528 research outputs found
Quantified CTL: Expressiveness and Complexity
While it was defined long ago, the extension of CTL with quantification over
atomic propositions has never been studied extensively. Considering two
different semantics (depending whether propositional quantification refers to
the Kripke structure or to its unwinding tree), we study its expressiveness
(showing in particular that QCTL coincides with Monadic Second-Order Logic for
both semantics) and characterise the complexity of its model-checking and
satisfiability problems, depending on the number of nested propositional
quantifiers (showing that the structure semantics populates the polynomial
hierarchy while the tree semantics populates the exponential hierarchy)
Reasoning About Strategies: On the Model-Checking Problem
In open systems verification, to formally check for reliability, one needs an
appropriate formalism to model the interaction between agents and express the
correctness of the system no matter how the environment behaves. An important
contribution in this context is given by modal logics for strategic ability, in
the setting of multi-agent games, such as ATL, ATL\star, and the like.
Recently, Chatterjee, Henzinger, and Piterman introduced Strategy Logic, which
we denote here by CHP-SL, with the aim of getting a powerful framework for
reasoning explicitly about strategies. CHP-SL is obtained by using first-order
quantifications over strategies and has been investigated in the very specific
setting of two-agents turned-based games, where a non-elementary model-checking
algorithm has been provided. While CHP-SL is a very expressive logic, we claim
that it does not fully capture the strategic aspects of multi-agent systems. In
this paper, we introduce and study a more general strategy logic, denoted SL,
for reasoning about strategies in multi-agent concurrent games. We prove that
SL includes CHP-SL, while maintaining a decidable model-checking problem. In
particular, the algorithm we propose is computationally not harder than the
best one known for CHP-SL. Moreover, we prove that such a problem for SL is
NonElementarySpace-hard. This negative result has spurred us to investigate
here syntactic fragments of SL, strictly subsuming ATL\star, with the hope of
obtaining an elementary model-checking problem. Among the others, we study the
sublogics SL[NG], SL[BG], and SL[1G]. They encompass formulas in a special
prenex normal form having, respectively, nested temporal goals, Boolean
combinations of goals and, a single goal at a time. About these logics, we
prove that the model-checking problem for SL[1G] is 2ExpTime-complete, thus not
harder than the one for ATL\star
Temporalized logics and automata for time granularity
Suitable extensions of the monadic second-order theory of k successors have
been proposed in the literature to capture the notion of time granularity. In
this paper, we provide the monadic second-order theories of downward unbounded
layered structures, which are infinitely refinable structures consisting of a
coarsest domain and an infinite number of finer and finer domains, and of
upward unbounded layered structures, which consist of a finest domain and an
infinite number of coarser and coarser domains, with expressively complete and
elementarily decidable temporal logic counterparts.
We obtain such a result in two steps. First, we define a new class of
combined automata, called temporalized automata, which can be proved to be the
automata-theoretic counterpart of temporalized logics, and show that relevant
properties, such as closure under Boolean operations, decidability, and
expressive equivalence with respect to temporal logics, transfer from component
automata to temporalized ones. Then, we exploit the correspondence between
temporalized logics and automata to reduce the task of finding the temporal
logic counterparts of the given theories of time granularity to the easier one
of finding temporalized automata counterparts of them.Comment: Journal: Theory and Practice of Logic Programming Journal Acronym:
TPLP Category: Paper for Special Issue (Verification and Computational Logic)
Submitted: 18 March 2002, revised: 14 Januari 2003, accepted: 5 September
200
A system of relational syllogistic incorporating full Boolean reasoning
We present a system of relational syllogistic, based on classical
propositional logic, having primitives of the following form:
Some A are R-related to some B;
Some A are R-related to all B;
All A are R-related to some B;
All A are R-related to all B.
Such primitives formalize sentences from natural language like `All students
read some textbooks'. Here A and B denote arbitrary sets (of objects), and R
denotes an arbitrary binary relation between objects. The language of the logic
contains only variables denoting sets, determining the class of set terms, and
variables denoting binary relations between objects, determining the class of
relational terms. Both classes of terms are closed under the standard Boolean
operations. The set of relational terms is also closed under taking the
converse of a relation. The results of the paper are the completeness theorem
with respect to the intended semantics and the computational complexity of the
satisfiability problem.Comment: Available at
http://link.springer.com/article/10.1007/s10849-012-9165-
Complexity Hierarchies Beyond Elementary
We introduce a hierarchy of fast-growing complexity classes and show its
suitability for completeness statements of many non elementary problems. This
hierarchy allows the classification of many decision problems with a
non-elementary complexity, which occur naturally in logic, combinatorics,
formal languages, verification, etc., with complexities ranging from simple
towers of exponentials to Ackermannian and beyond.Comment: Version 3 is the published version in TOCT 8(1:3), 2016. I will keep
updating the catalogue of problems from Section 6 in future revision
Model checking Branching-Time Properties of Multi-Pushdown Systems is Hard
We address the model checking problem for shared memory concurrent programs
modeled as multi-pushdown systems. We consider here boolean programs with a
finite number of threads and recursive procedures. It is well-known that the
model checking problem is undecidable for this class of programs. In this
paper, we investigate the decidability and the complexity of this problem under
the assumption of bounded context-switching defined by Qadeer and Rehof, and of
phase-boundedness proposed by La Torre et al. On the model checking of such
systems against temporal logics and in particular branching time logics such as
the modal -calculus or CTL has received little attention. It is known that
parity games, which are closely related to the modal -calculus, are
decidable for the class of bounded-phase systems (and hence for bounded-context
switching as well), but with non-elementary complexity (Seth). A natural
question is whether this high complexity is inevitable and what are the ways to
get around it. This paper addresses these questions and unfortunately, and
somewhat surprisingly, it shows that branching model checking for MPDSs is
inherently an hard problem with no easy solution. We show that parity games on
MPDS under phase-bounding restriction is non-elementary. Our main result shows
that model checking a context bounded MPDS against a simple fragment of
CTL, consisting of formulas that whose temporal operators come from the set
{\EF, \EX}, has a non-elementary lower bound
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