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 $k$-pebble transducer or by a $k$-dimensional MSO interpretation has growth rate $O(n^k)$. 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 $\mathbb{N}$-weighted automata. For any $k$, we also prove the stronger statement that there is some quadratic polyregular function whose output language differs from that of any $k$-fold composition of macro tree transducers (and which therefore cannot be computed by any $k$-pebble transducer). In the special case of unary input alphabets, we show that $k$ pebbles suffice to compute polyregular functions of growth $O(n^k)$. 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

Synthesizing Nested Relational Queries from Implicit Specifications

Derived datasets can be defined implicitly or explicitly. An implicit definition (of dataset in terms of datasets ) is a logical specification involving the source data and the interface data . It is a valid definition of in terms of , if any two models of the specification agreeing on agree on . In contrast, an explicit definition is a query that produces from . Variants of Beth’s theorem state that one can convert implicit definitions to explicit ones.Further, this conversion can be done effectively given a proof witnessing implicit definability in a suitable proof system. We prove the analogous effective implicit-to-explicit result for nested relations: implicit definitions, given in the natural logic for nested relations,can be effectively converted to explicit definitions in the nested relational calculus (NRC). As a consequence, we can effectively extract rewritings of NRC queries in terms of NRC views, given aproof witnessing that the query is determined by the views

Cantor-Bernstein implies Excluded Middle

Update: fixed an error on the applicability of thm 1, added some acks and a refWe prove in constructive logic that the statement of the Cantor-Bernstein theorem implies excluded middle. This establishes that the Cantor-Bernstein theorem can only be proven assuming the full power of classical logic. The key ingredient is a theorem of Martín Escardó stating that quantification over a particular subset of the Cantor space ℕ → 2, the so-called one-point compactification of ℕ, preserves decidable predicates

From normal functors to logarithmic space queries

International audienceWe introduce a new approach to implicit complexity in linear logic, inspired by functional database query languages and using recent developments in effective denotational semantics of polymorphism. We give the first sub-polynomial upper bound in a type system with impredicative polymorphism; adding restrictions on quantifiers yields a characterization of logarithmic space, for which extensional completeness is established via descriptive complexity

Implicit automata in typed λ-calculi I: aperiodicity in a non-commutative logic

International audienceWe give a characterization of star-free languages in a λ-calculus with support for non-commutative affine types (in the sense of linear logic), via the algebraic characterization of the former using aperiodic monoids. When the type system is made commutative, we show that we get regular languages instead. A key ingredient in our approach – that it shares with higher-order model checking – is the use of Church encodings for inputs and outputs. Our result is, to our knowledge, the first use of non-commutativity in implicit computational complexity

Comparison-free polyregular functions

International audienceThis paper introduces a new automata-theoretic class of string-to-string functions with polynomial growth. Several equivalent definitions are provided: a machine model which is a restricted variant of pebble transducers, and a few inductive definitions that close the class of regular functions under certain operations. Our motivation for studying this class comes from another characterization, which we merely mention here but prove elsewhere, based on a λ-calculus with a linear type system.As their name suggests, these comparison-free polyregular functions form a subclass of polyregular functions; we prove that the inclusion is strict. We also show that they are incomparable with HDT0L transductions, closed under usual function composition – but not under a certain “map” combinator – and satisfy a comparison-free version of the pebble minimization theorem.On the broader topic of polynomial growth transductions, we also consider the recently introduced layered streaming string transducers (SSTs), or equivalently k-marble transducers. We prove that a function can be obtained by composing such transducers together if and only if it is polyregular, and that k-layered SSTs (or k-marble transducers) are closed under “map” and equivalent to a corresponding notion of (k+1)-layered HDT0L systems