19 research outputs found
A Dialectica-Like Interpretation of a Linear MSO on Infinite Words
We devise a variant of Dialectica interpretation of intuitionistic linear logic for Open image in new window, a linear logic-based version MSO over infinite words. Open image in new window was known to be correct and complete w.r.t. Churchâs synthesis, thanks to an automata-based realizability model. Invoking BĂŒchi-Landweber Theorem and building on a complete axiomatization of MSO on infinite words, our interpretation provides us with a syntactic approach, without any further construction of automata on infinite words. Via Dialectica, as linear negation directly corresponds to switching players in games, we furthermore obtain a complete logic: either a closed formula or its linear negation is provable. This completely axiomatizes the theory of the realizability model of Open image in new window. Besides, this shows that in principle, one can solve Churchâs synthesis for a given ââ -formula by only looking for proofs of either that formula or its linear negation
Comparison-Free Polyregular Functions.
This paper introduces a new automata-theoretic class of string-to-string functions with polynomialgrowth. Several equivalent definitions are provided: a machine model which is a restricted variant ofpebble transducers, and a few inductive definitions that close the class of regular functions undercertain 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 polyregularfunctions; we prove that the inclusion is strict. We also show that they are incomparable withHDT0L transductions, closed under usual function composition â but not under a certain âmapâcombinator â and satisfy a comparison-free version of the pebble minimization theorem.On the broader topic of polynomial growth transductions, we also consider the recently introducedlayered streaming string transducers (SSTs), or equivalently k-marble transducers. We prove that afunction 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 âmapâ and equivalent to acorresponding notion of (k + 1)-layered HDT0L systems
Kleene Algebra with Hypotheses
We study the Horn theories of Kleene algebras and star continuous Kleene algebras, from the complexity point of view. While their equational theories coincide and are PSpace-complete, their Horn theories differ and are undecidable. We characterise the Horn theory of star continuous Kleene algebras in terms of downward closed languages and we show that when restricting the shape of allowed hypotheses, the problems lie in various levels of the arithmetical or analytical hierarchy. We also answer a question posed by Cohen about hypotheses of the form 1=S where S is a sum of letters: we show that it is decidable
From Normal Functors to Logarithmic Space Queries
We 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
Cantor-Bernstein implies Excluded Middle
We 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 MartĂn EscardĂł stating that quantification over a particular subset of the Cantor space â â 2, the so-called one-point compactification of â, preserves decidable predicates
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 a proof witnessing that the query is determined by
the views
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
A Curry-Howard Approach to Church's Synthesis
Church's synthesis problem asks whether there exists a finite-state stream
transducer satisfying a given input-output specification. For specifications
written in Monadic Second-Order Logic (MSO) over infinite words, Church's
synthesis can theoretically be solved algorithmically using automata and games.
We revisit Church's synthesis via the Curry-Howard correspondence by
introducing SMSO, an intuitionistic variant of MSO over infinite words, which
is shown to be sound and complete w.r.t. synthesis thanks to an automata-based
realizability model