752 research outputs found
Learning Probabilistic Termination Proofs
We present the first machine learning approach to the termination analysis
of probabilistic programs. Ranking supermartingales (RSMs) prove that
probabilistic programs halt, in expectation, within a finite number of steps.
While previously RSMs were directly synthesised from source code,
our method learns them from sampled execution traces.
We introduce the neural ranking supermartingale:
we let a neural network fit
an RSM over execution traces and then we
verify it over the source code using satisfiability modulo theories (SMT);
if the latter step produces a counterexample, we generate from it
new sample traces and repeat learning in a
counterexample-guided inductive synthesis loop,
until the SMT solver confirms the validity of the RSM.
The result is thus a sound witness of probabilistic termination.
Our learning strategy is agnostic to the source code and its
verification counterpart supports the widest range of probabilistic
single-loop programs that any existing tool can handle to date.
We demonstrate the efficacy of our method over a range of benchmarks that
include linear and polynomial programs with discrete, continuous, state-dependent,
multi-variate, hierarchical distributions, and distributions with
undefined moments
Stochastic Invariants for Probabilistic Termination
Termination is one of the basic liveness properties, and we study the
termination problem for probabilistic programs with real-valued variables.
Previous works focused on the qualitative problem that asks whether an input
program terminates with probability~1 (almost-sure termination). A powerful
approach for this qualitative problem is the notion of ranking supermartingales
with respect to a given set of invariants. The quantitative problem
(probabilistic termination) asks for bounds on the termination probability. A
fundamental and conceptual drawback of the existing approaches to address
probabilistic termination is that even though the supermartingales consider the
probabilistic behavior of the programs, the invariants are obtained completely
ignoring the probabilistic aspect.
In this work we address the probabilistic termination problem for
linear-arithmetic probabilistic programs with nondeterminism. We define the
notion of {\em stochastic invariants}, which are constraints along with a
probability bound that the constraints hold. We introduce a concept of {\em
repulsing supermartingales}. First, we show that repulsing supermartingales can
be used to obtain bounds on the probability of the stochastic invariants.
Second, we show the effectiveness of repulsing supermartingales in the
following three ways: (1)~With a combination of ranking and repulsing
supermartingales we can compute lower bounds on the probability of termination;
(2)~repulsing supermartingales provide witnesses for refutation of almost-sure
termination; and (3)~with a combination of ranking and repulsing
supermartingales we can establish persistence properties of probabilistic
programs.
We also present results on related computational problems and an experimental
evaluation of our approach on academic examples.Comment: Full version of a paper published at POPL 2017. 20 page
Ranking and Repulsing Supermartingales for Reachability in Probabilistic Programs
Computing reachability probabilities is a fundamental problem in the analysis
of probabilistic programs. This paper aims at a comprehensive and comparative
account on various martingale-based methods for over- and under-approximating
reachability probabilities. Based on the existing works that stretch across
different communities (formal verification, control theory, etc.), we offer a
unifying account. In particular, we emphasize the role of order-theoretic fixed
points---a classic topic in computer science---in the analysis of probabilistic
programs. This leads us to two new martingale-based techniques, too. We give
rigorous proofs for their soundness and completeness. We also make an
experimental comparison using our implementation of template-based synthesis
algorithms for those martingales
Learning Provably Stabilizing Neural Controllers for Discrete-Time Stochastic Systems
We consider the problem of learning control policies in discrete-time
stochastic systems which guarantee that the system stabilizes within some
specified stabilization region with probability~. Our approach is based on
the novel notion of stabilizing ranking supermartingales (sRSMs) that we
introduce in this work. Our sRSMs overcome the limitation of methods proposed
in previous works whose applicability is restricted to systems in which the
stabilizing region cannot be left once entered under any control policy. We
present a learning procedure that learns a control policy together with an sRSM
that formally certifies probability~ stability, both learned as neural
networks. We show that this procedure can also be adapted to formally verifying
that, under a given Lipschitz continuous control policy, the stochastic system
stabilizes within some stabilizing region with probability~. Our
experimental evaluation shows that our learning procedure can successfully
learn provably stabilizing policies in practice.Comment: Accepted at ATVA 2023. Follow-up work of arXiv:2112.0949
Understanding Inquiry, an Inquiry into Understanding: a conception of Inquiry Based Learning in Mathematics
IBL (Inquiry Based Learning) is a group of educational approaches centered on the student and aiming at developing higher-level thinking, as well as an adequate set of Knowledge, Skills, and Attitudes (KSA). IBL is at the center of recent educational research and practice, and is expanding quickly outside of schools: in this research we propose such forms of instruction as Guided Self-Study, Guided Problem Solving, Inquiry Based Homeschooling, IB e-learning, and particularly a mixed (Inquiry-Expository) form of lecturing, named IBLecturing. The research comprises a thorough review of previous research in IBL; it clarifies what is and what is not Inquiry Based Learning, and the distinctions between its various forms: Inquiry Learning, Discovery Learning, Case Study, Problem Based Learning, Project Based Learning, Experiential Learning, etc. There is a continuum between Pure Inquiry and Pure Expository approaches, and the extreme forms are very infrequently encountered. A new cognitive taxonomy adapted to the needs of higher-level thinking and its promotion in the study of mathematics is also presented. This research comprises an illustration of the modeling by an expert (teacher, trainer, etc.) of the heuristics and of the cognitive and metacognitive strategies employed by mathematicians for solving problems and building proofs. A challenging problem has been administered to a group of gifted students from secondary school, in order to get more information about the possibility of implementing Guided Problem Solving. Various opportunities for further research are indicated, for example applying the recent advances of cognitive psychology on the role of Working Memory (WM) in higher-level thinking
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