1,862 research outputs found
On Equivalences, Metrics, and Polynomial Time
International audienceInteractive behaviors are ubiquitous in modern cryptography, but are also present in λ-calculi, in the form of higher-order constructions. Traditionally, however, typed λ-calculi simply do not fit well into cryptography , being both deterministic and too powerful as for the complexity of functions they can express. We study interaction in a λ-calculus for probabilistic polynomial time computable functions. In particular, we show how notions of context equivalence and context metric can both be characterized by way of traces when defined on linear contexts. We then give evidence on how this can be turned into a proof methodology for computational indistinguishability, a key notion in modern cryptography. We also hint at what happens if a more general notion of a context is used
Solving Tree Problems with Category Theory
Artificial Intelligence (AI) has long pursued models, theories, and
techniques to imbue machines with human-like general intelligence. Yet even the
currently predominant data-driven approaches in AI seem to be lacking humans'
unique ability to solve wide ranges of problems. This situation begs the
question of the existence of principles that underlie general problem-solving
capabilities. We approach this question through the mathematical formulation of
analogies across different problems and solutions. We focus in particular on
problems that could be represented as tree-like structures. Most importantly,
we adopt a category-theoretic approach in formalising tree problems as
categories, and in proving the existence of equivalences across apparently
unrelated problem domains. We prove the existence of a functor between the
category of tree problems and the category of solutions. We also provide a
weaker version of the functor by quantifying equivalences of problem categories
using a metric on tree problems.Comment: 10 pages, 4 figures, International Conference on Artificial General
Intelligence (AGI) 201
Bisimulations and Logical Characterizations on Continuous-time Markov Decision Processes
In this paper we study strong and weak bisimulation equivalences for
continuous-time Markov decision processes (CTMDPs) and the logical
characterizations of these relations with respect to the continuous-time
stochastic logic (CSL). For strong bisimulation, it is well known that it is
strictly finer than CSL equivalence. In this paper we propose strong and weak
bisimulations for CTMDPs and show that for a subclass of CTMDPs, strong and
weak bisimulations are both sound and complete with respect to the equivalences
induced by CSL and the sub-logic of CSL without next operator respectively. We
then consider a standard extension of CSL, and show that it and its sub-logic
without X can be fully characterized by strong and weak bisimulations
respectively over arbitrary CTMDPs.Comment: The conference version of this paper was published at VMCAI 201
On Equivalences, Metrics, and Computational Indistinguishability
The continuous technological progress and the constant growing of information flow we observe every day, brought us an urgent need to find a way to defend our data from malicious intruders; cryptography is the field of computer science that deals with security and studies techniques to protect communications from third parties,
but in the recent years there has been a crisis in proving the security of cryptographic protocols, due to the exponential increase in the complexity of modeling proofs.
In this scenario we study interactions in a typed lambda-calculus properly defined to fit well into the key aspects of a cryptographic proof: interaction, complexity and probability. This calculus, RSLR, is an extension of Hofmann's SLR for probabilistic polynomial time computations and it is perfect to model cryptographic primitives and adversaries. In particular, we characterize notions of context equivalence and
context metrics, when defined on linear contexts, by way of traces, making proofs easier. Furthermore we show how to use this techniqe to obtain a proof methodology
for computational indistinguishability, a key notion in modern cryptography; finally we give some motivating examples of concrete cryptographic schemes
Model checking Quantitative Linear Time Logic
This paper considers QLtl, a quantitative analagon of Ltl and presents algorithms for model checking QLtl over quantitative versions of Kripke structures and Markov chains
- …