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

Normalisoidun suurimman uskottavuuden menetelmiä ryvästykseen ja tiheyden arviointiin

The normalized maximum likelihood (NML) distribution has an important position in minimum description length based modelling. Given a set of possible models, the corresponding NML distribution enables optimal encoding according to the worst-case criterion. However, many model classes of practical interest do not have an NML distribution. This thesis introduces solutions for a selection of such cases, including for example one-dimensional normal, uniform and exponential model classes with unrestricted parameters. The new code length functions are based on minimal assumptions about the data, because an approach that would be completely free of any assumptions is not possible in these cases. We also use the new techniques in clustering, as well as in density and entropy estimation applications.Lyhimmän kuvailupituuden periaate on informaatioteoreettinen menetelmä, jolla etsitään oleellista tietoa numeerisesta aineistosta tutkimalla tämän tiivistyvyyttä. Tarkemmin sanottuna tehtävänä on valita aineistoon nähden sopivin todennäköisyysjakauma annetusta kokoelmasta. Normalisoidun suurimman uskottavuuden jakauman käyttäminen on erilaisista valintamenetelmistä erikoisasemassa optimaalisuusominaisuuksiensa ansiosta. Useissa käytännön kannalta tärkeissä asetelmissa tätä jakaumaa ei kuitenkaan ole olemassa. Väitöskirja ehdottaa näihin tilanteisiin menetelmiä, joilla on normalisoidun suurimman uskottavuuden jakauman tyyppisiä ominaisuuksia. Uusia menetelmiä sovelletetaan ryvästykseen sekä tiheyden ja entropian arvioimiseen

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

Kerr-Schild Approach to the Boosted Kerr Solution

Using a complex representation of the Debney-Kerr-Schild (DKS) solutions and the Kerr theorem we analyze the boosted Kerr geometries and give the exact and explicit expressions for the metrics, the principal null congruences, the coordinate systems and the location of the singularities for arbitrary value and orientation of the boost with respect to the angular momentum. In the limiting, ultrarelativistic case we obtain light-like solutions possessing diverging and twisting principal null congruences and having, contrary to the known pp-wave limiting solutions, a non-zero value of the total angular momentum. The implications of the above results in various related fields are discussed.Comment: 16 pages, LaTe

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

Weak models of distributed computing, with connections to modal logic

Peer reviewe

Eclogite-facies shear zones--deep crustal reflectors?

Strongly foliated eclogite-facies rocks in 30-150 m thick shear zones of Caledonian age occur within a Grenvillian garnet granulite-facies gabbro-anorthosite terrain in the Bergen Arcs of Norway. The predominant eclogite-facies mineral assemblages in the shear zones are omphacite + garnet + zoisite + kyanite in gabbroic anorthosite and omphacite + garnet in gabbro. Eclogite-facies rocks in shear zones are generally fine-grained; alternating omphacite/garnet- and kyanite/clinozoisite-rich layers define gneissic layering. A strong shape preferred orientation of omphacite, kyanite, and white mica (phengitic muscovite and/or paragonite) define the foliation. The anorthositic eclogites show omphacite b-axis maxima approximately normal to the foliation and c-axis girdles within the foliation plane. P-wave velocities (Vp) determined at confining pressures to 600 MPa for samples from eclogite-facies shear zones range from 8.3 to 8.5 km s-1 and anisotropy ranges from 1 to 7%. The few samples with more pronounced anisotropy tend to be approximately transversely isotropic with minimum velocities for propagation directions normal to foliation and maximum velocities for propagation directions parallel to foliation. The fast propagation direction lies within the c-axis girdles (parallel to foliation) and the slow propagation direction is parallel to the b-axis concentration (normal to foliation) in samples for which omphacite crystallographic preferred orientation was determined. Vp for the granulite-facies protoliths average about 7.5 km s-1. High calculated reflection coefficients for these shear zones, 0.04-0.14, indicate that they are excellent candidates for deep crustal reflectors in portions of crust that experienced high-pressure conditions but escaped thermal reactivation.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31633/1/0000567.pd