22,301 research outputs found
Model checking quantum Markov chains
Although the security of quantum cryptography is provable based on the
principles of quantum mechanics, it can be compromised by the flaws in the
design of quantum protocols and the noise in their physical implementations.
So, it is indispensable to develop techniques of verifying and debugging
quantum cryptographic systems. Model-checking has proved to be effective in the
verification of classical cryptographic protocols, but an essential difficulty
arises when it is applied to quantum systems: the state space of a quantum
system is always a continuum even when its dimension is finite. To overcome
this difficulty, we introduce a novel notion of quantum Markov chain, specially
suited to model quantum cryptographic protocols, in which quantum effects are
entirely encoded into super-operators labelling transitions, leaving the
location information (nodes) being classical. Then we define a quantum
extension of probabilistic computation tree logic (PCTL) and develop a
model-checking algorithm for quantum Markov chains.Comment: Journal versio
Fixpoint Games on Continuous Lattices
Many analysis and verifications tasks, such as static program analyses and
model-checking for temporal logics reduce to the solution of systems of
equations over suitable lattices. Inspired by recent work on lattice-theoretic
progress measures, we develop a game-theoretical approach to the solution of
systems of monotone equations over lattices, where for each single equation
either the least or greatest solution is taken. A simple parity game, referred
to as fixpoint game, is defined that provides a correct and complete
characterisation of the solution of equation systems over continuous lattices,
a quite general class of lattices widely used in semantics. For powerset
lattices the fixpoint game is intimately connected with classical parity games
for -calculus model-checking, whose solution can exploit as a key tool
Jurdzi\'nski's small progress measures. We show how the notion of progress
measure can be naturally generalised to fixpoint games over continuous lattices
and we prove the existence of small progress measures. Our results lead to a
constructive formulation of progress measures as (least) fixpoints. We refine
this characterisation by introducing the notion of selection that allows one to
constrain the plays in the parity game, enabling an effective (and possibly
efficient) solution of the game, and thus of the associated verification
problem. We also propose a logic for specifying the moves of the existential
player that can be used to systematically derive simplified equations for
efficiently computing progress measures. We discuss potential applications to
the model-checking of latticed -calculi and to the solution of fixpoint
equations systems over the reals
Epistemic virtues, metavirtues, and computational complexity
I argue that considerations about computational complexity show that all finite agents need characteristics like those that have been called epistemic virtues. The necessity of these virtues follows in part from the nonexistence of shortcuts, or efficient ways of finding shortcuts, to cognitively expensive routines. It follows that agents must possess the capacities – metavirtues –of developing in advance the cognitive virtues they will need when time and memory are at a premium
The parameterized space complexity of model-checking bounded variable first-order logic
The parameterized model-checking problem for a class of first-order sentences
(queries) asks to decide whether a given sentence from the class holds true in
a given relational structure (database); the parameter is the length of the
sentence. We study the parameterized space complexity of the model-checking
problem for queries with a bounded number of variables. For each bound on the
quantifier alternation rank the problem becomes complete for the corresponding
level of what we call the tree hierarchy, a hierarchy of parameterized
complexity classes defined via space bounded alternating machines between
parameterized logarithmic space and fixed-parameter tractable time. We observe
that a parameterized logarithmic space model-checker for existential bounded
variable queries would allow to improve Savitch's classical simulation of
nondeterministic logarithmic space in deterministic space .
Further, we define a highly space efficient model-checker for queries with a
bounded number of variables and bounded quantifier alternation rank. We study
its optimality under the assumption that Savitch's Theorem is optimal
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