14 research outputs found
A refinement-based approach to large scale reflection for algebra
National audienceLarge scale reflection tactics are often implemented with ad-hoc data-structures and in a way which is specific to the problematic. This makes it hard to add improvements and to implement variations without writing an extensive theory of the specific data-structures involved. We suggest to replace the core of such tactics with procedures that are proven correct using CoqEAL refinement framework, and to build a modular methodology around it. This refinement framework addresses the problem of duplication by promoting the use of one extensive proof-oriented library together with one or several more efficient implementations, with a reduced amount of proofs, but destined to computation and proven correct with regard to the proof-oriented library. We show on the example of the ring tactic of Coq that this gain in flexibility opens the door to different improvements. This is a presentation to trigger discussion about ideas for a prototype based on the existing but improved CoqEAL refinement framework, described in [5] and [3]
Univalence for free
We present an internalization of the 2-groupoid interpretation of the calculus of construction that allows to realize the univalence axiom, proof irrelevance and reasoning modulo. As an example, we show that in our setting, the type of Church integers is equal to the inductive type of natural numbers
Playing Games in the Baire Space
We solve a generalized version of Church's Synthesis Problem where a play is
given by a sequence of natural numbers rather than a sequence of bits; so a
play is an element of the Baire space rather than of the Cantor space. Two
players Input and Output choose natural numbers in alternation to generate a
play. We present a natural model of automata ("N-memory automata") equipped
with the parity acceptance condition, and we introduce also the corresponding
model of "N-memory transducers". We show that solvability of games specified by
N-memory automata (i.e., existence of a winning strategy for player Output) is
decidable, and that in this case an N-memory transducer can be constructed that
implements a winning strategy for player Output.Comment: In Proceedings Cassting'16/SynCoP'16, arXiv:1608.0017
Domain-Aware Session Types
We develop a generalization of existing Curry-Howard interpretations of (binary) session types by relying on an extension of linear logic with features from hybrid logic, in particular modal worlds that indicate domains. These worlds govern domain migration, subject to a parametric accessibility relation familiar from the Kripke semantics of modal logic. The result is an expressive new typed process framework for domain-aware, message-passing concurrency. Its logical foundations ensure that well-typed processes enjoy session fidelity, global progress, and termination. Typing also ensures that processes only communicate with accessible domains and so respect the accessibility relation.
Remarkably, our domain-aware framework can specify scenarios in which domain information is available only at runtime; flexible accessibility relations can be cleanly defined and statically enforced. As a specific application, we introduce domain-aware multiparty session types, in which global protocols can express arbitrarily nested sub-protocols via domain migration. We develop a precise analysis of these multiparty protocols by reduction to our binary domain-aware framework: complex domain-aware protocols can be reasoned about at the right level of abstraction, ensuring also the principled transfer of key correctness properties from the binary to the multiparty setting
A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank
For even , the matchings connectivity matrix encodes which
pairs of perfect matchings on vertices form a single cycle. Cygan et al.
(STOC 2013) showed that the rank of over is
and used this to give an
time algorithm for counting Hamiltonian cycles modulo on graphs of
pathwidth . The same authors complemented their algorithm by an
essentially tight lower bound under the Strong Exponential Time Hypothesis
(SETH). This bound crucially relied on a large permutation submatrix within
, which enabled a "pattern propagation" commonly used in previous
related lower bounds, as initiated by Lokshtanov et al. (SODA 2011).
We present a new technique for a similar pattern propagation when only a
black-box lower bound on the asymptotic rank of is given; no
stronger structural insights such as the existence of large permutation
submatrices in are needed. Given appropriate rank bounds, our
technique yields lower bounds for counting Hamiltonian cycles (also modulo
fixed primes ) parameterized by pathwidth.
To apply this technique, we prove that the rank of over the
rationals is . We also show that the rank of
over is for any prime
and even for some primes.
As a consequence, we obtain that Hamiltonian cycles cannot be counted in time
for any unless SETH fails. This
bound is tight due to a time algorithm by Bodlaender et
al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be
counted modulo primes in time , indicating
that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in
SODA 201
Smooth Approximations and Relational Width Collapses
We prove that relational structures admitting specific polymorphisms (namely,
canonical pseudo-WNU operations of all arities ) have low relational
width. This implies a collapse of the bounded width hierarchy for numerous
classes of infinite-domain CSPs studied in the literature. Moreover, we obtain
a characterization of bounded width for first-order reducts of unary structures
and a characterization of MMSNP sentences that are equivalent to a Datalog
program, answering a question posed by Bienvenu, ten Cate, Lutz, and Wolter. In
particular, the bounded width hierarchy collapses in those cases as well