From Linear to Branching-Time Temporal Logics: Transfer of Semantics and Definability

This paper investigates logical aspects of combining linear orders as semantics for modal and temporal logics, with modalities for possible paths, resulting in a variety of branching time logics over classes of trees. Here we adopt a unified approach to the Priorean, Peircean and Ockhamist semantics for branching time logics, by considering them all as fragments of the latter, obtained as combinations, in various degrees, of languages and semantics for linear time with a modality for possible paths. We then consider a hierarchy of natural classes of trees and bundled trees arising from a given class of linear orders and show that in general they provide different semantics. We also discuss transfer of definability from linear orders to trees and introduce a uniform translation from Priorean to Peircean formulae which transfers definability of properties of linear orders to definability of properties of all paths in tree

Flow Logic

Flow networks have attracted a lot of research in computer science. Indeed, many questions in numerous application areas can be reduced to questions about flow networks. Many of these applications would benefit from a framework in which one can formally reason about properties of flow networks that go beyond their maximal flow. We introduce Flow Logics: modal logics that treat flow functions as explicit first-order objects and enable the specification of rich properties of flow networks. The syntax of our logic BFL* (Branching Flow Logic) is similar to the syntax of the temporal logic CTL*, except that atomic assertions may be flow propositions, like $> \gamma$ or $\geq \gamma$, for $\gamma \in \mathbb{N}$, which refer to the value of the flow in a vertex, and that first-order quantification can be applied both to paths and to flow functions. We present an exhaustive study of the theoretical and practical aspects of BFL*, as well as extensions and fragments of it. Our extensions include flow quantifications that range over non-integral flow functions or over maximal flow functions, path quantification that ranges over paths along which non-zero flow travels, past operators, and first-order quantification of flow values. We focus on the model-checking problem and show that it is PSPACE-complete, as it is for CTL*. Handling of flow quantifiers, however, increases the complexity in terms of the network to ${\rm P}^{\rm NP}$, even for the LFL and BFL fragments, which are the flow-counterparts of LTL and CTL. We are still able to point to a useful fragment of BFL* for which the model-checking problem can be solved in polynomial time. Finally, we introduce and study the query-checking problem for BFL*, where under-specified BFL* formulas are used for network exploration

Lightweight description logics and branching time: a troublesome marriage

We study branching-time temporal description logics (BTDLs) based on the temporal logic CTL in the presence of rigid (time-invariant) roles and general TBoxes. There is evidence that, if full CTL is combined with the classical ALC in this way, reasoning becomes undecidable. In this paper, we begin by substantiating this claim, establishing undecidability for a fragment of this combination. In view of this negative result, we then investigate BTDLs that emerge from combining fragments of CTL with lightweight DLs from the EL and DL-Lite families. We show that even rather inexpressive BTDLs based on EL exhibit very high complexity. Most notably, we identify two convex fragments which are undecidable and hard for non-elementary time, respectively. For BTDLs based on DL-LiteN bool, we obtain tight complexity bounds that range from PSPACE to EXPTIME

The Hierarchy of Hyperlogics

Hyperproperties, which generalize trace properties by relating multiple traces, are widely studied in information-flow security. Recently, a number of logics for hyperproperties have been proposed, and there is a need to understand their decidability and relative expressiveness. The new logics have been obtained from standard logics with two principal extensions: temporal logics, like LTL and CTL$^*$, have been generalized to hyperproperties by adding variables for traces or paths. First-order and second-order logics, like monadic first-order logic of order and MSO, have been extended with the equal-level predicate. We study the impact of the two extensions across the spectrum of linear-time and branching-time logics, in particular for logics with quantification over propositions. The resulting hierarchy of hyperlogics differs significantly from the classical hierarchy, suggesting that the equal-level predicate adds more expressiveness than trace and path variables. Within the hierarchy of hyperlogics, we identify new boundaries on the decidability of the satisfiability problem. Specifically, we show that while HyperQPTL and HyperCTL$^*$ are both undecidable in general, formulas within their $\exists^*\forall^*$ fragments are decidable.Comment: Originally published at LICS 201

Computational Aspects of Dependence Logic

In this thesis (modal) dependence logic is investigated. It was introduced in 2007 by Jouko V\"a\"aan\"anen as an extension of first-order (resp. modal) logic by the dependence operator =(). For first-order (resp. propositional) variables x_1,...,x_n, =(x_1,...,x_n) intuitively states that the value of x_n is determined by those of x_1,...,x_n-1. We consider fragments of modal dependence logic obtained by restricting the set of allowed modal and propositional connectives. We classify these fragments with respect to the complexity of their satisfiability and model-checking problems. For satisfiability we obtain complexity degrees from P over NP, Sigma_P^2 and PSPACE up to NEXP, while for model-checking we only classify the fragments with respect to their tractability, i.e. we either show NP-completeness or containment in P. We then study the extension of modal dependence logic by intuitionistic implication. For this extension we again classify the complexity of the model-checking problem for its fragments. Here we obtain complexity degrees from P over NP and coNP up to PSPACE. Finally, we analyze first-order dependence logic, independence-friendly logic and their two-variable fragments. We prove that satisfiability for two-variable dependence logic is NEXP-complete, whereas for two-variable independence-friendly logic it is undecidable; and use this to prove that the latter is also more expressive than the former.Comment: PhD thesis; 138 pages (110 main matter

Branching within Time: an Expressively Complete and Elementarily Decidable Temporal Logic for Time Granularity

Suitable extensions of monadic second-order theories of k successors have been proposed in the literature to specify in a concise way reactive systems whose behaviour can be naturally modeled with respect to a (possibly infinite) set of differently-grained temporal domains. This is the case, for instance, of the wide-ranging class of real-time reactive systems whose components have dynamic behaviours regulated by very different time constants, e.g., days, hours, and seconds. In this paper, we focus on the theory of k-refinable downward unbounded layered structures MSO[<_{tot},(\downarrow_i)_{i=0}^{k-1}], that is, the theory of infinitely refinable structures consisting of a coarsest domain and an infinite number of finer and finer domains, whose satisfiability problem is nonelementarily decidable. We define a propositional temporal logic counterpart of MSO[<_{tot},(\downarrow_i)_{i=0}^{k-1}], with set quantification restricted to infinite paths, called CTSL, which features an original mix of linear and branching temporal operators. We prove the expressive completeness of CTSL with respect to such a path fragment of MSO[<_{tot}, (\downarrow_i)_{i=0}^{k-1}], and show that its satisfiability problem is 2EXPTIME-complete

Complexity Results for Modal Dependence Logic

Modal dependence logic was introduced recently by V\"a\"an\"anen. It enhances the basic modal language by an operator =(). For propositional variables p_1,...,p_n, =(p_1,...,p_(n-1);p_n) intuitively states that the value of p_n is determined by those of p_1,...,p_(n-1). Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic is complete for nondeterministic exponential time. In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfibility for poor man's dependence logic, the language consisting of formulas built from literals and dependence atoms using conjunction, necessity and possibility (i.e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness. We also extend V\"a\"an\"anen's language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satistiability is complete for the second level of the polynomial hierarchy. In this way we completely classify the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by V\"a\"an\"anen and Sevenster.Comment: 22 pages, full version of CSL 2010 pape

How hard is it to verify flat affine counter systems with the finite monoid property ?

We study several decision problems for counter systems with guards defined by convex polyhedra and updates defined by affine transformations. In general, the reachability problem is undecidable for such systems. Decidability can be achieved by imposing two restrictions: (i) the control structure of the counter system is flat, meaning that nested loops are forbidden, and (ii) the set of matrix powers is finite, for any affine update matrix in the system. We provide tight complexity bounds for several decision problems of such systems, by proving that reachability and model checking for Past Linear Temporal Logic are complete for the second level of the polynomial hierarchy $\Sigma^P_2$, while model checking for First Order Logic is PSPACE-complete

Decidable fragments of first-order temporal logics

Accepted versio

Satisfiability in Strategy Logic can be Easier than Model Checking

In the design of complex systems, model-checking and satisfiability arise as two prominent decision problems. While model-checking requires the designed system to be provided in advance, satisfiability allows to check if such a system even exists. With very few exceptions, the second problem turns out to be harder than the first one from a complexity-theoretic standpoint. In this paper, we investigate the connection between the two problems for a non-trivial fragment of Strategy Logic (SL, for short). SL extends LTL with first-order quantifications over strategies, thus allowing to explicitly reason about the strategic abilities of agents in a multi-agent system. Satisfiability for the full logic is known to be highly undecidable, while model-checking is non-elementary.The SL fragment we consider is obtained by preventing strategic quantifications within the scope of temporal operators. The resulting logic is quite powerful, still allowing to express important game-theoretic properties of multi-agent systems, such as existence of Nash and immune equilibria, as well as to formalize the rational synthesis problem. We show that satisfiability for such a fragment is PSPACE-COMPLETE, while its model-checking complexity is 2EXPTIME-HARD. The result is obtained by means of an elegant encoding of the problem into the satisfiability of conjunctive-binding first-order logic, a recently discovered decidable fragment of first-order logic