8 research outputs found
Relational semantics of linear logic and higher-order model-checking
In this article, we develop a new and somewhat unexpected connection between
higher-order model-checking and linear logic. Our starting point is the
observation that once embedded in the relational semantics of linear logic, the
Church encoding of any higher-order recursion scheme (HORS) comes together with
a dual Church encoding of an alternating tree automata (ATA) of the same
signature. Moreover, the interaction between the relational interpretations of
the HORS and of the ATA identifies the set of accepting states of the tree
automaton against the infinite tree generated by the recursion scheme. We show
how to extend this result to alternating parity automata (APT) by introducing a
parametric version of the exponential modality of linear logic, capturing the
formal properties of colors (or priorities) in higher-order model-checking. We
show in particular how to reunderstand in this way the type-theoretic approach
to higher-order model-checking developed by Kobayashi and Ong. We briefly
explain in the end of the paper how his analysis driven by linear logic results
in a new and purely semantic proof of decidability of the formulas of the
monadic second-order logic for higher-order recursion schemes.Comment: 24 pages. Submitte
Profinite lambda-terms and parametricity
Combining ideas coming from Stone duality and Reynolds parametricity, we
formulate in a clean and principled way a notion of profinite lambda-term
which, we show, generalizes at every type the traditional notion of profinite
word coming from automata theory. We start by defining the Stone space of
profinite lambda-terms as a projective limit of finite sets of usual
lambda-terms, considered modulo a notion of equivalence based on the finite
standard model. One main contribution of the paper is to establish that,
somewhat surprisingly, the resulting notion of profinite lambda-term coming
from Stone duality lives in perfect harmony with the principles of Reynolds
parametricity. In addition, we show that the notion of profinite lambda-term is
compositional by constructing a cartesian closed category of profinite
lambda-terms, and we establish that the embedding from lambda-terms modulo
beta-eta-conversion to profinite lambda-terms is faithful using Statman's
finite completeness theorem. Finally, we prove that the traditional Church
encoding of finite words into lambda-terms can be extended to profinite words,
and leads to a homeomorphism between the space of profinite words and the space
of profinite lambda-terms of the corresponding Church type
Domains for Higher-Order Games
We study two-player inclusion games played over word-generating higher-order
recursion schemes. While inclusion checks are known to capture verification
problems, two-player games generalize this relationship to program synthesis.
In such games, non-terminals of the grammar are controlled by opposing players.
The goal of the existential player is to avoid producing a word that lies
outside of a regular language of safe words.
We contribute a new domain that provides a representation of the winning
region of such games. Our domain is based on (functions over) potentially
infinite Boolean formulas with words as atomic propositions. We develop an
abstract interpretation framework that we instantiate to abstract this domain
into a domain where the propositions are replaced by states of a finite
automaton. This second domain is therefore finite and we obtain, via standard
fixed-point techniques, a direct algorithm for the analysis of two-player
inclusion games. We show, via a second instantiation of the framework, that our
finite domain can be optimized, leading to a (k+1)EXP algorithm for order-k
recursion schemes. We give a matching lower bound, showing that our approach is
optimal. Since our approach is based on standard Kleene iteration, existing
techniques and tools for fixed-point computations can be applied.Comment: Conference version accepted for presentation and publication at the
42nd International Symposium on Mathematical Foundations of Computer Science
(MFCS 2017
On the Termination Problem for Probabilistic Higher-Order Recursive Programs
In the last two decades, there has been much progress on model checking of
both probabilistic systems and higher-order programs. In spite of the emergence
of higher-order probabilistic programming languages, not much has been done to
combine those two approaches. In this paper, we initiate a study on the
probabilistic higher-order model checking problem, by giving some first
theoretical and experimental results. As a first step towards our goal, we
introduce PHORS, a probabilistic extension of higher-order recursion schemes
(HORS), as a model of probabilistic higher-order programs. The model of PHORS
may alternatively be viewed as a higher-order extension of recursive Markov
chains. We then investigate the probabilistic termination problem -- or,
equivalently, the probabilistic reachability problem. We prove that almost sure
termination of order-2 PHORS is undecidable. We also provide a fixpoint
characterization of the termination probability of PHORS, and develop a sound
(but possibly incomplete) procedure for approximately computing the termination
probability. We have implemented the procedure for order-2 PHORSs, and
confirmed that the procedure works well through preliminary experiments that
are reported at the end of the article
LambdaY-Calculus With Priorities
International audienceThe lambdaY-calculus with priorities is a variant of the simply-typed lambda calculus designed for higher-order model-checking. The higher-order model-checking problem asks if a given parity tree automaton accepts the Böhm tree of a given term of the simply-typed lambda calculus with recursion. We show that this problem can be reduced to the same question but for terms of lambdaY-calculus with priorities and visibly parity automata; a subclass of parity automata. The latter question can be answered by evaluating terms in a simple powerset model with least and greatest fixpoints. We prove that the recognizing power of powerset models and visibly parity automata are the same. So, up to conversion to the lambdaY-calculus with priorities, powerset models with least and greatest fixpoints are indeed the right semantic framework for the model-checking problem. The reduction to lambdaY-calculus with priorities is also efficient algorithmically: it gives an algorithm of the same complexity as direct approaches to the higher-order model-checking problem. This indicates that the task of calculating the value of a term in a powerset model is a central algo-rithmic problem for higher-order model-checking
Lambda-calculus and formal language theory
Formal and symbolic approaches have offered computer science many application fields. The rich and fruitful connection between logic, automata and algebra is one such approach. It has been used to model natural languages as well as in program verification. In the mathematics of language it is able to model phenomena ranging from syntax to phonology while in verification it gives model checking algorithms to a wide family of programs. This thesis extends this approach to simply typed lambda-calculus by providing a natural extension of recognizability to programs that are representable by simply typed terms. This notion is then applied to both the mathematics of language and program verification. In the case of the mathematics of language, it is used to generalize parsing algorithms and to propose high-level methods to describe languages. Concerning program verification, it is used to describe methods for verifying the behavioral properties of higher-order programs. In both cases, the link that is drawn between finite state methods and denotational semantics provide the means to mix powerful tools coming from the two worlds