9 research outputs found
Complexity of correctness for pomset logic proof nets
We show that it is coNP-complete to decide whether a given proof structure of
pomset logic is a correct proof net, using the graph-theoretic used in a
previous paper of ours (arXiv:1901.10247).Comment: Fully subsumed by arXiv:2209.07825 (which contains a lot more
material and has an additional coauthor
A System of Interaction and Structure III: The Complexity of BV and Pomset Logic
Pomset logic and BV are both logics that extend multiplicative linear logic
(with Mix) with a third connective that is self-dual and non-commutative.
Whereas pomset logic originates from the study of coherence spaces and proof
nets, BV originates from the study of series-parallel orders, cographs, and
proof systems. Both logics enjoy a cut-admissibility result, but for neither
logic can this be done in the sequent calculus. Provability in pomset logic can
be checked via a proof net correctness criterion and in BV via a deep inference
proof system. It has long been conjectured that these two logics are the same.
In this paper we show that this conjecture is false. We also investigate the
complexity of the two logics, exhibiting a huge gap between the two. Whereas
provability in BV is NP-complete, provability in pomset logic is
-complete. We also make some observations with respect to possible
sequent systems for the two logics
Implicit automata in typed -calculi II: streaming transducers vs categorical semantics
We characterize regular string transductions as programs in a linear
-calculus with additives. One direction of this equivalence is proved
by encoding copyless streaming string transducers (SSTs), which compute regular
functions, into our -calculus. For the converse, we consider a
categorical framework for defining automata and transducers over words, which
allows us to relate register updates in SSTs to the semantics of the linear
-calculus in a suitable monoidal closed category. To illustrate the
relevance of monoidal closure to automata theory, we also leverage this notion
to give abstract generalizations of the arguments showing that copyless SSTs
may be determinized and that the composition of two regular functions may be
implemented by a copyless SST. Our main result is then generalized from strings
to trees using a similar approach. In doing so, we exhibit a connection between
a feature of streaming tree transducers and the multiplicative/additive
distinction of linear logic.
Keywords: MSO transductions, implicit complexity, Dialectica categories,
Church encodingsComment: 105 pages, 24 figure
Two-way automata and transducers with planar behaviours are aperiodic
We consider a notion of planarity for two-way finite automata and
transducers, inspired by Temperley-Lieb monoids of planar diagrams. We show
that this restriction captures star-free languages and first-order
transductions.Comment: 18 pages, DMTCS submissio
Revisiting the growth of polyregular functions: output languages, weighted automata and unary inputs
Polyregular functions are the class of string-to-string functions definable
by pebble transducers (an extension of finite automata) or equivalently by MSO
interpretations (a logical formalism). Their output length is bounded by a
polynomial in the input length: a function computed by a -pebble transducer
or by a -dimensional MSO interpretation has growth rate .
Boja\'nczyk has recently shown that the converse holds for MSO
interpretations, but not for pebble transducers. We give significantly
simplified proofs of those two results, extending the former to first-order
interpretations by reduction to an elementary property of -weighted
automata. For any , we also prove the stronger statement that there is some
quadratic polyregular function whose output language differs from that of any
-fold composition of macro tree transducers (and which therefore cannot be
computed by any -pebble transducer).
In the special case of unary input alphabets, we show that pebbles
suffice to compute polyregular functions of growth . This is obtained
as a corollary of a basis of simple word sequences whose ultimately periodic
combinations generate all polyregular functions with unary input. Finally, we
study polyregular and polyblind functions between unary alphabets (i.e. integer
sequences), as well as their first-order subclasses.Comment: 27 pages, not submitted ye