17 research outputs found
On the correctness of monadic backward induction
In control theory, to solve a finite-horizon sequential decision problem (SDP) commonly means to find a list of decision rules that result in an optimal expected total reward (or cost) when taking a given number of decision steps. SDPs are routinely solved using Bellman\u27s backward induction. Textbook authors (e.g. Bertsekas or Puterman) typically give more or less formal proofs to show that the backward induction algorithm is correct as solution method for deterministic and stochastic SDPs. Botta, Jansson and Ionescu propose a generic framework for finite horizon, monadic SDPs together with a monadic version of backward induction for solving such SDPs. In monadic SDPs, the monad captures a generic notion of uncertainty, while a generic measure function aggregates rewards. In the present paper, we define a notion of correctness for monadic SDPs and identify three conditions that allow us to prove a correctness result for monadic backward induction that is comparable to textbook correctness proofs for ordinary backward induction. The conditions that we impose are fairly general and can be cast in category-theoretical terms using the notion of Eilenberg-Moore algebra. They hold in familiar settings like those of deterministic or stochastic SDPs, but we also give examples in which they fail. Our results show that backward induction can safely be employed for a broader class of SDPs than usually treated in textbooks. However, they also rule out certain instances that were considered admissible in the context of Botta et al. \u27s generic framework. Our development is formalised in Idris as an extension of the Botta et al. framework and the sources are available as supplementary material
On the logical structure of choice and bar induction principles
We develop an approach to choice principles and their contrapositive
bar-induction principles as extensionality schemes connecting an "intensional"
or "effective" view of respectively ill-and well-foundedness properties to an
"extensional" or "ideal" view of these properties. After classifying and
analysing the relations between different intensional definitions of
ill-foundedness and well-foundedness, we introduce, for a domain , a
codomain and a "filter" on finite approximations of functions from
to , a generalised form GDC of the axiom of dependent choice and
dually a generalised bar induction principle GBI such that:
GDC intuitionistically captures the strength of
the general axiom of choice expressed as when is a
filter that derives point-wise from a relation on without
introducing further constraints,
the Boolean Prime Filter Theorem / Ultrafilter Theorem if is
the two-element set (for a constructive definition of prime
filter),
the axiom of dependent choice if ,
Weak K{\"o}nig's Lemma if and (up
to weak classical reasoning)
GBI intuitionistically captures the strength of
G{\"o}del's completeness theorem in the form validity implies
provability for entailment relations if ,
bar induction when ,
the Weak Fan Theorem when and .
Contrastingly, even though GDC and GBI smoothly capture
several variants of choice and bar induction, some instances are inconsistent,
e.g. when is and is .Comment: LICS 2021 - 36th Annual Symposium on Logic in Computer Science, Jun
2021, Rome / Virtual, Ital
Semantic verification of dynamic programming
We prove that the generic framework for specifying and solving
finite-horizon, monadic sequential decision problems proposed in (Botta et
al.,2017) is semantically correct. By semantically correct we mean that, for a
problem specification and for any initial state compatible with ,
the verified optimal policies obtained with the framework maximize the
-measure of the -sums of the -rewards along all the possible
trajectories rooted in . In short, we prove that, given , the verified
computations encoded in the framework are the correct computations to do. The
main theorem is formulated as an equivalence between two value functions: the
first lies at the core of dynamic programming as originally formulated in
(Bellman,1957) and formalized by Botta et al. in Idris (Brady,2017), and the
second is a specification. The equivalence requires the two value functions to
be extensionally equal. Further, we identify and discuss three requirements
that measures of uncertainty have to fulfill for the main theorem to hold.
These turn out to be rather natural conditions that the expected-value measure
of stochastic uncertainty fulfills. The formal proof of the main theorem
crucially relies on a principle of preservation of extensional equality for
functors. We formulate and prove the semantic correctness of dynamic
programming as an extension of the Botta et al. Idris framework. However, the
theory can easily be implemented in Coq or Agda.Comment: Manuscript ID: JFP-2020-003
Extensional equality preservation and verified generic programming
In verified generic programming, one cannot exploit the structure of concrete
data types but has to rely on well chosen sets of specifications or abstract
data types (ADTs). Functors and monads are at the core of many applications of
functional programming. This raises the question of what useful ADTs for
verified functors and monads could look like. The functorial map of many
important monads preserves extensional equality. For instance, if are extensionally equal, that is, , then and are also
extensionally equal. This suggests that preservation of extensional equality
could be a useful principle in verified generic programming. We explore this
possibility with a minimalist approach: we deal with (the lack of) extensional
equality in Martin-L\"of's intensional type theories without extending the
theories or using full-fledged setoids. Perhaps surprisingly, this minimal
approach turns out to be extremely useful. It allows one to derive simple
generic proofs of monadic laws but also verified, generic results in dynamical
systems and control theory. In turn, these results avoid tedious code
duplication and ad-hoc proofs. Thus, our work is a contribution towards
pragmatic, verified generic programming.Comment: Manuscript ID: JFP-2020-003
On the logical structure of choice and bar induction principles
International audienceWe develop an approach to choice principles and their contrapositive bar-induction principles as extensionality schemes connecting an "intensional" or "effective" view of respectively ill-and well-foundedness properties to an "extensional" or "ideal" view of these properties. After classifying and analysing the relations between different intensional definitions of ill-foundedness and well-foundedness, we introduce, for a domain , a codomain and a "filter" on finite approximations of functions from to , a generalised form GDC of the axiom of dependent choice and dually a generalised bar induction principle GBI such that:- GDC intuitionistically captures the strength ofâą the general axiom of choice expressed as when is a filter that derives point-wise from a relation on without introducing further constraints,âą the Boolean Prime Filter Theorem / Ultrafilter Theorem if is the two-element set (for a constructive definition of prime filter),âą the axiom of dependent choice if ,âą Weak Königâs Lemma if and (up to weak classical reasoning)- GBI intuitionistically captures the strength ofâą Gödelâs completeness theorem in the form validity implies provability for entailment relations if ,âą bar induction when ,âą the Weak Fan Theorem when and .Contrastingly, even though GDC and GBI smoothly capture several variants of choice and bar induction, some instances are inconsistent, e.g. when is and is
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