36,815 research outputs found
Proving uniformity and independence by self-composition and coupling
Proof by coupling is a classical proof technique for establishing
probabilistic properties of two probabilistic processes, like stochastic
dominance and rapid mixing of Markov chains. More recently, couplings have been
investigated as a useful abstraction for formal reasoning about relational
properties of probabilistic programs, in particular for modeling
reduction-based cryptographic proofs and for verifying differential privacy. In
this paper, we demonstrate that probabilistic couplings can be used for
verifying non-relational probabilistic properties. Specifically, we show that
the program logic pRHL---whose proofs are formal versions of proofs by
coupling---can be used for formalizing uniformity and probabilistic
independence. We formally verify our main examples using the EasyCrypt proof
assistant
Quantitative Separation Logic - A Logic for Reasoning about Probabilistic Programs
We present quantitative separation logic (). In contrast to
classical separation logic, employs quantities which evaluate to
real numbers instead of predicates which evaluate to Boolean values. The
connectives of classical separation logic, separating conjunction and
separating implication, are lifted from predicates to quantities. This
extension is conservative: Both connectives are backward compatible to their
classical analogs and obey the same laws, e.g. modus ponens, adjointness, etc.
Furthermore, we develop a weakest precondition calculus for quantitative
reasoning about probabilistic pointer programs in . This calculus
is a conservative extension of both Reynolds' separation logic for
heap-manipulating programs and Kozen's / McIver and Morgan's weakest
preexpectations for probabilistic programs. Soundness is proven with respect to
an operational semantics based on Markov decision processes. Our calculus
preserves O'Hearn's frame rule, which enables local reasoning. We demonstrate
that our calculus enables reasoning about quantities such as the probability of
terminating with an empty heap, the probability of reaching a certain array
permutation, or the expected length of a list
Toward a probability theory for product logic: states, integral representation and reasoning
The aim of this paper is to extend probability theory from the classical to
the product t-norm fuzzy logic setting. More precisely, we axiomatize a
generalized notion of finitely additive probability for product logic formulas,
called state, and show that every state is the Lebesgue integral with respect
to a unique regular Borel probability measure. Furthermore, the relation
between states and measures is shown to be one-one. In addition, we study
geometrical properties of the convex set of states and show that extremal
states, i.e., the extremal points of the state space, are the same as the
truth-value assignments of the logic. Finally, we axiomatize a two-tiered modal
logic for probabilistic reasoning on product logic events and prove soundness
and completeness with respect to probabilistic spaces, where the algebra is a
free product algebra and the measure is a state in the above sense.Comment: 27 pages, 1 figur
Local Reasoning about Probabilistic Behaviour for Classical-Quantum Programs
Verifying the functional correctness of programs with both classical and
quantum constructs is a challenging task. The presence of probabilistic
behaviour entailed by quantum measurements and unbounded while loops complicate
the verification task greatly. We propose a new quantum Hoare logic for local
reasoning about probabilistic behaviour by introducing distribution formulas to
specify probabilistic properties. We show that the proof rules in the logic are
sound with respect to a denotational semantics. To demonstrate the
effectiveness of the logic, we formally verify the correctness of non-trivial
quantum algorithms including the HHL and Shor's algorithms.Comment: 27 pages. arXiv admin note: text overlap with arXiv:2107.0080
Relational Neural Machines
Deep learning has been shown to achieve impressive results in several tasks
where a large amount of training data is available. However, deep learning
solely focuses on the accuracy of the predictions, neglecting the reasoning
process leading to a decision, which is a major issue in life-critical
applications. Probabilistic logic reasoning allows to exploit both statistical
regularities and specific domain expertise to perform reasoning under
uncertainty, but its scalability and brittle integration with the layers
processing the sensory data have greatly limited its applications. For these
reasons, combining deep architectures and probabilistic logic reasoning is a
fundamental goal towards the development of intelligent agents operating in
complex environments. This paper presents Relational Neural Machines, a novel
framework allowing to jointly train the parameters of the learners and of a
First--Order Logic based reasoner. A Relational Neural Machine is able to
recover both classical learning from supervised data in case of pure
sub-symbolic learning, and Markov Logic Networks in case of pure symbolic
reasoning, while allowing to jointly train and perform inference in hybrid
learning tasks. Proper algorithmic solutions are devised to make learning and
inference tractable in large-scale problems. The experiments show promising
results in different relational tasks
Beyond the grounding bottleneck: Datalog techniques for inference in probabilistic logic programs
State-of-the-art inference approaches in probabilistic logic programming
typically start by computing the relevant ground program with respect to the
queries of interest, and then use this program for probabilistic inference
using knowledge compilation and weighted model counting. We propose an
alternative approach that uses efficient Datalog techniques to integrate
knowledge compilation with forward reasoning with a non-ground program. This
effectively eliminates the grounding bottleneck that so far has prohibited the
application of probabilistic logic programming in query answering scenarios
over knowledge graphs, while also providing fast approximations on classical
benchmarks in the field
- …