20 research outputs found
The Expressive Power of Modal Dependence Logic
We study the expressive power of various modal logics with team semantics. We
show that exactly the properties of teams that are downward closed and closed
under team k-bisimulation, for some finite k, are definable in modal logic
extended with intuitionistic disjunction. Furthermore, we show that the
expressive power of modal logic with intuitionistic disjunction and extended
modal dependence logic coincide. Finally we establish that any translation from
extended modal dependence logic into modal logic with intuitionistic
disjunction increases the size of some formulas exponentially.Comment: 19 page
Regular Representations of Uniform TC^0
The circuit complexity class DLOGTIME-uniform AC^0 is known to be a modest
subclass of DLOGTIME-uniform TC^0. The weakness of AC^0 is caused by the fact
that AC^0 is not closed under restricting AC^0-computable queries into simple
subsequences of the input. Analogously, in descriptive complexity, the logics
corresponding to DLOGTIME-uniform AC^0 do not have the relativization property
and hence they are not regular. This weakness of DLOGTIME-uniform AC^0 has been
elaborated in the line of research on the Crane Beach Conjecture. The
conjecture (which was refuted by Barrington, Immerman, Lautemann, Schweikardt
and Th{\'e}rien) was that if a language L has a neutral letter, then L can be
defined in first-order logic with the collection of all numerical built-in
relations, if and only if L can be already defined in FO with order.
In the first part of this article we consider logics in the range of AC^0 and
TC^0. First we formulate a combinatorial criterion for a cardinality quantifier
C_S implying that all languages in DLOGTIME-uniform TC^0 can be defined in
FO(C_S). For instance, this criterion is satisfied by C_S if S is the range of
some polynomial with positive integer coefficients of degree at least two. In
the second part of the paper we first adapt the key properties of abstract
logics to accommodate built-in relations. Then we define the regular interior
R-int(L) and regular closure R-cl(L), of a logic L, and show that the Crane
Beach Conjecture can be interpreted as a statement concerning the regular
interior of first-order logic with built-in relations B. We show that if B={+},
or B contains only unary relations besides the order, then R-int(FO_B)
collapses to FO with order. In contrast, our results imply that if B contains
the order and the range of a polynomial of degree at least two, then R-cl(FO_B)
includes all languages in DLOGTIME-uniform TC^0
Weak models of distributed computing, with connections to modal logic
This work presents a classification of weak models of distributed computing. We focus on deterministic distributed algorithms, and we study models of computing that are weaker versions of the widely-studied port-numbering model. In the port-numbering model, a node of degree d receives messages through d input ports and it sends messages through d output ports, both numbered with 1, 2,..., d. In this work, VVc is the class of all graph problems that can be solved in the standard port-numbering model. We study the following subclasses of VVc: VV: Input port i and output port i are not necessarily connected to the same neighbour. MV: Input ports are not numbered; algorithms receive a multiset of messages. SV: Input ports are not numbered; algorithms receive a set of messages. VB: Output ports are not numbered; algorithms send the same message to all output ports. MB: Combination of MV and VB. SB: Combination of SV and VB. Now we have many trivial containment relations, such as SB ⊆ MB ⊆ VB ⊆ VV ⊆ VVc, but it is not obvious if, e.g., either of VB ⊆ SV or SV ⊆ VB should hold. Nevertheless, it turns out that we can identify a linear order on these classes. We prove that SB � MB = VB � SV = MV = VV � VVc. The same holds for the constant-time versions of these classes. We also show that the constant-time variants of these classes can be characterised by a corresponding modal logic. Hence the linear order identified in this work has direct implications in the study of the expressibility of modal logic. Conversely, we can use tools from modal logic to study these classes
Ultrametric spaces bi-Lipschitz embeddable in
It is proved that if an ultrametric space can be bi-Lipschitz embedded in , then its Assouad dimension is less than n. Together with a result of Luukkainen and Movahedi-Lankarani, where the converse was shown, this gives a characterization in terms of Assouad dimension of the ultrametric spaces which are bi-Lipschitz embeddable in