3 research outputs found
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
Liveness Properties in Geometric Logic for Domain-Theoretic Streams
We devise a version of Linear Temporal Logic (LTL) on a denotational domain
of streams. We investigate this logic in terms of domain theory, (point-free)
topology and geometric logic. This yields the first steps toward an extension
of the "Domain Theory in Logical Form" paradigm to temporal liveness
properties. We show that the negation-free formulae of LTL induce sober
subspaces of streams, but that this is in general not the case in presence of
negation. We propose a direct, inductive, translation of negation-free LTL to
geometric logic. This translation reflects the approximations used to compute
the usual fixpoint representations of LTL modalities. As a motivating example,
we handle a natural input-output specification for the usual filter function on
streams