Karsten Borgwardt (born 1980) is a German computer scientist and biologist specializing in machine learning and computational biology. Since February 2023, he has been a director at the Max Planck Institute of Biochemistry in Martinsried, Germany, where he leads the Department of Machine Learning and Systems Biology. == Education and career == Borgwardt was born in Kaiserslautern. He obtained a Diplom (equivalent to a master’s degree) in computer science from LMU Munich in 2004 and a Master of Science in biology from the University of Oxford in 2003. In 2007, he obtained his PhD from LMU Munich in computer science. Following a postdoctoral position at the University of Cambridge, he became a research group leader for machine learning and computational biology at the Max Planck Institute for Biological Cybernetics and the former Max Planck Institute for Developmental Biology in Tübingen in 2008. In 2011, Borgwardt was appointed professor of data mining in the life sciences at the University of Tübingen. In 2014, he joined ETH Zurich as an associate professor in the Department of Biosystems Science and Engineering (D-BSSE) and was promoted to full professor in 2017. During his tenure at ETH Zurich, he coordinated significant research programs, including two Marie Curie Innovative Training Networks and the Personalized Swiss Sepsis Study, focusing on the prediction of sepsis using machine learning. In 2023, he was appointed as Scientific Member of the Max Planck Society and as Director at the Max Planck Institute of Biochemistry in Martinsried. == Research contributions == Borgwardt’s research integrates big data analysis with biomedical research. He develops novel machine learning algorithms to detect patterns and statistical dependencies in large biological and medical datasets. His work aims to enable the automatic generation of new knowledge from big data and to understand the relationship between the function of biological systems and their molecular properties, which is fundamental for personalized medicine. == Awards and honors == During his studies, he was a scholar of the Stiftung Maximilianeum, and the Bavarian Foundation for the Promotion of the Gifted. Borgwardt received scholarships from the Studienstiftung des deutschen Volkes in 2002 and 2007. His PhD dissertation received the Heinz Schwärtzel Dissertation Award for Foundations of Computer Science in 2007. As a professor in Tübingen, he was awarded the Alfried-Krupp-Förderpreis for Young Professors in 2013. In 2015, he received an SNSF Starting Grant. In 2014, 2015 and 2016, he was listed in “Top 40 under 40” in Germany rankings selected by Capital magazine. In 2018, Borgwardt was named among “25 individuals who have the potential to shape the next 25 years” by Focus magazine. In 2023, Borgwardt received an honorary professorship from LMU Munich by the Faculty of Chemistry and Pharmacy. Publications from Borgwardt's group have received the Outstanding Student Paper Award in NIPS in 2009, the SIB Graduate Paper Award in 2020 and SIB Remarkable Output Awards in 2020 and 2021 from the Swiss Institute of Bioinformatics (SIB). == Selected publications == Weisfeiler-Lehman Graph Kernels (’‘Journal of Machine Learning Research’’, 2011): Introduced an efficient graph kernel based on the Weisfeiler-Lehman algorithm. “Direct antimicrobial resistance prediction from clinical MALDI-TOF mass spectra using machine learning” (’‘Nature Medicine’’, 2022): showcased the feasibility of predicting antimicrobial resistance from readily collected mass spectrometry data in the hospital. The new method is able to identify antibiotic resistance 24 hours earlier than previous methods.
Version space learning
Version space learning is a logical approach to machine learning, specifically binary classification. Version space learning algorithms search a predefined space of hypotheses, viewed as a set of logical sentences. Formally, the hypothesis space is a disjunction H 1 ∨ H 2 ∨ . . . ∨ H n {\displaystyle H_{1}\lor H_{2}\lor ...\lor H_{n}} (i.e., one or more of hypotheses 1 through n are true). A version space learning algorithm is presented with examples, which it will use to restrict its hypothesis space; for each example x, the hypotheses that are inconsistent with x are removed from the space. This iterative refining of the hypothesis space is called the candidate elimination algorithm, the hypothesis space maintained inside the algorithm, its version space. == The version space algorithm == In settings where there is a generality-ordering on hypotheses, it is possible to represent the version space by two sets of hypotheses: (1) the most specific consistent hypotheses, and (2) the most general consistent hypotheses, where "consistent" indicates agreement with observed data. The most specific hypotheses (i.e., the specific boundary SB) cover the observed positive training examples, and as little of the remaining feature space as possible. These hypotheses, if reduced any further, exclude a positive training example, and hence become inconsistent. These minimal hypotheses essentially constitute a (pessimistic) claim that the true concept is defined just by the positive data already observed: Thus, if a novel (never-before-seen) data point is observed, it should be assumed to be negative. (I.e., if data has not previously been ruled in, then it's ruled out.) The most general hypotheses (i.e., the general boundary GB) cover the observed positive training examples, but also cover as much of the remaining feature space without including any negative training examples. These, if enlarged any further, include a negative training example, and hence become inconsistent. These maximal hypotheses essentially constitute a (optimistic) claim that the true concept is defined just by the negative data already observed: Thus, if a novel (never-before-seen) data point is observed, it should be assumed to be positive. (I.e., if data has not previously been ruled out, then it's ruled in.) Thus, during learning, the version space (which itself is a set – possibly infinite – containing all consistent hypotheses) can be represented by just its lower and upper bounds (maximally general and maximally specific hypothesis sets), and learning operations can be performed just on these representative sets. After learning, classification can be performed on unseen examples by testing the hypothesis learned by the algorithm. If the example is consistent with multiple hypotheses, a majority vote rule can be applied. == Historical background == The notion of version spaces was introduced by Mitchell in the early 1980s as a framework for understanding the basic problem of supervised learning within the context of solution search. Although the basic "candidate elimination" search method that accompanies the version space framework is not a popular learning algorithm, there are some practical implementations that have been developed (e.g., Sverdlik & Reynolds 1992, Hong & Tsang 1997, Dubois & Quafafou 2002). A major drawback of version space learning is its inability to deal with noise: any pair of inconsistent examples can cause the version space to collapse, i.e., become empty, so that classification becomes impossible. One solution of this problem is proposed by Dubois and Quafafou that proposed the Rough Version Space, where rough sets based approximations are used to learn certain and possible hypothesis in the presence of inconsistent data.
Ordered weighted averaging
In applied mathematics, specifically in fuzzy logic, the ordered weighted averaging (OWA) operators provide a parameterized class of mean type aggregation operators. They were introduced by Ronald R. Yager. Many notable mean operators such as the max, arithmetic average, median and min, are members of this class. They have been widely used in computational intelligence because of their ability to model linguistically expressed aggregation instructions. == Definition == An OWA operator of dimension n {\displaystyle \ n} is a mapping F : R n → R {\displaystyle F:\mathbb {R} ^{n}\rightarrow \mathbb {R} } that has an associated collection of weights W = [ w 1 , … , w n ] {\displaystyle \ W=[w_{1},\ldots ,w_{n}]} lying in the unit interval and summing to one and with F ( a 1 , … , a n ) = ∑ j = 1 n w j b j {\displaystyle F(a_{1},\ldots ,a_{n})=\sum _{j=1}^{n}w_{j}b_{j}} where b j {\displaystyle b_{j}} is the jth largest of the a i {\displaystyle a_{i}} . By choosing different W one can implement different aggregation operators. The OWA operator is a non-linear operator as a result of the process of determining the bj. == Notable OWA operators == F ( a 1 , … , a n ) = max ( a 1 , … , a n ) {\displaystyle \ F(a_{1},\ldots ,a_{n})=\max(a_{1},\ldots ,a_{n})} if w 1 = 1 {\displaystyle \ w_{1}=1} and w j = 0 {\displaystyle \ w_{j}=0} for j ≠ 1 {\displaystyle j\neq 1} F ( a 1 , … , a n ) = min ( a 1 , … , a n ) {\displaystyle \ F(a_{1},\ldots ,a_{n})=\min(a_{1},\ldots ,a_{n})} if w n = 1 {\displaystyle \ w_{n}=1} and w j = 0 {\displaystyle \ w_{j}=0} for j ≠ n {\displaystyle j\neq n} F ( a 1 , … , a n ) = a v e r a g e ( a 1 , … , a n ) {\displaystyle \ F(a_{1},\ldots ,a_{n})=\mathrm {average} (a_{1},\ldots ,a_{n})} if w j = 1 n {\displaystyle \ w_{j}={\frac {1}{n}}} for all j ∈ [ 1 , n ] {\displaystyle j\in [1,n]} == Properties == The OWA operator is a mean operator. It is bounded, monotonic, symmetric, and idempotent, as defined below. == Characterizing features == Two features have been used to characterize the OWA operators. The first is the attitudinal character, also called orness. This is defined as A − C ( W ) = 1 n − 1 ∑ j = 1 n ( n − j ) w j . {\displaystyle A-C(W)={\frac {1}{n-1}}\sum _{j=1}^{n}(n-j)w_{j}.} It is known that A − C ( W ) ∈ [ 0 , 1 ] {\displaystyle A-C(W)\in [0,1]} . In addition A − C(max) = 1, A − C(ave) = A − C(med) = 0.5 and A − C(min) = 0. Thus the A − C goes from 1 to 0 as we go from Max to Min aggregation. The attitudinal character characterizes the similarity of aggregation to OR operation(OR is defined as the Max). The second feature is the dispersion. This defined as H ( W ) = − ∑ j = 1 n w j ln ( w j ) . {\displaystyle H(W)=-\sum _{j=1}^{n}w_{j}\ln(w_{j}).} An alternative definition is E ( W ) = ∑ j = 1 n w j 2 . {\displaystyle E(W)=\sum _{j=1}^{n}w_{j}^{2}.} The dispersion characterizes how uniformly the arguments are being used. == Type-1 OWA aggregation operators == The above Yager's OWA operators are used to aggregate the crisp values. Can we aggregate fuzzy sets in the OWA mechanism? The Type-1 OWA operators have been proposed for this purpose. So the type-1 OWA operators provides us with a new technique for directly aggregating uncertain information with uncertain weights via OWA mechanism in soft decision making and data mining, where these uncertain objects are modelled by fuzzy sets. The type-1 OWA operator is defined according to the alpha-cuts of fuzzy sets as follows: Given the n linguistic weights { W i } i = 1 n {\displaystyle \left\{{W^{i}}\right\}_{i=1}^{n}} in the form of fuzzy sets defined on the domain of discourse U = [ 0 , 1 ] {\displaystyle U=[0,\;\;1]} , then for each α ∈ [ 0 , 1 ] {\displaystyle \alpha \in [0,\;1]} , an α {\displaystyle \alpha } -level type-1 OWA operator with α {\displaystyle \alpha } -level sets { W α i } i = 1 n {\displaystyle \left\{{W_{\alpha }^{i}}\right\}_{i=1}^{n}} to aggregate the α {\displaystyle \alpha } -cuts of fuzzy sets { A i } i = 1 n {\displaystyle \left\{{A^{i}}\right\}_{i=1}^{n}} is given as Φ α ( A α 1 , … , A α n ) = { ∑ i = 1 n w i a σ ( i ) ∑ i = 1 n w i | w i ∈ W α i , a i ∈ A α i , i = 1 , … , n } {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\ldots ,A_{\alpha }^{n}}\right)=\left\{{{\frac {\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}}}{\sum \limits _{i=1}^{n}{w_{i}}}}\left|{w_{i}\in W_{\alpha }^{i},\;a_{i}}\right.\in A_{\alpha }^{i},\;i=1,\ldots ,n}\right\}} where W α i = { w | μ W i ( w ) ≥ α } , A α i = { x | μ A i ( x ) ≥ α } {\displaystyle W_{\alpha }^{i}=\{w|\mu _{W_{i}}(w)\geq \alpha \},A_{\alpha }^{i}=\{x|\mu _{A_{i}}(x)\geq \alpha \}} , and σ : { 1 , … , n } → { 1 , … , n } {\displaystyle \sigma :\{\;1,\ldots ,n\;\}\to \{\;1,\ldots ,n\;\}} is a permutation function such that a σ ( i ) ≥ a σ ( i + 1 ) , ∀ i = 1 , … , n − 1 {\displaystyle a_{\sigma (i)}\geq a_{\sigma (i+1)},\;\forall \;i=1,\ldots ,n-1} , i.e., a σ ( i ) {\displaystyle a_{\sigma (i)}} is the i {\displaystyle i} th largest element in the set { a 1 , … , a n } {\displaystyle \left\{{a_{1},\ldots ,a_{n}}\right\}} . The computation of the type-1 OWA output is implemented by computing the left end-points and right end-points of the intervals Φ α ( A α 1 , … , A α n ) {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\ldots ,A_{\alpha }^{n}}\right)} : Φ α ( A α 1 , … , A α n ) − {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\ldots ,A_{\alpha }^{n}}\right)_{-}} and Φ α ( A α 1 , … , A α n ) + , {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\ldots ,A_{\alpha }^{n}}\right)_{+},} where A α i = [ A α − i , A α + i ] , W α i = [ W α − i , W α + i ] {\displaystyle A_{\alpha }^{i}=[A_{\alpha -}^{i},A_{\alpha +}^{i}],W_{\alpha }^{i}=[W_{\alpha -}^{i},W_{\alpha +}^{i}]} . Then membership function of resulting aggregation fuzzy set is: μ G ( x ) = ∨ α : x ∈ Φ α ( A α 1 , ⋯ , A α n ) α α {\displaystyle \mu _{G}(x)=\mathop {\vee } _{\alpha :x\in \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{\alpha }}\alpha } For the left end-points, we need to solve the following programming problem: Φ α ( A α 1 , ⋯ , A α n ) − = min W α − i ≤ w i ≤ W α + i A α − i ≤ a i ≤ A α + i ∑ i = 1 n w i a σ ( i ) / ∑ i = 1 n w i {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{-}=\min \limits _{\begin{array}{l}W_{\alpha -}^{i}\leq w_{i}\leq W_{\alpha +}^{i}A_{\alpha -}^{i}\leq a_{i}\leq A_{\alpha +}^{i}\end{array}}\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}/\sum \limits _{i=1}^{n}{w_{i}}}} while for the right end-points, we need to solve the following programming problem: Φ α ( A α 1 , ⋯ , A α n ) + = max W α − i ≤ w i ≤ W α + i A α − i ≤ a i ≤ A α + i ∑ i = 1 n w i a σ ( i ) / ∑ i = 1 n w i {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{+}=\max \limits _{\begin{array}{l}W_{\alpha -}^{i}\leq w_{i}\leq W_{\alpha +}^{i}A_{\alpha -}^{i}\leq a_{i}\leq A_{\alpha +}^{i}\end{array}}\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}/\sum \limits _{i=1}^{n}{w_{i}}}} Zhou et al. presented a fast method to solve two programming problem so that the type-1 OWA aggregation operation can be performed efficiently. == OWA for committee voting == Amanatidis, Barrot, Lang, Markakis and Ries present voting rules for multi-issue voting, based on OWA and the Hamming distance. Barrot, Lang and Yokoo study the manipulability of these rules.
Blocks world
The blocks world is a planning domain in artificial intelligence. It consists of a set of wooden blocks of various shapes and colors sitting on a table. The goal is to build one or more vertical stacks of blocks. Only one block may be moved at a time: it may either be placed on the table or placed atop another block. Because of this, any blocks that are, at a given time, under another block cannot be moved. Moreover, some kinds of blocks cannot have other blocks stacked on top of them. The simplicity of this toy world lends itself readily to classical symbolic artificial intelligence approaches, in which the world is modeled as a set of abstract symbols which may be reasoned about. == Motivation == Artificial Intelligence can be researched in theory and with practical applications. The problem with most practical applications is that the engineers don't know how to program an AI system. Instead of rejecting the challenge at all the idea is to invent an easy to solve domain which is called a toy problem. Toy problems were invented with the aim to program an AI which can solve it. The blocks world domain is an example of a toy problem. Its major advantage over more realistic AI applications is that many algorithms and software programs are available which can handle the situation. This allows comparing different theories against each other. In its basic form, the blocks world problem consists of cubes of the same size which have all the color black. A mechanical robot arm has to pick and place the cubes. More complicated derivatives of the problem consist of cubes of different sizes, shapes and colors. From an algorithmic perspective, blocks world is an NP-hard search and planning problem. The task is to bring the system from an initial state into a goal state. Automated planning and scheduling problems are usually described in the Planning Domain Definition Language (PDDL) notation which is an AI planning language for symbolic manipulation tasks. If something was formulated in the PDDL notation, it is called a domain. Therefore, the task of stacking blocks is a blocks world domain which stands in contrast to other planning problems like the dock worker robot domain and the monkey and banana problem. == Theses/projects which took place in a blocks world == Terry Winograd's SHRDLU Patrick Winston's Learning Structural Descriptions from Examples and Copy Demo Gerald Jay Sussman's Sussman anomaly Decision problem (Gupta and Nau, 1992): Given a starting Blocks World, an ending Blocks World, and an integer L > 0, is there a way to move the blocks to change the starting position to the ending position with L or less steps? This decision problem is NP-hard.
Residuated lattice
In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice x ≤ y and a monoid x•y that admits operations x\z and z/y, loosely analogous to division or implication, when x•y is viewed as multiplication or conjunction, respectively. Called respectively right and left residuals, these operations coincide when the monoid is commutative. The general concept was introduced by Morgan Ward and Robert P. Dilworth in 1939. Examples, some of which existed prior to the general concept, include Boolean algebras, Heyting algebras, residuated Boolean algebras, relation algebras, and MV-algebras. Residuated semilattices omit the meet operation ∧, for example Kleene algebras and action algebras. == Definition == In mathematics, a residuated lattice is an algebraic structure L = (L, ≤, •, I) such that (i) (L, ≤) is a lattice. (ii) (L, •, I) is a monoid. (iii) For all z there exists for every x a greatest y, and for every y a greatest x, such that x•y ≤ z (the residuation properties). In (iii), the "greatest y", being a function of z and x, is denoted x\z and called the right residual of z by x. Think of it as what remains of z on the right after "dividing" z on the left by x. Dually, the "greatest x" is denoted z/y and called the left residual of z by y. An equivalent, more formal statement of (iii) that uses these operations to name these greatest values is (iii)' for all x, y, z in L, y ≤ x\z ⇔ x•y ≤ z ⇔ x ≤ z/y. As suggested by the notation, the residuals are a form of quotient. More precisely, for a given x in L, the unary operations x• and x\ are respectively the lower and upper adjoints of a Galois connection on L, and dually for the two functions •y and /y. By the same reasoning that applies to any Galois connection, we have yet another definition of the residuals, namely, x•(x\y) ≤ y ≤ x\(x•y), and (y/x)•x ≤ y ≤ (y•x)/x, together with the requirement that x•y be monotone in x and y. (When axiomatized using (iii) or (iii)' monotonicity becomes a theorem and hence not required in the axiomatization.) These give a sense in which the functions x• and x\ are pseudoinverses or adjoints of each other, and likewise for •x and /x. This last definition is purely in terms of inequalities, noting that monotonicity can be axiomatized as x • y ≤ (x∨z) • y and similarly for the other operations and their arguments. Moreover, any inequality x ≤ y can be expressed equivalently as an equation, either x∧y = x or x∨y = y. This along with the equations axiomatizing lattices and monoids then yields a purely equational definition of residuated lattices, provided the requisite operations are adjoined to the signature (L, ≤, •, I) thereby expanding it to (L, ∧, ∨, •, I, /, \). When thus organized, residuated lattices form an equational class or variety, whose homomorphisms respect the residuals as well as the lattice and monoid operations. Note that distributivity x • (y ∨ z) = (x • y) ∨ (x • z) and x•0 = 0 are consequences of these axioms and so do not need to be made part of the definition. This necessary distributivity of • over ∨ does not in general entail distributivity of ∧ over ∨, that is, a residuated lattice need not be a distributive lattice. However distributivity of ∧ over ∨ is entailed when • and ∧ are the same operation, a special case of residuated lattices called a Heyting algebra. Alternative notations for x•y include x◦y, x;y (relation algebra), and x⊗y (linear logic). Alternatives for I include e and 1'. Alternative notations for the residuals are x → y for x\y and y ← x for y/x, suggested by the similarity between residuation and implication in logic, with the multiplication of the monoid understood as a form of conjunction that need not be commutative. When the monoid is commutative the two residuals coincide. When not commutative, the intuitive meaning of the monoid as conjunction and the residuals as implications can be understood as having a temporal quality: x•y means x and then y, x → y means had x (in the past) then y (now), and y ← x means if-ever x (in the future) then y (at that time), as illustrated by the natural language example at the end of the examples. == Examples == One of the original motivations for the study of residuated lattices was the lattice of (two-sided) ideals of a ring. Given a ring R, the ideals of R, denoted Id(R), forms a complete lattice with set intersection acting as the meet operation and "ideal addition" acting as the join operation. The monoid operation • is given by "ideal multiplication", and the element R of Id(R) acts as the identity for this operation. Given two ideals A and B in Id(R), the residuals are given by A / B := { r ∈ R ∣ r B ⊆ A } {\displaystyle A/B:=\{r\in R\mid rB\subseteq A\}} B ∖ A := { r ∈ R ∣ B r ⊆ A } {\displaystyle B\setminus A:=\{r\in R\mid Br\subseteq A\}} It is worth noting that {0}/B and B\{0} are respectively the left and right annihilators of B. This residuation is related to the conductor (or transporter) in commutative algebra written as (A:B)=A/B. One difference in usage is that B need not be an ideal of R: it may just be a subset. Boolean algebras and Heyting algebras are commutative residuated lattices in which x•y = x∧y (whence the unit I is the top element 1 of the algebra) and both residuals x\y and y/x are the same operation, namely implication x → y. The second example is quite general since Heyting algebras include all finite distributive lattices, as well as all chains or total orders, for example the unit interval [0,1] in the real line, or the integers and ± ∞ {\displaystyle \pm \infty } . The structure (Z, min, max, +, 0, −, −) (the integers with subtraction for both residuals) is a commutative residuated lattice such that the unit of the monoid is not the greatest element (indeed there is no least or greatest integer), and the multiplication of the monoid is not the meet operation of the lattice. In this example the inequalities are equalities because − (subtraction) is not merely the adjoint or pseudoinverse of + but the true inverse. Any totally ordered group under addition such as the rationals or the reals can be substituted for the integers in this example. The nonnegative portion of any of these examples is an example provided min and max are interchanged and − is replaced by monus, defined (in this case) so that x-y = 0 when x ≤ y and otherwise is ordinary subtraction. A more general class of examples is given by the Boolean algebra of all binary relations on a set X, namely the power set of X2, made a residuated lattice by taking the monoid multiplication • to be composition of relations and the monoid unit to be the identity relation I on X consisting of all pairs (x,x) for x in X. Given two relations R and S on X, the right residual R\S of S by R is the binary relation such that x(R\S)y holds just when for all z in X, zRx implies zSy (notice the connection with implication). The left residual is the mirror image of this: y(S/R)x holds just when for all z in X, xRz implies ySz. This can be illustrated with the binary relations < and > on {0,1} in which 0 < 1 and 1 > 0 are the only relationships that hold. Then x(>\<)y holds just when x = 1, while x()y holds just when y = 0, showing that residuation of < by > is different depending on whether we residuate on the right or the left. This difference is a consequence of the difference between <•> and >•<, where the only relationships that hold are 0(<•>)0 (since 0<1>0) and 1(>•<)1 (since 1>0<1). Had we chosen ≤ and ≥ instead of < and >, ≥\≤ and ≤/≥ would have been the same because ≤•≥ = ≥•≤, both of which always hold between all x and y (since x≤1≥y and x≥0≤y). The Boolean algebra 2Σ of all formal languages over an alphabet (set) Σ forms a residuated lattice whose monoid multiplication is language concatenation LM and whose monoid unit I is the language {ε} consisting of just the empty string ε. The right residual M\L consists of all words w over Σ such that Mw ⊆ L. The left residual L/M is the same with wM in place of Mw. The residuated lattice of all binary relations on X is finite just when X is finite, and commutative just when X has at most one element. When X is empty the algebra is the degenerate Boolean algebra in which 0 = 1 = I. The residuated lattice of all languages on Σ is commutative just when Σ has at most one letter. It is finite just when Σ is empty, consisting of the two languages 0 (the empty language {}) and the monoid unit I = {ε} = 1. The examples forming a Boolean algebra have special properties treated in the article on residuated Boolean algebras. == Residuated semilattice == A residuated semilattice is defined almost identically for residuated lattices, omitting just the meet operation ∧. Thus it is an algebraic structure L = (L, ∨, •, 1, /, \) satisfying all the residuated lattice equations as specified above except those containing an occurrence of the symbol ∧. The option of defining x ≤ y as x∧y = x is then not available, leaving on
Software engineering demographics
Software engineers make up a significant portion of the global workforce. As of 2022, there are an estimated 26.9 million professional software engineers worldwide, up from 21 million in 2016. == By country == === United States === In 2023, there were an estimated 1.6 million professional software developers in North America. There are 166 million people employed in the US workforce, making software developers 0.96% of the total workforce. ==== Summary ==== ==== Software engineers vs. traditional engineers ==== The following two tables compare the number of software engineers (611,900 in 2002) versus the number of traditional engineers (1,157,020 in 2002). There are another 1,500,000 people in system analysis, system administration, and computer support, many of whom might be called software engineers. Many systems analysts manage software development teams, and as analysis is an important software engineering role, many of them may be considered software engineers in the near future. This means that the number of software engineers may actually be much higher. It is important to note that the number of software engineers declined by 5 to 10 percent from 2000 to 2002. ==== Computer managers vs. construction and engineering managers ==== Computer and information system managers (264,790) manage software projects, as well as computer operations. Similarly, Construction and engineering managers (413,750) oversee engineering projects, manufacturing plants, and construction sites. Computer management is 64% the size of construction and engineering management. ==== Software engineering educators vs. engineering educators ==== Most people working in the field of computer science, whether making software systems (software engineering) or studying the theoretical and mathematical facts of software systems (computer science), acquire degrees in computer science. According to the U.S. Bureau of Labor Statistics (May 2023 data), there were approximately 44,800 postsecondary computer science teachers and 50,300 engineering teachers, indicating that the computer science educator workforce is nearly 89% as large as that of engineering educators. The combined number of postsecondary chemistry (25,400) and physics (17,100) teachers totaled 42,500, slightly less than the number of computer science educators. ==== Other software and engineering roles ==== ==== Relation to IT demographics ==== Software engineers are part of the much larger software, hardware, application, and operations community. In 2000 in the U.S., there were about 680,000 software engineers and about 10,000,000 IT workers. As of early 2025, there are an estimated 47.2 million software developers worldwide, representing a 50% increase from 31 million in Q1 2022. There are no numbers on testers in the BLS data. === India === There has been a healthy growth in the number of India's IT professionals over the past few years. From a base of 6,800 knowledge workers in 1985–86, the number increased to 522,000 software and services professionals by the end of 2001–02. It is estimated that out of these 528,000 knowledge workers, almost 170,000 are working in the IT software and services export industry; nearly 106,000 are working in the IT enabled services and over 230,000 in user organizations. === Australia === In May 2024, the Australian government reported that 169,300 Australians are employed as software and applications programmers, 17% of who are women. The role grew annually by 8,300 workers. === Russia === According to the Russian government, the number of IT specialists in the country increased by 13% in 2023, reaching approximately 857,000. During the initial phase of the 2022 invasion of Ukraine, an estimated 100,000 IT specialists left Russia.
Vague set
In mathematics, vague sets are an extension of fuzzy sets. In a fuzzy set, each object is assigned a single value in the interval [0,1] reflecting its grade of membership. This single value does not allow a separation of evidence for membership and evidence against membership. Gau et al. proposed the notion of vague sets, where each object is characterized by two different membership functions: a true membership function and a false membership function. This kind of reasoning is also called interval membership, as opposed to point membership in the context of fuzzy sets. == Mathematical definition == A vague set V {\displaystyle V} is characterized by its true membership function t v ( x ) {\displaystyle t_{v}(x)} its false membership function f v ( x ) {\displaystyle f_{v}(x)} with 0 ≤ t v ( x ) + f v ( x ) ≤ 1 {\displaystyle 0\leq t_{v}(x)+f_{v}(x)\leq 1} The grade of membership for x is not a crisp value anymore, but can be located in [ t v ( x ) , 1 − f v ( x ) ] {\displaystyle [t_{v}(x),1-f_{v}(x)]} . This interval can be interpreted as an extension to the fuzzy membership function. The vague set degenerates to a fuzzy set, if 1 − f v ( x ) = t v ( x ) {\displaystyle 1-f_{v}(x)=t_{v}(x)} for all x. The uncertainty of x is the difference between the upper and lower bounds of the membership interval; it can be computed as ( 1 − f v ( x ) ) − t v ( x ) {\displaystyle (1-f_{v}(x))-t_{v}(x)} .