Catie Cuan is an artist, entrepeuneur, and innovator in the field of robotic art and human-robot interaction, where she specializes in choreorobotics, an emerging field at the intersection of choreographic dance and robotics. Catie Cuan is currently one of the academic researchers pioneering the field of choreorobotics and currently holds a post-doctoral fellowship at Stanford University. == Career == Catie Cuan earned a bachelor's degree from the University of California, Berkeley. She graduated with a Ph.D. from the Department of Mechanical Engineering at Stanford University, focusing in robotics. Her most cited publication is about how to improve robotic expressive systems using tools from dance theory, such as the Laban/Bartenieff Movement Analysis. In her most recent research projects, she explores a predictive model of imitation learning for robots moving around humans, a project that advances the field of social robotics. Cuan credits her work in robotics to the experience with her father when he had a stroke and was surrounded by many medical machines, which made her think about how people might feel empowered and hopeful rather than afraid. As a ballet dancer and choreographer, she has performed with the Metropolitan Opera Ballet and the Lyric Opera of Chicago. In 2020, she was the dancer and choreographer of the show Output, which was part of a collaboration with ThoughtWorks Arts and the Pratt Institute. In the production, she danced with an ABB IRB 6700 industrial robot. In 2022, she was named as an IF/THEN ambassador for the American Association for the Advancement of Science. The same year, she was appointed Futurist-in-Residence at the Smithsonian Arts and Industries Building, where she performed at the closing ceremonies of the FUTURES exhibit on July 6, 2022. Cuan has also contributed to product designs, working with IDEO and Dutch interior design firm moooi on their Piro project, which launched a dancing scent diffuser robot during Milan Design Week in June 2022. She is a TED speaker with talks about how to teach robots to dance, and what is coming up for dancing robots in the AI era.
Scale space implementation
In the areas of computer vision, image analysis and signal processing, the notion of scale-space representation is used for processing measurement data at multiple scales, and specifically enhance or suppress image features over different ranges of scale (see the article on scale space). A special type of scale-space representation is provided by the Gaussian scale space, where the image data in N dimensions is subjected to smoothing by Gaussian convolution. Most of the theory for Gaussian scale space deals with continuous images, whereas one when implementing this theory will have to face the fact that most measurement data are discrete. Hence, the theoretical problem arises concerning how to discretize the continuous theory while either preserving or well approximating the desirable theoretical properties that lead to the choice of the Gaussian kernel (see the article on scale-space axioms). This article describes basic approaches for this that have been developed in the literature, see also for an in-depth treatment regarding the topic of approximating the Gaussian smoothing operation and the Gaussian derivative computations in scale-space theory, and for a complementary treatment regarding hybrid discretization methods. == Statement of the problem == The Gaussian scale-space representation of an N-dimensional continuous signal, f C ( x 1 , ⋯ , x N , t ) , {\displaystyle f_{C}\left(x_{1},\cdots ,x_{N},t\right),} is obtained by convolving fC with an N-dimensional Gaussian kernel: g N ( x 1 , ⋯ , x N , t ) . {\displaystyle g_{N}\left(x_{1},\cdots ,x_{N},t\right).} In other words: L ( x 1 , ⋯ , x N , t ) = ∫ u 1 = − ∞ ∞ ⋯ ∫ u N = − ∞ ∞ f C ( x 1 − u 1 , ⋯ , x N − u N , t ) ⋅ g N ( u 1 , ⋯ , u N , t ) d u 1 ⋯ d u N . {\displaystyle L\left(x_{1},\cdots ,x_{N},t\right)=\int _{u_{1}=-\infty }^{\infty }\cdots \int _{u_{N}=-\infty }^{\infty }f_{C}\left(x_{1}-u_{1},\cdots ,x_{N}-u_{N},t\right)\cdot g_{N}\left(u_{1},\cdots ,u_{N},t\right)\,du_{1}\cdots du_{N}.} However, for implementation, this definition is impractical, since it is continuous. When applying the scale space concept to a discrete signal fD, different approaches can be taken. This article is a brief summary of some of the most frequently used methods. == Separability == Using the separability property of the Gaussian kernel g N ( x 1 , … , x N , t ) = G ( x 1 , t ) ⋯ G ( x N , t ) {\displaystyle g_{N}\left(x_{1},\dots ,x_{N},t\right)=G\left(x_{1},t\right)\cdots G\left(x_{N},t\right)} the N-dimensional convolution operation can be decomposed into a set of separable smoothing steps with a one-dimensional Gaussian kernel G along each dimension L ( x 1 , ⋯ , x N , t ) = ∫ u 1 = − ∞ ∞ ⋯ ∫ u N = − ∞ ∞ f C ( x 1 − u 1 , ⋯ , x N − u N , t ) G ( u 1 , t ) d u 1 ⋯ G ( u N , t ) d u N , {\displaystyle L(x_{1},\cdots ,x_{N},t)=\int _{u_{1}=-\infty }^{\infty }\cdots \int _{u_{N}=-\infty }^{\infty }f_{C}(x_{1}-u_{1},\cdots ,x_{N}-u_{N},t)G(u_{1},t)\,du_{1}\cdots G(u_{N},t)\,du_{N},} where G ( x , t ) = 1 2 π t e − x 2 2 t {\displaystyle G(x,t)={\frac {1}{\sqrt {2\pi t}}}e^{-{\frac {x^{2}}{2t}}}} and the standard deviation of the Gaussian σ is related to the scale parameter t according to t = σ2. Separability will be assumed in all that follows, even when the kernel is not exactly Gaussian, since separation of the dimensions is the most practical way to implement multidimensional smoothing, especially at larger scales. Therefore, the rest of the article focuses on the one-dimensional case. == The sampled Gaussian kernel == When implementing the one-dimensional smoothing step in practice, the presumably simplest approach is to convolve the discrete signal fD with a sampled Gaussian kernel: L ( x , t ) = ∑ n = − ∞ ∞ f ( x − n ) G ( n , t ) {\displaystyle L(x,t)=\sum _{n=-\infty }^{\infty }f(x-n)\,G(n,t)} where G ( n , t ) = 1 2 π t e − n 2 2 t {\displaystyle G(n,t)={\frac {1}{\sqrt {2\pi t}}}e^{-{\frac {n^{2}}{2t}}}} (with t = σ2) which in turn is truncated at the ends to give a filter with finite impulse response L ( x , t ) = ∑ n = − M M f ( x − n ) G ( n , t ) {\displaystyle L(x,t)=\sum _{n=-M}^{M}f(x-n)\,G(n,t)} for M chosen sufficiently large (see error function) such that 2 ∫ M ∞ G ( u , t ) d u = 2 ∫ M t ∞ G ( v , 1 ) d v < ε . {\displaystyle 2\int _{M}^{\infty }G(u,t)\,du=2\int _{\frac {M}{\sqrt {t}}}^{\infty }G(v,1)\,dv<\varepsilon .} A common choice is to set M to a constant C times the standard deviation of the Gaussian kernel M = C σ + 1 = C t + 1 {\displaystyle M=C\sigma +1=C{\sqrt {t}}+1} where C is often chosen somewhere between 3 and 6. Using the sampled Gaussian kernel can, however, lead to implementation problems, in particular when computing higher-order derivatives at finer scales by applying sampled derivatives of Gaussian kernels. When accuracy and robustness are primary design criteria, alternative implementation approaches should therefore be considered. For small values of ε (10−6 to 10−8) the errors introduced by truncating the Gaussian are usually negligible. For larger values of ε, however, there are many better alternatives to a rectangular window function. For example, for a given number of points, a Hamming window, Blackman window, or Kaiser window will do less damage to the spectral and other properties of the Gaussian than a simple truncation will. Notwithstanding this, since the Gaussian kernel decreases rapidly at the tails, the main recommendation is still to use a sufficiently small value of ε such that the truncation effects are no longer important. == The discrete Gaussian kernel == A more refined approach is to convolve the original signal with the discrete Gaussian kernel T(n, t) L ( x , t ) = ∑ n = − ∞ ∞ f ( x − n ) T ( n , t ) {\displaystyle L(x,t)=\sum _{n=-\infty }^{\infty }f(x-n)\,T(n,t)} where T ( n , t ) = e − t I n ( t ) {\displaystyle T(n,t)=e^{-t}I_{n}(t)} and I n ( t ) {\displaystyle I_{n}(t)} denotes the modified Bessel functions of integer order, n. This is the discrete counterpart of the continuous Gaussian in that it is the solution to the discrete diffusion equation (discrete space, continuous time), just as the continuous Gaussian is the solution to the continuous diffusion equation. This filter can be truncated in the spatial domain as for the sampled Gaussian L ( x , t ) = ∑ n = − M M f ( x − n ) T ( n , t ) {\displaystyle L(x,t)=\sum _{n=-M}^{M}f(x-n)\,T(n,t)} or can be implemented in the Fourier domain using a closed-form expression for its discrete-time Fourier transform: T ^ ( θ , t ) = ∑ n = − ∞ ∞ T ( n , t ) e − i θ n = e t ( cos θ − 1 ) . {\displaystyle {\widehat {T}}(\theta ,t)=\sum _{n=-\infty }^{\infty }T(n,t)\,e^{-i\theta n}=e^{t(\cos \theta -1)}.} With this frequency-domain approach, the scale-space properties transfer exactly to the discrete domain, or with excellent approximation using periodic extension and a suitably long discrete Fourier transform to approximate the discrete-time Fourier transform of the signal being smoothed. Moreover, higher-order derivative approximations can be computed in a straightforward manner (and preserving scale-space properties) by applying small support central difference operators to the discrete scale space representation. As with the sampled Gaussian, a plain truncation of the infinite impulse response will in most cases be a sufficient approximation for small values of ε, while for larger values of ε it is better to use either a decomposition of the discrete Gaussian into a cascade of generalized binomial filters or alternatively to construct a finite approximate kernel by multiplying by a window function. If ε has been chosen too large such that effects of the truncation error begin to appear (for example as spurious extrema or spurious responses to higher-order derivative operators), then the options are to decrease the value of ε such that a larger finite kernel is used, with cutoff where the support is very small, or to use a tapered window. == Recursive filters == Since computational efficiency is often important, low-order recursive filters are often used for scale-space smoothing. For example, Young and van Vliet use a third-order recursive filter with one real pole and a pair of complex poles, applied forward and backward to make a sixth-order symmetric approximation to the Gaussian with low computational complexity for any smoothing scale. By relaxing a few of the axioms, Lindeberg concluded that good smoothing filters would be "normalized Pólya frequency sequences", a family of discrete kernels that includes all filters with real poles at 0 < Z < 1 and/or Z > 1, as well as with real zeros at Z < 0. For symmetry, which leads to approximate directional homogeneity, these filters must be further restricted to pairs of poles and zeros that lead to zero-phase filters. To match the transfer function curvature at zero frequency of the discrete Gaussian, which ensures an approximate semi-group property of additive t, two poles at Z = 1 + 2 t − ( 1 + 2 t ) 2 − 1 {\displaystyle
Corpus-assisted discourse studies
Corpus-assisted discourse studies (abbr.: CADS) is related historically and methodologically to the discipline of corpus linguistics. The principal endeavor of corpus-assisted discourse studies is the investigation, and comparison of features of particular discourse types, integrating into the analysis the techniques and tools developed within corpus linguistics. These include the compilation of specialised corpora and analyses of word and word-cluster frequency lists, comparative keyword lists and, above all, concordances. A broader conceptualisation of corpus-assisted discourse studies would include any study that aims to bring together corpus linguistics and discourse analysis. Such research is often labelled as corpus-based or corpus-assisted discourse analysis, with the term CADS coined by a research group in Italy (Partington 2004) for a specific type of corpus-assisted discourse analysis (see the section 'in different countries' below). == Aims == Corpus-assisted discourse studies aim to uncover non-obvious meaning, that is, meaning which might not be readily available to naked-eye perusal. Much of what carries meaning in texts is not open to direct observation: “you cannot understand the world just by looking at it” (Stubbs [after Gellner 1959] 1996: 92). We use language “semi-automatically”, in the sense that speakers and writers make semi-conscious choices within the various complex overlapping systems of which language is composed, including those of transitivity, modality (Michael Halliday 1994), lexical sets (e.g. freedom, liberty, deliverance), modification, and so on. Authors themselves are, famously, generally unaware of all the meanings their texts convey. By combining the quantitative research approach, that is, statistical analysis of large amounts of the discourse in question - more precisely, large numbers of tokens of the discourse type under study contained in a corpus - with the more qualitative research approach typical of discourse analysis, that is, the close, detailed examination of particular stretches of discourse it may be possible to better understand the processes at play in the discourse type and to gain access to non-obvious meanings. Aims can differ in other types of corpus-based or corpus-assisted discourse analysis; but in general such studies combine quantitative and qualitative research and aim to shed light on discourses, registers, discourse patterns, etc., with the help of a corpus linguistic approach. Specific aims and techniques depend on the relevant project. == In different countries == In German-speaking countries: Pioneering work in corpus-based discourse analysis was conducted in Europe, in particular by Hardt-Mautner/Mautner (1995, 2000) and Stubbs (1996, 2001). CADS and other types of corpus-based discourse analysis are inspired by this important early work. In Italy: A considerable body of research has been conducted in Italy either by individual researchers or under the aegis of combined inter-university projects such as Newspool (Partington et al. 2004) and CorDis (Morley and Bayley eds, 2009). It has concentrated on political and media language, mainly because a nucleus of linguists in Italian universities work in Political Science faculties and are increasingly interested in the use of corpus techniques to conduct a particular type of sociopolitical discourse analysis, including the unearthing of noteworthy ideological metaphors and motifs in the language of political figures and institutions. Italian researchers also developed Modern diachronic corpus-assisted discourse studies (MD-CADS). This approach contrasts the language contained in comparable corpora from different but recent points in time in order to track changes in modern language usage but also social, cultural and political changes over modern times, as reflected - and shared among people - in language. It is this Italian body of research that makes most use of the label CADS. In the UK: Linguists in the UK tend to undertake corpus-based critical discourse analysis (CDA). CDA generally adopts a leftist political stance, focusing on the ways that social and political domination is reproduced by text and talk. This type of corpus-based research was originally associated with Lancaster University (Baker et al. 2008), but has spread more widely since. Such work typically studies the discourses around particular groups of people (e.g. Muslims, people with disabilities) or concepts/events (e.g. feminism, same-sex marriage). In Australia: Corpus-based discourse analysis is undertaken by a growing number of Australian researchers, most often on media texts. Some of this work aims to elucidate specific features of discourse types (news, social media, television series, etc.), while other work is rooted in the tradition of corpus-based critical discourse analysis. == Comparison with traditional corpus linguistics == Traditional corpus linguistics has, quite naturally, tended to privilege the quantitative approach. In the drive to produce more authentic dictionaries and grammars of a language, it has been characterised by the compilation of some very large corpora of heterogeneric discourse types in the desire to obtain an overview of the greatest quantity and variety of discourse types possible, in other words, of the chimerical but useful fiction called the “general language” (“general English”, “general Italian”, and so on). This has led to the construction of immensely valuable research tools such as the Bank of English and the British National Corpus. Some branches of corpus linguistics have also promoted an approach that is "corpus-driven", in which we need, grammatically speaking, a mental tabula rasa to free ourselves of the baleful prejudice exerted by traditional models and allow the data to speak entirely for itself. The aim of corpus-assisted discourse studies and related approaches is radically different. Here the aim of the exercise is to acquaint oneself as much as possible with the discourse type(s) in hand. Researchers typically engage with their corpus in a variety of ways. As well as via wordlists and concordancing, intuitions for further research can also arise from reading or watching or listening to parts of the data-set, a process which can help provide a feel for how things are done linguistically in the discourse-type being studied. Corpus-assisted discourse analysis is also typically characterised by the compilation of ad hoc specialised corpora, since very frequently there exists no previously available collection of the discourse type in question. Often, other corpora are utilized in the course of a study for purposes of comparison. These may include pre-existing corpora or may themselves need to be compiled by the researcher. In some sense, all work with corpora – just as all work with discourse - is properly comparative. Even when a single corpus is employed, it is used to test the data it contains against another body of data. This may consist of the researcher's intuitions, or the data found in reference works such as dictionaries and grammars, or it may be statements made by previous authors in the field. == CADS as a specific type of corpus-based discourse analysis == Researchers in Italy have developed CADS as a specific type of corpus-based discourse analysis, creating a standard set of methods: 'A basic, standard methodology in CADS may resemble the following:' Step 1: Decide upon the research question; Step 2: Choose, compile or edit an appropriate corpus; Step 3: Choose, compile or edit an appropriate reference corpus / corpora; Step 4: Make frequency lists and run a keywords comparison of the corpora; Step 5: Determine the existence of sets of key items; Step 6: Concordance interesting key items (with differing quantities of co-text); Step 7: (Possibly) refine the research question and return to Step 2. This basic procedure can of course vary according to individual research circumstances and requirements. A particular way of conceptualising research questions has also been proposed in such CADS projects: Given that P is a discourse participant (or possibly an institution) and G is a goal, often a political goal: How does P achieve G with language? What does this tell us about P? Comparative studies: how do P1 and P2 differ in their use of language? Does this tell us anything about their different principles and objectives? A second general type of CADS research question, which might be asked of interactive discourse data, has been conceptualised as follows: Given that P(x) is a particular participant or set of participants, DT is the discourse type, and R is an observed relationship between or among participants: How do {P(a), P(b)...P(n)} achieve / maintain R in DT [using language]? Another common type of research question has been conceptualised thus: Given that A is an author, Ph(x) is a phenomenon or practice or behaviour, and DT(x) is a particular discourse type. A has said P
Tree transducer
In theoretical computer science and formal language theory, a tree transducer (TT) is an abstract machine taking as input a tree, and generating output – generally other trees, but models producing words or other structures exist. Roughly speaking, tree transducers extend tree automata in the same way that word transducers extend word automata. Manipulating tree structures instead of words enable TT to model syntax-directed transformations of formal or natural languages. However, TT are not as well-behaved as their word counterparts in terms of algorithmic complexity, closure properties, etcetera. In particular, most of the main classes are not closed under composition. The main classes of tree transducers are: == Top-Down Tree Transducers (TOP) == A TOP T is a tuple (Q, Σ, Γ, I, δ) such that: Q is a finite set, the set of states; Σ is a finite ranked alphabet, called the input alphabet; Γ is a finite ranked alphabet, called the output alphabet; I is a subset of Q, the set of initial states; and δ is a set of rules of the form q ( f ( x 1 , … , x n ) ) → u {\displaystyle q(f(x_{1},\dots ,x_{n}))\to u} , where f is a symbol of Σ, n is the arity of f, q is a state, and u is a tree on Γ and Q × 1.. n {\displaystyle Q\times 1..n} , such pairs being nullary. === Examples of rules and intuitions on semantics === For instance, q ( f ( x 1 , … , x 3 ) ) → g ( a , q ′ ( x 1 ) , h ( q ″ ( x 3 ) ) ) {\displaystyle q(f(x_{1},\dots ,x_{3}))\to g(a,q'(x_{1}),h(q''(x_{3})))} is a rule – one customarily writes q ( x i ) {\displaystyle q(x_{i})} instead of the pair ( q , x i ) {\displaystyle (q,x_{i})} – and its intuitive semantics is that, under the action of q, a tree with f at the root and three children is transformed into g ( a , q ′ ( x 1 ) , h ( q ″ ( x 3 ) ) ) {\displaystyle g(a,q'(x_{1}),h(q''(x_{3})))} where, recursively, q ′ ( x 1 ) {\displaystyle q'(x_{1})} and q ″ ( x 3 ) {\displaystyle q''(x_{3})} are replaced, respectively, with the application of q ′ {\displaystyle q'} on the first child and with the application of q ″ {\displaystyle q''} on the third. === Semantics as term rewriting === The semantics of each state of the transducer T, and of T itself, is a binary relation between input trees (on Σ) and output trees (on Γ). A way of defining the semantics formally is to see δ {\displaystyle \delta } as a term rewriting system, provided that in the right-hand sides the calls are written in the form q ( x i ) {\displaystyle q(x_{i})} , where states q are unary symbols. Then the semantics [ [ q ] ] {\displaystyle [\![q]\!]} of a state q is given by [ [ q ] ] = { u ↦ v ∣ u is a tree on Σ , v is a tree on Γ , and q ( u ) → δ ∗ v } . {\displaystyle [\![q]\!]=\{u\mapsto v\mid u{\text{ is a tree on }}\Sigma ,\ v{\text{ is a tree on }}\Gamma {\text{, and }}q(u)\to _{\delta }^{}v\}.} The semantics of T is then defined as the union of the semantics of its initial states: [ [ T ] ] = ⋃ q ∈ I [ [ q ] ] . {\displaystyle [\![T]\!]=\bigcup _{q\in I}[\![q]\!].} === Determinism and domain === As with tree automata, a TOP is said to be deterministic (abbreviated DTOP) if no two rules of δ share the same left-hand side, and there is at most one initial state. In that case, the semantics of the DTOP is a partial function from input trees (on Σ) to output trees (on Γ), as are the semantics of each of the DTOP's states. The domain of a transducer is the domain of its semantics. Likewise, the image of a transducer is the image of its semantics. === Properties of DTOP === DTOP are not closed under union: this is already the case for deterministic word transducers. The domain of a DTOP is a regular tree language. Furthermore, the domain is recognisable by a deterministic top-down tree automaton (DTTA) of size at most exponential in that of the initial DTOP. That the domain is DTTA-recognizable is not surprising, considering that the left-hand sides of DTOP rules are the same as for DTTA. As for the reason for the exponential explosion in the worst case (that does not exist in the word case), consider the rule q ( f ( x 1 , x 2 ) ) → g ( p 1 ( x 1 ) , p 2 ( x 1 ) , p 3 ( x 2 ) ) {\displaystyle q(f(x_{1},x_{2}))\to g(p_{1}(x_{1}),p_{2}(x_{1}),p_{3}(x_{2}))} . In order for the computation to succeed, it must succeed for both children. That means that the right child must be in the domain of p 3 {\displaystyle p_{3}} . As for the left child, it must be in the domain of both p 1 {\displaystyle p_{1}} and p 2 {\displaystyle p_{2}} . Generally, since subtrees can be copied, a single subtree can be evaluated by multiple states during a run, despite the determinism, and unlike DTTA. Thus the construction of the DTTA recognising the domain of a DTOP must account for sets of states and compute the intersections of their domains, hence the exponential. In the special case of linear DTOP, that is to say DTOP where each x i {\displaystyle x_{i}} appears at most once in the right-hand side of each rule, the construction is linear in time and space. The image of a DTOP is not a regular tree language. Consider the transducer coding the transformation f ( x ) → g ( x , x ) {\displaystyle f(x)\to g(x,x)} ; that is, duplicate the child of the input. This is easily done by a rule q ( f ( x 1 ) ) → g ( p ( x 1 ) , p ( x 1 ) ) {\displaystyle q(f(x_{1}))\to g(p(x_{1}),p(x_{1}))} , where p encodes the identity. Then, absent any restrictions on the first child of the input, the image is a classical non-regular tree language. However, the domain of a DTOP cannot be restricted to a regular tree language. That is to say, given a DTOP T and a language L, one cannot in general build a DTOP T ′ {\displaystyle T'} such that the semantics of T ′ {\displaystyle T'} is that of T, restricted to L. This property is linked to the reason deterministic top-down tree automata are less expressive than bottom-up automata: once you go down a given path, information from other paths is inaccessible. Consider the transducer coding the transformation f ( x , y ) → y {\displaystyle f(x,y)\to y} ; that is, output the right child of the input. This is easily done by a rule q ( f ( x 1 , x 2 ) ) → p ( x 2 ) {\displaystyle q(f(x_{1},x_{2}))\to p(x_{2})} , where p encodes the identity. Now let's say we want to restrict this transducer to the finite (and thus, in particular, regular) domain { f ( c , a ) , f ( c , b ) } {\displaystyle \{f(c,a),\ f(c,b)\}} . We must use the rules q ( f ( x 1 , x 2 ) ) → p ( x 2 ) , p ( a ) → a , p ( b ) → b {\displaystyle q(f(x_{1},x_{2}))\to p(x_{2}),\ p(a)\to a,\ p(b)\to b} . But in the first rule, x 1 {\displaystyle x_{1}} does not appear at all, since nothing is produced from the left child. Thus, it is not possible to test that the left child is c. In contrast, since we produce from the right child, we can test that it is a or b. In general, the criterion is that DTOP cannot test properties of subtrees from which they do not produce output. DTOP are not closed under composition. However this problem can be solved by the addition of a lookahead: a tree automaton, coupled to the transducer, that can perform tests on the domain which the transducer is incapable of. This follows from the point about domain restriction: composing the DTOP encoding identity on { f ( c , a ) , f ( c , b ) } {\displaystyle \{f(c,a),\ f(c,b)\}} with the one encoding f ( x , y ) → y {\displaystyle f(x,y)\to y} must yield a transducer with the semantics { f ( c , a ) ↦ a , f ( c , b ) ↦ b } {\displaystyle \{f(c,a)\mapsto a,\ f(c,b)\mapsto b\}} , which we know is not expressible by a DTOP. The typechecking problem—testing whether the image of a regular tree language is included in another regular tree language—is decidable. The equivalence problem—testing whether two DTOP define the same functions—is decidable. == Bottom-Up Tree Transducers (BOT) == As in the simpler case of tree automata, bottom-up tree transducers are defined similarly to their top-down counterparts, but proceed from the leaves of the tree to the root, instead of from the root to the leaves. Thus the main difference is in the form of the rules, which are of the form f ( q 1 ( x 1 ) , … , q n ( x n ) ) → q ( u ) {\displaystyle f(q_{1}(x_{1}),\dots ,q_{n}(x_{n}))\to q(u)} .
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C3D Toolkit
C3D Toolkit is a proprietary cross-platform geometric modeling kit software developed by Russian C3D Labs (previously part of ASCON Group). It's written in C++ . It can be licensed by other companies for use in their 3D computer graphics software products. The most widely known software in which C3D Toolkit is typically used are computer aided design (CAD), computer-aided manufacturing (CAM), and computer-aided engineering (CAE) systems. C3D Toolkit provides routines for 3D modeling, 3D constraint solving, polygonal mesh-to-B-rep conversion, 3D visualization, and 3D file conversions etc. == History == Nikolai Golovanov is a graduate of the Mechanical Engineering department of Bauman Moscow State Technical University as a designer of space launch vehicles. Upon his graduation, he began with the Kolomna Engineering Design bureau, which at the time employed the future founders of ASCON, Alexander Golikov and Tatiana Yankina. While at the bureau, Dr Golovanov developed software for analyzing the strength and stability of shell structures. In 1989, Alexander Golikov and Tatiana Yankina left Kolomna to start up ASCON as a private company. Although they began with just an electronic drawing board, even then they were already conceiving the idea of three-dimensional parametric modeling. This radical concept eventually changed flat drawings into three-dimensional models. The ASCON founders shared their ideas with Nikolai Golovanov, and in 1996 he moved to take up his current position with ASCON. As of 2012 he was involved in developing algorithms for C3D Toolkit. In 2012 the earliest version of the C3D Modeller kernel was extracted from KOMPAS-3D CAD. It was later adopted to a range of different platforms and advertised as a separate product. == Overview == It incorporates five modules: C3D Modeler constructs geometric models, generates flat projections of models, performs triangulations, calculates the inertial characteristics of models, and determines whether collisions occur between the elements of models; C3D Modeler for ODA enables advanced 3D modeling operations through the ODA's standard "OdDb3DSolid" API from the Open Design Alliance; C3D Solver makes connections between the elements of geometric models, and considers the geometric constraints of models being edited; C3D B-Shaper converts polygonal models to boundary representation (B-rep) bodies; C3D Vision controls the quality of rendering for 3D models using mathematical apparatus and software, and the workstation hardware; C3D Converter reads and writes geometric models in a variety of standard exchange formats. == Features == == Development == == Applications == Since 2013 - the date the company started issuing a license for the toolkit -, several companies have adopted C3D software components for their products, users include: Recently, C3D Modeler has been adapted to ODA Platform. In April 2017, C3D Viewer was launched for end users. The application allows to read 3D models in common formats and write it to the C3D file format. Free version is available.
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