. We further introduce f-hybrid knowledge bases, a hybrid framework whereknowledge bases and FoLPs coexist, and we show that reasoning with such knowledge bases can be reduced to reasoning with FoLPs only. We note that f-hybrid knowledge bases do not require the usual (weakly) DL-safety of the rule component, thus providing a genuine alternative approach to current integration approaches of ontologies and rules.

Paraconsistent Computation Tree Logic

It is known that paraconsistent logical systems are more appropriate for inconsistency-tolerant and uncertainty reasoning than other types of logical systems. In this paper, a paraconsistent computation tree logic, PCTL, is obtained by adding paraconsistent negation to the standard computation tree logic CTL. PCTL can be used to appropriately formalize inconsistency-tolerant temporal reasoning. A theorem for embedding PCTL into CTL is proved. The validity, satisfiability, and model-checking problems of PCTL are shown to be decidable. The embedding and decidability results indicate that we can reuse the existing CTL-based algorithms for validity, satisfiability, and model-checking. An illustrative example of medical reasoning involving the use of PCTL is presented.

A Decidable Extension of Data Automata

Data automata on data words is a decidable model proposed by Boja\'nczyk et al. in 2006. Class automata, introduced recently by Boja\'nczyk and Lasota, is an extension of data automata which unifies different automata models on data words. The nonemptiness of class automata is undecidable, since class automata can simulate two-counter machines. In this paper, a decidable model called class automata with priority class condition, which restricts class automata but strictly extends data automata, is proposed. The decidability of this model is obtained by establishing a correspondence with priority multicounter automata. This correspondence also completes the picture of the links between various class conditions of class automata and various models of counter machines. Moreover, this model is applied to extend a decidability result of Alur, Cern\'y and Weinstein on the algorithmic analysis of array-accessing programs.

Containment of Conjunctive Queries on Annotated Relations

We study containment and equivalence of (unions of) conjunctive queries on relations annotated with elements of a commutative semiring. Such relations and the semantics of positive relational queries on them were introduced in a recent paper as a generalization of set semantics, bag semantics, incomplete databases, and databases annotated with various kinds of provenance information. We obtain positive decidability results and complexity characterizations for databases with lineage, why-provenance, and provenance polynomial annotations, for both conjunctive queries and unions of conjunctive queries. At least one of these results is surprising given that provenance polynomial annotations seem “more expressive” than bag semantics and under the latter, containment of unions of conjunctive queries is known to be undecidable. The decision procedures rely on interesting variations on the notion of containment mappings. We also show that for any positive semiring (a very large class) and conjunctive queries without self-joins, equivalence is the same as isomorphism.

View-based query answering in Description Logics: Semantics and complexity

View-based query answering is the problem of answering a query based only on the precomputed answers to a set of views. While this problem has been widely investigated in databases, it is largely unexplored in the context of Description Logic ontologies. Differently from traditional databases, Description Logics may express several forms of incomplete information, and this poses challenging problems in characterizing the semantics of views. In this paper, we first present a general framework for view-based query answering, where we address the above semantical problems by providing two notions of view-based query answering over ontologies, all based on the idea that the precomputed answers to views are the certain answers to the corresponding queries. We also relate such notions to privacy-aware access to ontologies. Then, we provide decidability results, algorithms, and data complexity characterizations for view-based query answering in several Description Logics, ranging from those with limited modeling capability to highly expressive ones.

The tree width of auxiliary storage

We propose a generalization of results on the decidability of emptiness for several restricted classes of sequential and distributed automata with auxiliary storage (stacks, queues) that have recently been proved. Our generalization relies on reducing emptiness of these automata to finite-state graph automata (without storage) restricted to monadic second-order (MSO) definable graphs of bounded tree-width, where the graph structure encodes the mechanism provided by the auxiliary storage. Our results outline a uniform mechanism to derive emptiness algorithms for automata, explaining and simplifying several existing results, as well as proving new decidability results.

Average-price-per-reward games on hybrid automata with strong resets

We study price-per-reward games on hybrid automata with strong resets. They generalise average-price games previously studied and have applications in scheduling. We obtain decidability results by a translation to a novel class of finite graphs with price and reward information, and games assigned to edges. The cost and reward of following an edge are determined by the outcome of the edge game that is assigned to it.

Block insertion and deletion on trajectories

In this paper, we introduce block insertion and deletion on trajectories, which provide us with a new framework to study properties of language operations. With the parallel syntactical constraint provided by trajectories, these operations properly generalize several sequential as well as parallel binary language operations such as catenation, sequential insertion, k -insertion, parallel insertion, quotient, sequential deletion, k -deletion, etc. We establish some relationships between the new operations and shuffle and deletion on trajectories, and obtain several closure properties of the families of regular and context-free languages under the new operations. Moreover, we obtain several decidability results of three types of language equation problems which involve the new operations. The first one is to answer, given languages L 1 , L 2 , L 3 and a trajectory set T , whether the result of an operation between L 1 and L 2 on the trajectory set T is equal to L 3 . The second one is to answer, for three given languages L 1 , L 2 , L 3 , whether there exists a set of trajectories such that the block insertion or deletion between L 1 and L 2 on this trajectory set is equal to L 3 . The third problem is similar to the second one, but the language L 1 is unknown while languages L 2 , L 3 as well as a trajectory set T are given.

Decidability and Combination Results for Two Notions of Knowledge in Security Protocols

In formal approaches, messages sent over a network are usually modeled by terms together with an equational theory, axiomatizing the properties of the cryptographic functions (encryption, exclusive or, ...). The analysis of cryptographic protocols requires a precise understanding of the attacker knowledge. Two standard notions are usually considered: deducibility and indistinguishability. Those notions are well-studied and several decidability results already exist to deal with a variety of equational theories. Most of the existing results are dedicated to specific equational theories and only few results, especially in the case of indistinguishability, have been obtained for equational theories with associative and commutative properties $(\textsf{AC})$ . In this paper, we show that existing decidability results can be easily combined for any disjoint equational theories: if the deducibility and indistinguishability relations are decidable for two disjoint theories, they are also decidable for their union. We also propose a general setting for solving deducibility and indistinguishability for an important class (called monoidal) of equational theories involving $\textsf{AC}$ operators. As a consequence of these two results, new decidability and complexity results can be obtained for many relevant equational theories.

Nominals, Inverses, Counting, and Conjunctive Queries or: Why Infinity is your Friend!

Description Logics are knowledge representation formalisms that provide, for example, the logical underpinning of the W3C OWL standards. Conjunctive queries, the standard query language in databases, have recently gained significant attention as an expressive formalism for querying Description Logic knowledge bases. Several different techniques for deciding conjunctive query entailment are available for a wide range of DLs. Nevertheless, the combination of nominals, inverse roles, and number restrictions in OWL 1 and OWL 2 DL causes unsolvable problems for the techniques hitherto available. We tackle this problem and present a decidability result for entailment of unions of conjunctive queries in the DL ALCHOIQb that contains all three problematic constructors simultaneously. Provided that queries contain only simple roles, our result also shows decidability of entailment of (unions of) conjunctive queries in the logic that underpins OWL 1 DL and we believe that the presented results will pave the way for further progress towards conjunctive query entailment decision procedures for the Description Logics underlying the OWL standards.

A decidability result for the dominating set problem

We study the following question: given a finite collection of graphs G1,…,Gk, is the dominating set problem polynomial-time solvable in the class of (G1,…,Gk)-free graphs? In this paper, we prove the existence of an efficient algorithm that answers this question for k=2.

Computing Knowledge in Security Protocols Under Convergent Equational Theories

The analysis of security protocols requires reasoning about the knowledge an attacker acquires by eavesdropping on network traffic. In formal approaches, the messages exchanged over the network are modelled by a term algebra equipped with an equational theory axiomatising the properties of the cryptographic primitives (e.g. encryption, signature). In this context, two classical notions of knowledge, deducibility and indistinguishability, yield corresponding decision problems. We propose a procedure for both problems under arbitrary convergent equational theories. Since the underlying problems are undecidable we cannot guarantee termination. Nevertheless, our procedure terminates on a wide range of equational theories. In particular, we obtain a new decidability result for a theory we encountered when studying electronic voting protocols. We also provide a prototype implementation.

Termination of Rewriting with Right-Flat Rules Modulo Permutative Theories

We present decidability results for termination of classes of term rewriting systems modulo permutative theories. Termination and innermost termination modulo permutative theories are shown to be decidable for term rewrite systems (TRS) whose right-hand side terms are restricted to be shallow (variables occur at depth at most one) and linear (each variable occurs at most once). Innermost termination modulo permutative theories is also shown to be decidable for shallow TRS. We first show that a shallow TRS can be transformed into a flat (only variables and constants occur at depth one) TRS while preserving termination and innermost termination. The decidability results are then proved by showing that (a) for right-flat right-linear (flat) TRS, non-termination (respectively, innermost non-termination) implies non-termination starting from flat terms, and (b) for right-flat TRS, the existence of non-terminating derivations starting from a given term is decidable. On the negative side, we show PSPACE-hardness of termination and innermost termination for shallow right-linear TRS, and undecidability of termination for flat TRS.

A Decidability Result for the Model Checking of Infinite-State Systems

We present a decidability result for the model checking of a certain class of properties that can be conveniently expressed as ground formulae of a first-order temporal fragment. The decidability result is obtained by importing into the context of model-checking problems some techniques developed for the combination of decision procedures for the satisfiability of constraints. The general decidability result is then specialized for checking properties of particular interest, such as liveness and safety, and, for the latter case, a more optimized algorithm has been proposed.