In web development, the CSS box model refers to how HTML elements are modeled in browser engines and how the dimensions of those HTML elements are derived from CSS properties. It is a fundamental concept for the composition of HTML webpages. The guidelines of the box model are described by web standards World Wide Web Consortium (W3C) specifically the CSS Working Group. For much of the late-1990s and early 2000s there had been non-standard compliant implementations of the box model in mainstream browsers. With the advent of CSS2 in 1998, which introduced the box-sizing property, the problem had mostly been resolved. == Specifics == The Cascading Style Sheets (CSS) specification describes how elements of web pages are displayed by graphical browsers. Section 4 of the CSS1 specification defines a "formatting model" that gives block-level elements—such as p and blockquote—a width and height, and three levels of boxes surrounding it: padding, borders, and margins. While the specification never uses the term "box model" explicitly, the term has become widely used by web developers and web browser vendors. All HTML elements can be considered "boxes", this includes div tag, p tag, or a tag. Each of those boxes has five modifiable dimensions: the height and width describe dimensions of the actual content of the box (text, images, ...) the padding describes the space between this content and the border of the box the border is any kind of line (solid, dotted, dashed...) surrounding the box, if present the margin is the space around the border According to the CSS1 specification, released by W3C in 1996 and revised in 1999, when a width or height is explicitly specified for any block-level element, it should determine only the width or height of the visible element, with the padding, borders, and margins applied afterward. Before CSS3, this box model was known as W3C box model, in CSS3, it is known as the content-box. The total width of a box is therefore margin-left + border-left + padding-left + width + padding-right + border-right + margin-right. Similarly, the total height of a box equals margin-top + border-top + padding-top + height + padding-bottom + border-bottom + margin-bottom. For example, the following CSS code would specify the box dimensions of each block belonging to 'my-class'. Moreover, each such box will have total height 140px and width 240px. CSS3 introduced the Internet Explorer box model to the standard, known referred to as border-box. == History == Before HTML 4 and CSS, very few HTML elements supported both border and padding, so the definition of the width and height of an element was not very contentious. However, it varied depending on the element. The HTML width attribute of a table defined the width of the table including its border. On the other hand, the HTML width attribute of an image defined the width of the image itself (inside any border). The only element to support padding in those early days was the table cell. Width for the cell was defined as "the suggested width for a cell content in pixels excluding the cell padding." In 1996, CSS introduced margin, border and padding for many more elements. It adopted a definition width in relation to content, border, margin and padding similar to that for a table cell. This has since become known as the W3C box model. At the time, very few browser vendors implemented the W3C box model to the letter. The two major browsers at the time, Netscape 4.0 and Internet Explorer 4.0 both defined width and height as the distance from border to border. This has been referred to as the traditional or the Internet Explorer box model. Internet Explorer in "quirks mode" includes the content, padding and borders within a specified width or height; this results in a narrower or shorter rendering of a box than would result following the standard behavior. The Internet Explorer box model behavior was often considered a bug, because of the way in which earlier versions of Internet Explorer handle the box model or sizing of elements in a web page, which differs from the standard way recommended by the W3C for the Cascading Style Sheets language. As of Internet Explorer 6, the browser supports an alternative rendering mode (called the "standards-compliant mode") which solves this discrepancy. However, for backward compatibility reasons, all versions still behave in the usual, non-standard way by default (see quirks mode). Internet Explorer for Mac is not affected by this non-standard behavior. === Workarounds === Internet Explorer versions 6 and onward are not affected by the bug if the page contains certain HTML document type declarations. These versions maintain the buggy behavior when in quirks mode for reasons of backward compatibility. For example, quirks mode is triggered: When the document type declaration is absent or incomplete; When an HTML 3 or earlier document is encountered; When an HTML 4.0 Transitional or Frameset document type declaration is used and a system identifier (URI) is not present; When an SGML comment or other unrecognized content appears before the document type declaration Internet Explorer 6 also uses quirks mode if there is an XML declaration prior to the document type declaration. Various workarounds have been devised to force Internet Explorer versions 5 and earlier to display Web pages using the W3C box model. These workarounds generally exploit unrelated bugs in Internet Explorer's CSS selector processing in order to hide certain rules from the browser. The best known of these workarounds is the "box model hack" developed by Tantek Çelik, a former Microsoft employee who developed this idea while working on Internet Explorer for the Macintosh. It involves specifying a width declaration for Internet Explorer for Windows, and then overriding it with another width declaration for CSS-compliant browsers. This second declaration is hidden from Internet Explorer for Windows by exploiting other bugs in the way that it parses CSS rules. The implementation of these CSS “hacks” has been further complicated by the public release of Internet Explorer 7, which has had some issues fixed, but not others, causing undesired results in pages using these hacks. Box model hacks have proven unreliable because they rely on bugs in browsers' CSS support that may be fixed in later versions. For this reason, some Web developers have instead recommended either avoiding specifying both width and padding for the same element or using conditional comment and/or CSS filters to work around the box model bug in older versions of Internet Explorer. == Support for Internet Explorer's box model == Web designer Doug Bowman has said that the original Internet Explorer box model represents a better, more logical approach. Peter-Paul Koch gives the example of a physical box, whose dimensions always refer to the box itself, including potential padding, but never its content. He says that this box model is more useful for graphic designers, who create designs based on the visible width of boxes rather than the width of their content. Bernie Zimmermann says that the Internet Explorer box model is closer to the definition of cell dimensions and padding used in the HTML table model. The W3C has included a "box-sizing" property in CSS3. When box-sizing: border-box; is specified for an element, any padding or border of the element is drawn inside the specified width and height, "as commonly implemented by legacy HTML user agents". Internet Explorer 8, WebKit browsers such as Apple Safari 5.1+ and Google Chrome, Gecko-based browsers such as Mozilla Firefox 29.0 and later, Opera 7.0 and later, and Konqueror 3.3.2 and later support the CSS3 box-sizing property. Gecko browsers previous than 29.0 support the same functionality using the browser-specific -moz-box-sizing property. border-box is the default box model used in Bootstrap framework.
Description logic
Description logics (DL) are a family of formal knowledge representation languages. Many DLs are more expressive than propositional logic but less expressive than first-order logic. In contrast to the latter, the core reasoning problems for DLs are (usually) decidable, and efficient decision procedures have been designed and implemented for these problems. There are general, spatial, temporal, spatiotemporal, and fuzzy description logics, and each description logic features a different balance between expressive power and reasoning complexity by supporting different sets of mathematical constructors. DLs are used in artificial intelligence to describe and reason about the relevant concepts of an application domain (known as terminological knowledge). It is of particular importance in providing a logical formalism for ontologies and the Semantic Web: the Web Ontology Language (OWL) and its profiles are based on DLs. A major area of application of DLs and OWL is in biomedical informatics, where they assist in the codification of biomedical knowledge. DLs and OWL are also applied in other domains, including defense, climate modeling, and large-scale industrial knowledge graphs. == Introduction == A DL models concepts, roles and individuals, and their relationships. The fundamental modeling concept of a DL is the axiom—a logical statement relating roles and/or concepts. This is a key difference from the frames paradigm where a frame specification declares and completely defines a class. == Nomenclature == === Terminology compared to FOL and OWL === The description logic community uses different terminology than the first-order logic (FOL) community for operationally equivalent notions; some examples are given below. The Web Ontology Language (OWL) uses again a different terminology, also given in the table below. === Naming convention === There are many varieties of description logics and there is an informal naming convention, roughly describing the operators allowed. The expressivity is encoded in the label for a logic starting with one of the following basic logics: Followed by any of the following extensions: ==== Exceptions ==== Some canonical DLs that do not exactly fit this convention are: ==== Examples ==== As an example, A L C {\displaystyle {\mathcal {ALC}}} is a centrally important description logic from which comparisons with other varieties can be made. A L C {\displaystyle {\mathcal {ALC}}} is simply A L {\displaystyle {\mathcal {AL}}} with complement of any concept allowed, not just atomic concepts. A L C {\displaystyle {\mathcal {ALC}}} is used instead of the equivalent A L U E {\displaystyle {\mathcal {ALUE}}} . A further example, the description logic S H I Q {\displaystyle {\mathcal {SHIQ}}} is the logic A L C {\displaystyle {\mathcal {ALC}}} plus extended cardinality restrictions, and transitive and inverse roles. The naming conventions aren't purely systematic so that the logic A L C O I N {\displaystyle {\mathcal {ALCOIN}}} might be referred to as A L C N I O {\displaystyle {\mathcal {ALCNIO}}} and other abbreviations are also made where possible. The Protégé ontology editor supports S H O I N ( D ) {\displaystyle {\mathcal {SHOIN}}^{\mathcal {(D)}}} . Three major biomedical informatics terminology bases, SNOMED CT, GALEN, and GO, are expressible in E L {\displaystyle {\mathcal {EL}}} (with additional role properties). OWL 2 provides the expressiveness of S R O I Q ( D ) {\displaystyle {\mathcal {SROIQ}}^{\mathcal {(D)}}} , OWL-DL is based on S H O I N ( D ) {\displaystyle {\mathcal {SHOIN}}^{\mathcal {(D)}}} , and for OWL-Lite it is S H I F ( D ) {\displaystyle {\mathcal {SHIF}}^{\mathcal {(D)}}} . == History == Description logic was given its current name in the 1980s. Previous to this it was called (chronologically): terminological systems, and concept languages. === Knowledge representation === Frames and semantic networks lack formal (logic-based) semantics. DL was first introduced into knowledge representation (KR) systems to overcome this deficiency. The first DL-based KR system was KL-ONE (by Ronald J. Brachman and Schmolze, 1985). During the '80s other DL-based systems using structural subsumption algorithms were developed including KRYPTON (1983), LOOM (1987), BACK (1988), K-REP (1991) and CLASSIC (1991). This approach featured DL with limited expressiveness but relatively efficient (polynomial time) reasoning. In the early '90s, the introduction of a new tableau based algorithm paradigm allowed efficient reasoning on more expressive DL. DL-based systems using these algorithms — such as KRIS (1991) — show acceptable reasoning performance on typical inference problems even though the worst case complexity is no longer polynomial. From the mid '90s, reasoners were created with good practical performance on very expressive DL with high worst case complexity. Examples from this period include FaCT, RACER (2001), CEL (2005), and KAON 2 (2005). DL reasoners, such as FaCT, FaCT++, RACER, DLP and Pellet, implement the method of analytic tableaux. KAON2 is implemented by algorithms which reduce a SHIQ(D) knowledge base to a disjunctive datalog program. === Semantic web === The DARPA Agent Markup Language (DAML) and Ontology Inference Layer (OIL) ontology languages for the Semantic Web can be viewed as syntactic variants of DL. In particular, the formal semantics and reasoning in OIL use the S H I Q {\displaystyle {\mathcal {SHIQ}}} DL. The DAML+OIL DL was developed as a submission to—and formed the starting point of—the World Wide Web Consortium (W3C) Web Ontology Working Group. In 2004, the Web Ontology Working Group completed its work by issuing the OWL recommendation. The design of OWL is based on the S H {\displaystyle {\mathcal {SH}}} family of DL with OWL DL and OWL Lite based on S H O I N ( D ) {\displaystyle {\mathcal {SHOIN}}^{\mathcal {(D)}}} and S H I F ( D ) {\displaystyle {\mathcal {SHIF}}^{\mathcal {(D)}}} respectively. The W3C OWL Working Group began work in 2007 on a refinement of - and extension to - OWL. In 2009, this was completed by the issuance of the OWL2 recommendation. OWL2 is based on the description logic S R O I Q ( D ) {\displaystyle {\mathcal {SROIQ}}^{\mathcal {(D)}}} . Practical experience demonstrated that OWL DL lacked several key features necessary to model complex domains. == Modeling == === TBox vs Abox === In DL, a distinction is drawn between the so-called TBox (terminological box) and the ABox (assertional box). In general, the TBox contains sentences describing concept hierarchies (i.e., relations between concepts) while the ABox contains ground sentences stating where in the hierarchy, individuals belong (i.e., relations between individuals and concepts). For example, the statement: belongs in the TBox, while the statement: belongs in the ABox. Note that the TBox/ABox distinction is not significant, in the same sense that the two "kinds" of sentences are not treated differently in first-order logic (which subsumes most DL). When translated into first-order logic, a subsumption axiom like (1) is simply a conditional restriction to unary predicates (concepts) with only variables appearing in it. Clearly, a sentence of this form is not privileged or special over sentences in which only constants ("grounded" values) appear like (2). === Motivation for having Tbox and Abox === So why was the distinction introduced? The primary reason is that the separation can be useful when describing and formulating decision-procedures for various DL. For example, a reasoner might process the TBox and ABox separately, in part because certain key inference problems are tied to one but not the other one ('classification' is related to the TBox, 'instance checking' to the ABox). Another example is that the complexity of the TBox can greatly affect the performance of a given decision-procedure for a certain DL, independently of the ABox. Thus, it is useful to have a way to talk about that specific part of the knowledge base. The secondary reason is that the distinction can make sense from the knowledge base modeler's perspective. It is plausible to distinguish between our conception of terms/concepts in the world (class axioms in the TBox) and particular manifestations of those terms/concepts (instance assertions in the ABox). In the above example: when the hierarchy within a company is the same in every branch but the assignment to employees is different in every department (because there are other people working there), it makes sense to reuse the TBox for different branches that do not use the same ABox. There are two features of description logic that are not shared by most other data description formalisms: DL does not make the unique name assumption (UNA) or the closed-world assumption (CWA). Not having UNA means that two concepts with different names may be allowed by some inference to be shown to be equivalent. Not having CWA, or rather having the open world assumption (OWA) means that
Tang Xiao'ou
Tang Xiao'ou (汤晓鸥; 24 January 1968 – 15 December 2023) was a Chinese businessman and computer scientist. He was the founder and chairman of SenseTime, an AI company. He also served as professor of information engineering, associate dean of engineering, and outstanding fellow of engineering at the Chinese University of Hong Kong. Tang's research primarily focused on areas such as computer vision, pattern recognition, and video processing. Tang was honored with the Best Paper Award at the 2009 IEEE Conference on Computer Vision and Pattern Recognition. He served as the programme chair in 2009 and the general chair in 2019 for the IEEE International Conference on Computer Vision. His editorial contributions include roles as an Associate Editor for both the IEEE Transactions on Pattern Analysis and Machine Intelligence and the International Journal of Computer Vision. Additionally, Tang has been recognised as a Fellow of the IEEE. == Biography == Tang was born in Anshan, Liaoning, northeastern China in 1968. Tang received a Bachelor of Science with a major in computer science from the University of Science and Technology of China in 1990. He received a Master of Science from the University of Rochester in 1991 and a Doctor of Philosophy in ocean engineering from the Massachusetts Institute of Technology in 1996. He worked at MIT and Woods Hole Oceanographic Institution during his doctoral studies. Funders of his research included the Office of Naval Research of the United States Department of the Navy. After graduating from MIT, Tang taught in the Department of Information Engineering of the Chinese University of Hong Kong. In 2001, he founded the Multimedia Laboratory of the Chinese University of Hong Kong. From 2005 to 2008, he worked at Microsoft Research Asia. He served as Associate Dean of the Chinese University of Hong Kong. In 2014, he spearheaded the first facial recognition to beat human accuracy. Tang co-founded SenseTime with Xu Li in 2014. Upon SenseTime's IPO in December 2021, Tang was estimated to have a net worth of approximately $3.4 billion. Tang died on 15 December 2023, at the age of 55. SenseTime made the announcement the next day and changed the colour scheme of its website to black-and-white in mourning. The Chinese University of Hong Kong also changed his faculty page to a black-and-white theme.
Büchi automaton
In computer science and automata theory, a deterministic Büchi automaton is a theoretical machine which either accepts or rejects infinite inputs. Such a machine has a set of states and a transition function, which determines which state the machine should move to from its current state when it reads the next input character. Some states are accepting states and one state is the start state. The machine accepts an input if and only if it will pass through an accepting state infinitely many times as it reads the input. A non-deterministic Büchi automaton, later referred to just as a Büchi automaton, has a transition function which may have multiple outputs, leading to many possible paths for the same input; it accepts an infinite input if and only if some possible path is accepting. Deterministic and non-deterministic Büchi automata generalize deterministic finite automata and nondeterministic finite automata to infinite inputs. Each are types of ω-automata. Büchi automata recognize the ω-regular languages, the infinite word version of regular languages. They are named after the Swiss mathematician Julius Richard Büchi, who invented them in 1962. Büchi automata are often used in model checking as an automata-theoretic version of a formula in linear temporal logic. == Formal definition == Formally, a deterministic Büchi automaton is a tuple A = ( Q , Σ , δ , q 0 , F ) {\textstyle A=(Q,\Sigma ,\delta ,q_{0},\mathbf {F} )} that consists of the following components: Q {\textstyle Q} is a finite set. The elements of Q {\textstyle Q} are called the states of A {\textstyle A} . Σ {\textstyle \Sigma } is a finite set called the alphabet of A {\textstyle A} . δ : Q × Σ → Q {\textstyle \delta \colon Q\times \Sigma \to Q} is a function, called the transition function of A {\textstyle A} . q 0 {\textstyle q_{0}} is an element of Q {\textstyle Q} , called the initial state of A {\textstyle A} . F ⊆ Q {\textstyle \mathbf {F} \subseteq Q} is the acceptance condition. A run i _ = i 0 i 1 i 2 ⋯ ∈ Σ ω {\displaystyle {\underline {i}}=i_{0}i_{1}i_{2}\cdots \in \Sigma ^{\omega }} is an infinite string of inputs of A {\displaystyle A} . By calling δ {\displaystyle \delta } recursively, we can extend it to a function δ ω : Σ ω → Q ω {\displaystyle \delta ^{\omega }:\Sigma ^{\omega }\to Q^{\omega }} . A state q ∈ Q {\displaystyle q\in Q} is said to occur infinitely often for a run i _ {\displaystyle {\underline {i}}} when the set { n ∈ N ∣ δ ω ( i _ ) n = q } {\displaystyle \{n\in \mathbb {N} \mid \delta ^{\omega }({\underline {i}})_{n}=q\}} is infinite. Let I n f ( i _ ) {\displaystyle \mathrm {Inf} ({\underline {i}})} be the set of states occurring infinitely often for i _ {\displaystyle {\underline {i}}} . The language of A {\displaystyle A} is then the set of runs of A {\displaystyle A} in which at least one of the infinitely-often occurring states is in F {\textstyle \mathbf {F} } ; in symbols: L ( A ) = { i _ ∈ Σ ω ∣ I n f ( i _ ) ∩ F ≠ ∅ } . {\displaystyle L(A)=\{{\underline {i}}\in \Sigma ^{\omega }\mid \mathrm {Inf} ({\underline {i}})\cap \mathbf {F} \neq \varnothing \}.} In a (non-deterministic) Büchi automaton, the transition function δ {\textstyle \delta } is replaced with a transition relation Δ {\textstyle \Delta } that returns a set of states, and the single initial state q 0 {\textstyle q_{0}} is replaced by a set I {\textstyle I} of initial states. Generally, the term Büchi automaton without qualifier refers to non-deterministic Büchi automata. For more comprehensive formalism see also ω-automaton. == Closure properties == The set of Büchi automata is closed under the following operations. Let A = ( Q A , Σ , Δ A , I A , F A ) {\displaystyle A=(Q_{A},\Sigma ,\Delta _{A},I_{A},{F}_{A})} and B = ( Q B , Σ , Δ B , I B , F B ) {\displaystyle B=(Q_{B},\Sigma ,\Delta _{B},I_{B},{F}_{B})} be Büchi automata and C = ( Q C , Σ , Δ C , I C , F C ) {\displaystyle C=(Q_{C},\Sigma ,\Delta _{C},I_{C},{F}_{C})} be a finite automaton. Union: There is a Büchi automaton that recognizes the language L ( A ) ∪ L ( B ) . {\displaystyle L(A)\cup L(B).} Proof: If we assume, w.l.o.g., Q A ∩ Q B {\displaystyle Q_{A}\cap Q_{B}} is empty then L ( A ) ∪ L ( B ) {\displaystyle L(A)\cup L(B)} is recognized by the Büchi automaton ( Q A ∪ Q B , Σ ∪ Σ , Δ A ∪ Δ B , I A ∪ I B , F A ∪ F B ) . {\displaystyle (Q_{A}\cup Q_{B},\Sigma \cup \Sigma ,\Delta _{A}\cup \Delta _{B},I_{A}\cup I_{B},{F}_{A}\cup {F}_{B}).} Intersection: There is a Büchi automaton that recognizes the language L ( A ) ∩ L ( B ) . {\displaystyle L(A)\cap L(B).} Proof: The Büchi automaton A ′ = ( Q ′ , Σ , Δ ′ , I ′ , F ′ ) {\displaystyle A'=(Q',\Sigma ,\Delta ',I',F')} recognizes L ( A ) ∩ L ( B ) , {\displaystyle L(A)\cap L(B),} where Q ′ = Q A × Q B × { 1 , 2 } {\displaystyle Q'=Q_{A}\times Q_{B}\times \{1,2\}} Δ ′ = Δ 1 ∪ Δ 2 {\displaystyle \Delta '=\Delta _{1}\cup \Delta _{2}} Δ 1 = { ( ( q A , q B , 1 ) , a , ( q A ′ , q B ′ , i ) ) | ( q A , a , q A ′ ) ∈ Δ A and ( q B , a , q B ′ ) ∈ Δ B and if q A ∈ F A then i = 2 else i = 1 } {\displaystyle \Delta _{1}=\{((q_{A},q_{B},1),a,(q'_{A},q'_{B},i))|(q_{A},a,q'_{A})\in \Delta _{A}{\text{ and }}(q_{B},a,q'_{B})\in \Delta _{B}{\text{ and if }}q_{A}\in F_{A}{\text{ then }}i=2{\text{ else }}i=1\}} Δ 2 = { ( ( q A , q B , 2 ) , a , ( q A ′ , q B ′ , i ) ) | ( q A , a , q A ′ ) ∈ Δ A and ( q B , a , q B ′ ) ∈ Δ B and if q B ∈ F B then i = 1 else i = 2 } {\displaystyle \Delta _{2}=\{((q_{A},q_{B},2),a,(q'_{A},q'_{B},i))|(q_{A},a,q'_{A})\in \Delta _{A}{\text{ and }}(q_{B},a,q'_{B})\in \Delta _{B}{\text{ and if }}q_{B}\in F_{B}{\text{ then }}i=1{\text{ else }}i=2\}} I ′ = I A × I B × { 1 } {\displaystyle I'=I_{A}\times I_{B}\times \{1\}} F ′ = { ( q A , q B , 2 ) | q B ∈ F B } {\displaystyle F'=\{(q_{A},q_{B},2)|q_{B}\in F_{B}\}} By construction, r ′ = ( q A 0 , q B 0 , i 0 ) , ( q A 1 , q B 1 , i 1 ) , … {\displaystyle r'=(q_{A}^{0},q_{B}^{0},i^{0}),(q_{A}^{1},q_{B}^{1},i^{1}),\dots } is a run of automaton A' on input word w {\textstyle w} if r A = q A 0 , q A 1 , … {\displaystyle r_{A}=q_{A}^{0},q_{A}^{1},\dots } is run of A {\textstyle A} on w {\textstyle w} and r B = q B 0 , q B 1 , … {\displaystyle r_{B}=q_{B}^{0},q_{B}^{1},\dots } is run of B {\textstyle B} on w {\textstyle w} . r A {\textstyle r_{A}} is accepting and r B {\textstyle r_{B}} is accepting if r ′ {\textstyle r'} is concatenation of an infinite series of finite segments of 1-states (states with third component 1) and 2-states (states with third component 2) alternatively. There is such a series of segments of r ′ {\textstyle r'} if r ′ {\textstyle r'} is accepted by A ′ {\textstyle A'} . Concatenation: There is a Büchi automaton that recognizes the language L ( C ) ⋅ L ( A ) . {\displaystyle L(C)\cdot L(A).} Proof: If we assume, w.l.o.g., Q C ∩ Q A {\displaystyle Q_{C}\cap Q_{A}} is empty then the Büchi automaton A ′ = ( Q C ∪ Q A , Σ , Δ ′ , I ′ , F A ) {\displaystyle A'=(Q_{C}\cup Q_{A},\Sigma ,\Delta ',I',F_{A})} recognizes L ( C ) ⋅ L ( A ) {\displaystyle L(C)\cdot L(A)} , where Δ ′ = Δ A ∪ Δ C ∪ { ( q , a , q ′ ) | q ′ ∈ I A and ∃ f ∈ F C . ( q , a , f ) ∈ Δ C } {\displaystyle \Delta '=\Delta _{A}\cup \Delta _{C}\cup \{(q,a,q')|q'\in I_{A}{\text{ and }}\exists f\in F_{C}.(q,a,f)\in \Delta _{C}\}} if I C ∩ F C is empty then I ′ = I C otherwise I ′ = I C ∪ I A {\displaystyle {\text{ if }}I_{C}\cap F_{C}{\text{ is empty then }}I'=I_{C}{\text{ otherwise }}I'=I_{C}\cup I_{A}} ω-closure: If L ( C ) {\displaystyle L(C)} does not contain the empty word then there is a Büchi automaton that recognizes the language L ( C ) ω . {\displaystyle L(C)^{\omega }.} Proof: The Büchi automaton that recognizes L ( C ) ω {\displaystyle L(C)^{\omega }} is constructed in two stages. First, we construct a finite automaton A ′ {\textstyle A'} such that A ′ {\textstyle A'} also recognizes L ( C ) {\displaystyle L(C)} but there are no incoming transitions to initial states of A ′ {\textstyle A'} . So, A ′ = ( Q C ∪ { q new } , Σ , Δ ′ , { q new } , F C ) , {\displaystyle A'=(Q_{C}\cup \{q_{\text{new}}\},\Sigma ,\Delta ',\{q_{\text{new}}\},F_{C}),} where Δ ′ = Δ C ∪ { ( q new , a , q ′ ) | ∃ q ∈ I C . ( q , a , q ′ ) ∈ Δ C } . {\displaystyle \Delta '=\Delta _{C}\cup \{(q_{\text{new}},a,q')|\exists q\in I_{C}.(q,a,q')\in \Delta _{C}\}.} Note that L ( C ) = L ( A ′ ) {\displaystyle L(C)=L(A')} because L ( C ) {\displaystyle L(C)} does not contain the empty string. Second, we will construct the Büchi automaton A ″ {\textstyle A''} that recognize L ( C ) ω {\displaystyle L(C)^{\omega }} by adding a loop back to the initial state of A ′ {\textstyle A'} . So, A ″ = ( Q C ∪ { q new } , Σ , Δ ″ , { q new } , { q new } ) {\displaystyle A''=(Q_{C}\cup \{q_{\text{new}}\},\Sigma ,\Delta '',\{q_{\text{new}}\},\{q_{\text{new}}\})} , where Δ ″ = Δ ′ ∪ { ( q , a , q new ) | ∃ q ′ ∈ F C . ( q , a , q ′ ) ∈ Δ ′ } . {\displaystyle \Delta ''=\Delta '\cup \{(q,a,q_{\text{new}})|\exists q'\in F_{C}.(q,a,q')\in \Delta '\}.} Complementation:
Volker Markl
Volker Markl (born 1971) is a German computer scientist and database systems researcher. == Career == In 1999, Markl received his PhD in computer science under the direction of Rudolf Bayer at the Technical University of Munich. His doctoral research led to the development of the UB-Tree. From 1997 to 2000, he was research group leader at FORWISS, the Bavarian research center for knowledge-based systems. From 2001 to 2008, he was project leader at the IBM Almaden Research Center, Silicon Valley. Since 2008, he has been full professor and Chair of the Database Systems and Information Management Group at Technische Universität Berlin. Since 2014, he is head of the Intelligent Analytics for Massive Data Research Department at the German Research Centre for Artificial Intelligence (DFKI), Berlin. From 2014 to 2020, he was director of the Berlin Big Data Center (BBDC). From 2018 to 2020, he was co-director of the Berlin Machine Learning Center (BZML). Together with Klaus-Robert Müller he became director of the new Berlin Institute for the Foundations of Learning and Data (BIFOLD), after both BBDC and the BZML merged into BIFOLD in 2020. From 2010 through 2019, he led the DFG funded Stratosphere project, which led to the establishment of Apache Flink. In 2018, he was elected president of the VLDB Endowment for a six years period that ended in 2024. == Research == Markl’s research interests lie at the intersection of distributed systems, scalable data processing, and machine learning. == Awards and honors == Markl was elected member of the Berlin-Brandenburg Academy of Sciences and Humanities in 2021. Since 2026 he is member of the German National Academy of Sciences Leopoldina. His work was honoured with several awards, including: 2025 ICDE Best Paper Award 2021 ICDE Best Paper Award 2021 BTW Best Paper Award 2020 ACM SIGMOD Best Paper Award 2020 ACM Fellow 2019 EDBT Best Paper Award 2017 BTW Best Paper Award 2017 EDBT Best Demonstration Award 2016 ACM SIGMOD Research Highlight Award 2014 VLDB Best Paper Award 2012 IBM Faculty Award 2012 IBM Shared University Research Grant 2010 Hewlett Packard Open Innovation Award 2005 IBM Outstanding Technological Achievement Award 2005 IBM Pat Goldberg Best Paper Award
3D-Coat
3DCoat is a commercial digital sculpting program from Pilgway designed to create free-form organic and hard surfaced 3D models, with tools which enable users to sculpt, add polygonal topology (automatically or manually), create UV maps (automatically or manually), texture the resulting models with natural painting tools, and render static images or animated "turntable" movies. The program can also be used to modify imported 3D models from a number of commercial 3D software products by means of plugins called Applinks. Imported models can be converted into voxel objects for further refinement and for adding high resolution detail, complete UV unwrapping and mapping, as well as adding PBR textures for displacement, bump maps, specular and diffuse color maps. A live connection to a chosen external 3D application can be established through the Applink pipeline, allowing for the transfer of model and texture information. 3DCoat specializes in voxel sculpting and polygonal sculpting using dynamic patch tessellation technology and polygonal sculpting tools. It includes "auto-retopology", a proprietary skinning algorithm which generates a polygonal mesh skin over any voxel sculpture, composed primarily of quadrangles.
AI Headshot Generators Reviews: What Actually Works in 2026
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