16 research outputs found
Local Normal Forms for First-Order Logic with Applications to Games and Automata
Building on work of Gaifman [Gai82] it is shown that every first-order formula is logically equivalent to a formula of the form ∃ x_1,...,x_l, \forall y, φ where φ is r-local around y, i.e. quantification in φ is restricted to elements of the universe of distance at most r from y. \par From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs. \par As another application, automata models are defined that have, on arbitrary classes of relational structures, exactly the expressive power of first-order logic and existential monadic second-order logic, respectively
Subshifts as Models for MSO Logic
We study the Monadic Second Order (MSO) Hierarchy over colourings of the
discrete plane, and draw links between classes of formula and classes of
subshifts. We give a characterization of existential MSO in terms of
projections of tilings, and of universal sentences in terms of combinations of
"pattern counting" subshifts. Conversely, we characterise logic fragments
corresponding to various classes of subshifts (subshifts of finite type, sofic
subshifts, all subshifts). Finally, we show by a separation result how the
situation here is different from the case of tiling pictures studied earlier by
Giammarresi et al.Comment: arXiv admin note: substantial text overlap with arXiv:0904.245
An automaton over data words that captures EMSO logic
We develop a general framework for the specification and implementation of
systems whose executions are words, or partial orders, over an infinite
alphabet. As a model of an implementation, we introduce class register
automata, a one-way automata model over words with multiple data values. Our
model combines register automata and class memory automata. It has natural
interpretations. In particular, it captures communicating automata with an
unbounded number of processes, whose semantics can be described as a set of
(dynamic) message sequence charts. On the specification side, we provide a
local existential monadic second-order logic that does not impose any
restriction on the number of variables. We study the realizability problem and
show that every formula from that logic can be effectively, and in elementary
time, translated into an equivalent class register automaton
Subshifts as Models for MSO Logic
We study the Monadic Second Order (MSO) Hierarchy over colourings of the discrete plane, and draw links between classes of formula and classes of subshifts. We give a characterization of existential MSO in terms of projections of tilings, and of universal sentences in terms of combinations of ''pattern counting'' subshifts. Conversely, we characterise logic fragments corresponding to various classes of subshifts (subshifts of finite type, sofic subshifts, all subshifts). Finally, we show by a separation result how the situation here is different from the case of tiling pictures studied earlier by Giammarresi et al
Local Normal Forms for First-Order Logic with Applications to Games and Automata
Building on work of Gaifman [Gai82] it is shown that every first-order formula is logically equivalent to a formula of the form 9x 1 ; : : : ; x l 8y' where ' is r-local around y, i. e. quantification in ' is restricted to elements of the universe of distance at most r from y. From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs. As another application, automata models are defined that have, on arbitrary classes of relational structures, exactly the expressive power of firstorder logic and existential monadic second-order logic, respectively. 1 Introduction First-order (FO) logic and its extensions play an important role in many branches of (theoretical) computer science. ..
Local Normal Forms for First-Order Logic with Applications to Games and Automata
Building on work of Gaifman [Gai82] it is shown that every first-order formula is logically equivalent to a formula of the form ∃ x_1,...,x_l, \forall y, φ where φ is r-local around y, i.e. quantification in φ is restricted to elements of the universe of distance at most r from y. \par From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs. \par As another application, automata models are defined that have, on arbitrary classes of relational structures, exactly the expressive power of first-order logic and existential monadic second-order logic, respectively
Local Normal Forms for First-Order Logic with Applications to Games and Automata
this paper are automata theory and descriptive complexity. Since Buchi's and Elgot's famous characterization of the regular string languages as the sets of models of (existential) monadic second-order (MSO) sentences, (existential) MSO logic has been used as a guideline in the search for reasonable automata models for other kinds of structures like trees or graphs. In descriptive complexity, since Fagin [Fag74] showed that the complexity class NP coincides with the sets of models of existential secondorder (