37,639 research outputs found
A note on the expressive power of linear orders
This article shows that there exist two particular linear orders such that
first-order logic with these two linear orders has the same expressive power as
first-order logic with the Bit-predicate FO(Bit). As a corollary we obtain that
there also exists a built-in permutation such that first-order logic with a
linear order and this permutation is as expressive as FO(Bit)
The succinctness of first-order logic on linear orders
Succinctness is a natural measure for comparing the strength of different
logics. Intuitively, a logic L_1 is more succinct than another logic L_2 if all
properties that can be expressed in L_2 can be expressed in L_1 by formulas of
(approximately) the same size, but some properties can be expressed in L_1 by
(significantly) smaller formulas.
We study the succinctness of logics on linear orders. Our first theorem is
concerned with the finite variable fragments of first-order logic. We prove
that:
(i) Up to a polynomial factor, the 2- and the 3-variable fragments of
first-order logic on linear orders have the same succinctness. (ii) The
4-variable fragment is exponentially more succinct than the 3-variable
fragment. Our second main result compares the succinctness of first-order logic
on linear orders with that of monadic second-order logic. We prove that the
fragment of monadic second-order logic that has the same expressiveness as
first-order logic on linear orders is non-elementarily more succinct than
first-order logic
The succinctness of first-order logic on linear orders
Succinctness is a natural measure for comparing the strength of different logics. Intuitively, a logic L_1 is more succinct than another logic L_2 if all properties that can be expressed in L_2 can be expressed in L_1 by formulas of (approximately) the same size, but some properties can be expressed in L_1 by (significantly) smaller formulas.
We study the succinctness of logics on linear orders. Our first theorem is concerned with the finite variable fragments of first-order logic. We prove that:
(i) Up to a polynomial factor, the 2- and the 3-variable fragments of first-order logic on linear orders have the same succinctness. (ii) The 4-variable fragment is exponentially more succinct than the 3-variable fragment. Our second main result compares the succinctness of first-order logic on linear orders with that of monadic second-order logic. We prove that the fragment of monadic second-order logic that has the same expressiveness as first-order logic on linear orders is non-elementarily more succinct than first-order logic
Order Invariance on Decomposable Structures
Order-invariant formulas access an ordering on a structure's universe, but
the model relation is independent of the used ordering. Order invariance is
frequently used for logic-based approaches in computer science. Order-invariant
formulas capture unordered problems of complexity classes and they model the
independence of the answer to a database query from low-level aspects of
databases. We study the expressive power of order-invariant monadic
second-order (MSO) and first-order (FO) logic on restricted classes of
structures that admit certain forms of tree decompositions (not necessarily of
bounded width).
While order-invariant MSO is more expressive than MSO and, even, CMSO (MSO
with modulo-counting predicates), we show that order-invariant MSO and CMSO are
equally expressive on graphs of bounded tree width and on planar graphs. This
extends an earlier result for trees due to Courcelle. Moreover, we show that
all properties definable in order-invariant FO are also definable in MSO on
these classes. These results are applications of a theorem that shows how to
lift up definability results for order-invariant logics from the bags of a
graph's tree decomposition to the graph itself.Comment: Accepted for LICS 201
A Proof of Stavi's Theorem
Kamp's theorem established the expressive equivalence of the temporal logic
with Until and Since and the First-Order Monadic Logic of Order (FOMLO) over
the Dedekind-complete time flows. However, this temporal logic is not
expressively complete for FOMLO over the rationals. Stavi introduced two
additional modalities and proved that the temporal logic with Until, Since and
Stavi's modalities is expressively equivalent to FOMLO over all linear orders.
We present a simple proof of Stavi's theorem.Comment: arXiv admin note: text overlap with arXiv:1401.258
Order-Invariant First-Order Logic over Hollow Trees
We show that the expressive power of order-invariant first-order logic collapses to first-order logic over hollow trees. A hollow tree is an unranked ordered tree where every non leaf node has at most four adjacent nodes: two siblings (left and right) and its first and last children. In particular there is no predicate for the linear order among siblings nor for the descendant relation. Moreover only the first and last nodes of a siblinghood are linked to their parent node, and the parent-child relation cannot be completely reconstructed in first-order
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