Michael L. Scott , in Programming Language Pragmatics (Third Edition), 2009
2.4 Theoretical Foundations
Our understanding of the relative roles and computational power of scanners, parsers, regular expressions, and context-gratuitous grammars is based on the formalisms of automata theory. In automata theory, a formal language is a prepare of strings of symbols fatigued from a finite alphabet. A formal language can exist specified either by a ready of rules (such as regular expressions or a context-free grammar) that generates the language, or by a formal auto that accepts (recognizes) the linguistic communication. A formal machine takes strings of symbols equally input and outputs either "yes" or "no." A auto is said to accept a linguistic communication if information technology says "yep" to all and but those strings that are in the language. Alternatively, a language can exist defined as the set of strings for which a item motorcar says "yes."
Formal languages can be grouped into a serial of successively larger classes known as the Chomsky bureaucracy.14 Most of the classes tin can be characterized in two ways: by the types of rules that can exist used to generate the set of strings, or past the type of formal machine that is capable of recognizing the language. As we accept seen, regular languages are divers by using concatenation, alternation, and Kleene closure, and are recognized by a scanner. Context-complimentary languages are a proper superset of the regular languages. They are defined past using chain, alternation, and recursion (which subsumes Kleene closure), and are recognized by a parser. A scanner is a physical realization of a finite automaton, a type of formal machine. A parser is a concrete realization of a push-down automaton. But every bit context-free grammars add recursion to regular expressions, push-downwards automata add a stack to the memory of a finite automaton. There are boosted levels in the Chomsky hierarchy, but they are less straight applicable to compiler construction, and are non covered here.
Information technology can be proven, constructively, that regular expressions and finite automata are equivalent: one can construct a finite automaton that accepts the linguistic communication defined by a given regular expression, and vice versa. Similarly, it is possible to construct a button-downward automaton that accepts the language defined by a given context-gratuitous grammar, and vice versa. The grammar-to-automaton constructions are in fact performed by scanner and parser generators such every bit lex and yacc. Of form, a real scanner does not accept just one token; it is called in a loop so that it keeps accepting tokens repeatedly. As noted in the sidebar on folio 60, this particular is accommodated past having the scanner accept the alternation of all the tokens in the language (with distinguished final states), and by having information technology continue to consume characters until no longer token can be constructed.
In More Depth
On the PLP CD we consider finite and pushdown automata in more than detail. We give an algorithm to convert a DFA into an equivalent regular expression. Combined with the constructions in Section 2.2.1, this algorithm demonstrates the equivalence of regular expressions and finite automata. We also consider the sets of grammars and languages that tin and cannot exist parsed by the various linear-time parsing algorithms.
Carl J. Posy , in Handbook of the History of Logic, 2007
The morphology ChiliadC
Singular terms:
Constants : a, b, c, d, a1,…,b1,…,…
(A abiding is a "closed" singular term.)
Variables : 10, y, z, xone,…,y1,…,…
Predicates For each i(≥ 0) and each due north(≥ 0) nosotros have the n-place predicate letter .
Connectives
Sentential connectives : ∼, →
Quantifier: ∀
A formal language, LC, built from this morphology, volition contain all the variables and logical symbols together with a (non-empty) selection of relation symbols and a (possibly empty) collection of constants. In such a language the notion of a well formed formula ("wff") is divers every bit follows:
1.
If t1,…,tn are singular terms and Q is an n-place predicate letter of the alphabet, then Q(t1,…, tdue north) is a wff. Such a wff is an "diminutive" wff. (Note that the case in which n = 0 shows that every proposition letter standing alone is an atomic wff.)
2.
If A and B are wffs so too are ∼ A and (A → B).
3.
If A is a wff and v is a variable, then∀vA is a wff.
4.
There are no other wffs.
The scope of the quantifier in a wff and free and bound occurrences of variables are defined as usual, equally are the notions of open and airtight wff's. A closed wff is called a sentence.
Abbreviations
(A ∨ B) abbreviates (∼ A → B)
(A&B) abbreviates ∼ (∼ A∨∼ B)
∃vA abbreviates ∼∀v ∼ A.
At/due south is the wff derived from A past replacing all free occurrences of t by southward.
At//southward is the wff achieved by replacing some (or all) occurrences of t by s.
We will need several variations on this basic morphology:
Ronnie Cann , ... Daniel Wedgwood , in Philosophy of Linguistics, 2012
3.1 Formal languages and the form of grammar
The classical formal languages of propositional and predicate logic were defined non for the report of linguistic communication merely for the formal report of mathematical inference, though predicate logic incorporated a partial reflection of natural-linguistic communication structure in its articulation of subpropositional predicate-argument structure. Logic is the formal modelling of inference, which involves truth-dependence between wellformed formulae of the divers language. The objective in defining such a formal language is to capture all and only the valid inferences expressible in that language, via some concept of truth with respect to a model. The job is to posit a minimal number of appropriate units and construction-inducing processes that together yield all and simply the advisable outputs of the grammar, viz. the space set of wellformed strings, over which the inferences nether study tin be defined. The objective is to derive the infinite complexity of valid inference patterns from the interaction of a small number of primitive notions.
The perspective of mathematical inference imposes an of import restrictive chapters: information technology is global and static. In that location is no modelling of context external to the structure existence defined — mathematical truths are by definition independent of context. There is no modelling of growth of information or of its corollaries, underspecification of content and the concept of update. In fact, the catamenia of information is in exactly the opposite direction: rather than building information, inference involves just what follows from some data that is already given, the premises. There are therefore cardinal reasons to dubiety that the methodology of describing these formal languages could ever be sufficient for modelling natural languages. If the interpretation of expressions of tongue necessarily involves the building up of information relative to context, then a formal explication of this procedure is required. Models based in mathematical inference will not provide this, fifty-fifty though insights as to the nature of inference undoubtedly form a sub-part of a full characterisation of natural linguistic communication interpretation.
Despite its restrictiveness, the methodology for articulating formal languages has transformed the landscape within which formal frameworks for natural language and tongue grammars have adult; and the assumption of a truth-provisional ground to semantics for natural language is very widely adopted. In predicate logic, the grammar defines a system for inducing an infinite set up of propositional strings which are taken to exist truth-value denoting; and sentencesized units are defined equally having predicate-argument structure made up of predicate forms and individual constants, with naming and quantificational devices. Syntactic rules involve mappings from (sub)-formulae to (sub)-formulae making essential reference to structural properties: these rules constitute a finite set of principles inducing an infinite set of strings. Semantics is the provision of an algorithm for computing denotations for arbitrarily complex strings: the result is a pairing of strings and the objects they represent. This pairing is determined on the basis of some notion of content assigned to elementary parts, plus rules that determine how such contents are to be composed, through stepwise correspondence with syntax, yielding the values true and simulated as output.
The blueprint provided by such formal languages was famously extended to natural language semantics past Montague [1974], who argued that natural languages could exist characterised equally formal languages, with semantics defined in terms of reference/denotation/truth with respect to a model. To achieve this, Montague developed a program of logical types and formulations of content for expressions of the language which are divers and articulated in the lambda calculus. These were divers to enable the preservation of predicate logic insights into the meanings to be assigned to these expressions even while sustaining the view that limerick of such meanings is determined from some conventional analysis of the syntax of the language. Consequently, the tongue grammer, like a formal language, is conceived of as a arrangement that induces an infinite set of strings paired with denotations, where these denotations are determined past semantic rules which straight match the combinatorial operations that produce the strings themselves. For example, a functional abstruse can exist defined using a functional operator, the λ operator, which binds an open variable in some propositional formula to yield a function from individuals to propositional contents, as in λx.x smiled. If nosotros accept this to be the contribution of the intransitive verb smiled, and we take a abiding of type〈e〉, john, to be the contribution of the substantive phrase John, and so information technology is clear that this allows semantic limerick to mirror the combination of the words yielding the string John smiled. At the aforementioned time, a further functional abstruse can be defined in which a predicate variable is bound in some open up propositional formula to yield a office from backdrop to propositional contents λP.P(john) (semantically, the set of properties true of John, or the fix of classes that include John of type 〈〈e,t〉,t〉). This is as able to combine with the predicate λ10.x smiled to yield the proposition expressible by John smiled, only in this case the contribution of the subject is the functor and that of the verb is the statement.
As fifty-fifty these simple examples testify, there are in principle a number of different modes of combination bachelor, involving dissimilar functor/argument relations — with potentially many means of deriving the same string-meaning pair as more than complex sentences are considered. If this approach is applied with no independent constraints on syntactic analysis, the syntactic structure assigned to strings is effectively epiphenomenal, being no more than a vehicle for the semantic operations: this account is notably consort in categorial grammar formalisms [Moortgat, 1988; Morrill, 1994; 2010, this volume; Steedman, 2000]. These grammars are non-representationalist in that, on the one hand, the semantics of strings is divers in denotational terms (in terms of individuals, sets of individuals, and functions on those which ultimately map on to concepts of truth and inference); and, on the other, the rules of syntax constitute cipher more than than mappings from strings onto denotationally interpreted strings (that is to say, mappings from strings to strings suitably paired with mappings from denotational contents to denotational contents). Any invocation of structure is then no more than than a convenient way of talking about such pairwise mappings of strings and assignable denotational contents.
Fifty-fifty without adopting this strict variant of the Montague claim nearly natural languages equally formal languages, the influence of the formal-language methodology holds sway very generally. On a more pluralistic model of natural-language grammar — influenced by Montague solely with respect to semantics — a natural language grammar is a finite prepare of rules which assigns structure to sentences, and it is these syntactic structures to which denotational interpretations are assigned, defined in terms of truth with respect to a model. Semantic operations are thus defined in tandem with syntactic ones, almost transparently practical in the so-called 'Rule-to-Rule Hypothesis' whereby each structure-defining syntactic rule is paired with an appropriate semantic one. This is the ascendant model in work within theoretical linguistics that labels its object of study 'the syntax-semantics interface' (eastward.g. [Heim and Kratzer, 1998]) — which is to say near piece of work that purports to exist formally explicit almost both syntactic structure and its estimation. In terms of representationalism every bit a whole, this constitutes a mixed approach. The view of syntax is representationalist, in that there are assumed to be stock-still structures defined over strings of words. Just the semantics is not representational, at least if conceived of in terms of conventional logic formalisms, considering the semantic characterisation assigned to each syntactic structure is given in terms of denotation with respect to a model (or some equivalent).
It remains a matter of controversy in linguistic theory whether syntactic and semantic operations can be directly matched in this way. While for some analysts the direct assignment of content to syntactic structures remains an platonic worth striving for, others work on the basis that this is demonstrably impossible for natural languages. Broadly, speaking, there are ii common kinds of claim for the necessary departure of syntactic and semantic structures, necessitating multiple levels of representation. One is that the interpretation of natural languages requires representations of significant that are not directly interpretable in terms of denotations. This relates to bug of context-dependence, and we render to it in section 4. The other kind of claim is that natural language syntax has distinctive properties that are neither reducible to, nor reflected in, its semantics; this is our side by side topic of word.
Georg Buchgeher , Rainer Weinreich , in Agile Software Architecture, 2014
vii.3.3 Architecture clarification languages
ADLs are formal languages for describing the compages of a software system [26,27]. Each ADL defines a notation with precise syntax and semantics in which architecture models can be expressed, and provides a corresponding toolkit for working with the language.
ADLs include full general purpose languages similar xADL [28] and Summit [29], and domain-specific languages (DSLs) [30] like Koala [31], the Architecture Assay and Pattern Linguistic communication [32], and AUTOSAR [33]. A survey of available ADLs can be plant in [30]. Many ADLs are bookish research projects.
ADLs support the description of structural and selected behavioral aspects. An ADL describes a organisation at the component and connector abstraction level. A organisation is a configuration of components and connectors. Components are units of ciphering and data stores. Connectors depict interactions between components and the rules that govern these interactions [30]. The supported behavioral aspects are different for each ADL. For example, Wright [34] can be used for identifying interface incompatibilities and deadlocks.
ADLs primarily support architecture evaluation of selected quality attributes. In addition, architecture models tin be analyzed for completeness with respect to a modeling notation, and for consistency. Some ADLs, like ACME, likewise support compatibility analysis [35].
ADL-based architecture analysis is performed automatically using defended analysis tools. ADL-based architecture descriptions can also be used to simulate system behavior [v].
The cosmos of ADL-based architecture models is sometimes difficult and requires technical stakeholders with specific expertise [5]. This may be i reason why ADLs take not however constitute their way into mainstream software development. Additional reasons are listed by Woods and Hilliard [36] and include the restrictive nature of ADLs, the lack of multiple views, lack of adept tool support, their generic nature, and the lack of domain concepts.
In addition to ADLs, DSLs can be used to describe software architectures. Architecture-centric DSLs are typically developed for a particular domain or even a detail system and back up the automatic generation of the system implementation and specific kinds of automatic analysis [37].
George Boger , in Handbook of the History of Logic, 2004
6 Summary of Aristotle's Accomplishments in Prior Analytics
By the end of Prior Analytics A6 Aristotle had systematically worked through all possible patterns of two categorical sentences that could serve in the role of premisses to find "how every syllogism comes about". He established that among these possible patterns at that place are xiv that upshot, in their application to an object language, in something post-obit necessarily, that is, that effect in syllogisms. To reach this projection, Aristotle invented a formal language to devise a rudimentary model of his logic in Prior Analytic. In this way he was able to describe a deduction organization and demonstrate certain logical relationships among syllogistic rules, not the least of which accomplishment was establishing the independence of a pocket-sized prepare of deduction rules (A7). The formal language used in Prior Analytics congenital upon the foundation of his linguistic studies in Categories and On Estimation. Strictly speaking, Aristotle's formal language does not consist in sentences, every bit 'judgement' is defined in On Interpretation and equally 'protasis' is used in Prior Analytics. Rather, his formal language consists in relatively uninterpreted sentence patterns. By substituting non-logical constants — a predicate term and a subject term — for schematic letters, Aristotle could produce whatsoever number of object language sentences. Nosotros could easily call such sentences interpretations without baloney, as a modernistic logician understands this notion. This, however, would misrepresent Aristotle's logic. Nevertheless, this closeness to modern practice is not a superficial resemblance, but an indication of Aristotle'south genius and originality. Here we summarize some of his accomplishments and insights into logic with a synopsis of his model (§6.1) and with a summary of four proof-theoretic processes he employed in Prior Analytics (§half-dozen.2).
6.1 Synopsis of Aristotle's model
Aristotle invented his formal language with an aim to model scientific discourse. Such soapbox, and then, might be taken as its 'intended interpretation'. In any instance, using a modern mathematical template, we tin can re-present Aristotle's own model in the following way (Tabular array 44).
Table 44.
An ancient model of an underlying logic
Aristotle'due south ain model
Aristotle's model expressed by a modernistic note
LANGUAGE
Linguistic communication
Vocabulary
Vocabulary
i.
Four fully interpreted logical constants
1.
Logical constants
belongs to every
a
belongs to no
e
belongs to some
i
belongs not to every
o
2.
n schematic (upper case) letters intended to hold places for non-logical constants (terms)
ii.
northward schematic letters
A, B, C; M, N, X; P, R, Due south…
A, B, C…
Grammar
Grammer
1.
Sentences are the elements of a language. A categorical sentence is formed by concatenating a non-logical abiding with a logical constant with a different non-logical constant.
ane.
Categorical patterns
sentence
A belongs to every B.
AaB
A belongs to no B.
AeastwardB
A belongs to some B
AiB
A belongs not to every B.
AoB
2.
Relationships of opposite sentences
2.
Sentential relationships
Contradictories
Contradictories
A belongs to every B — to —
AaB — to — AoB
A belongs non to every B.
A belongs to no B — to — A belongs to some B
AeastwardB — to — AiB
Contraries
Contraries
A belongs to every B — to —
AaB — to — AeB
A belongs to no B.
iii.
Premiss germination
3.
Premiss germination
One-premiss statement
One-premiss argument
Take any ane of the four categorical sentences
AB
A10B
2-premiss argument
Two-premiss statement
Have whatsoever two of the four categorical sentences with 3 different terms, one in mutual
First figure:PMS
First figure:PMS
ane. PM
1. PxM
2. MS
2. KyDue south
2d figure:MPS
2nd figure:MPS
1. MP
1. KxP
2. MS
two. MyS
Tertiary effigy: PSM
Third figure: PSM
ane. PM
1. PxOne thousand
two. SM
ii. SyM
iv.
P-c argument formation
4.
P-c argument germination
I-premiss (conversion) statement
One-premiss argument
ane. AB
1. AtenB
?. BA
?. ByA
Two-premiss argument
Two-premiss argument
First figure:PMS
Start figure:PMS
1. PM
1. PxThou
2. MS
2. MySouthward
?. PS
?. PzS
Second figure:MPS
Second figure: MPS
1. MP
one. MxP
2. MS
ii. ChiliadyS
?. PS
?. PzSouthward
3rd figure: PSM
Third effigy: PSM
1. PM
1. PxThousand
2. SM
2. SyM
?. PS
?. PzS
DEDUCTION SYSTEM
DEDUCTION SYSTEM
one.
Deduction rules
1.
Deduction rules
One-premiss conversion rules
I-premiss rules
If A belongs to every B, then B belongs to some A.
1. AaB ane. AiB one. AeB
If A belongs to some B, so B belongs to some A.
∴ BiA ∴ BiA ∴ BeA
If A belongs to no B, then B belongs to no A.
Ii-premiss syllogism rules (reduced system)
2-premiss rules
If A belongs to every B and B belongs to every C, then A belongs to every C.
1. AaB 1. Adue eastB
If A belongs to no B and B belongs to every C, so A belongs to no C.
2. BaC two. BaC
∴ AaC ∴ AeastC
two.
Types of deduction
2.
Types of deduction
Direct deduction [See section 5.2]
Direct deduction
Indirect deduction [Come across Section 5.2]
Indirect deduction
SEMANTICS
SEMANTICS
1.
Meanings of sentences
1.
Meanings of sentences
AaB: universal attributive: Every B has property A
[Same]
AeB: universal privative: No B has property A.
AiB: partial attributive: Some B has holding A
AoB: partial privative: Some B does not accept holding A.
2.
Truth-values of sentences
ii.
Truth-values of sentences
AaB is true iff every B has property A.
[Same]
Adue eastB is true iff no B has belongings A.
Aib is truthful iff some B has property A.
AoB is true iff some B does non have property A.
3.
Logical Consequence
3.
Logical consequence
Information technology is impossible that a false sentence follows necessarily from truthful sentences.
[Same]
6.2 Iv proof-theoretic processes in Prior Analytics
Aristotle did not describe deductions in Prior Analytics A4–vi but showed how every syllogism comes about. He besides explained how syllogisms do not come up about and he refined his system. He described a natural deduction system of an underlying logic in ways suggestive of modern methods, and he proved sure properties of this deduction system. His handling of his logic is thoroughly metalogical. Here we briefly summarize four proof-theoretic processes used in Prior Analytics. All four processes have their counterparts in one or another object linguistic communication.
Deciding concludence: the method of completion
Completion (teleiôsis, teleiousthai) is a proof-theoretic, deductive process that establishes knowledge that a given statement pattern with places for two premisses having places for three different terms is panvalid by using the patterns of the teleioi sullogismoi every bit deduction rules. Completion is an epistemic procedure. In Prior Analytics A4–vi Aristotle established the preeminence of the patterns of the teleioi sullogismoi among the panvalid patterns or, conversely, he implicitly established that the patterns of the ateleis sullogismoi are redundant rules in his deduction system. The process of completion per se does not plant that any dominion of deduction is redundant. Nor does completion involve transforming a given argument pattern into some other argument pattern, since in the procedure of deduction a given argument pattern is not itself transformed only shown to be panvalid through a chain of reasoning cogent in context, which chain of reasoning is generated by means of specified deduction rules. Aristotle'south metalogical theorem apropos completion is that "all the ateleis sullogismoi are completed past means of the first effigy syllogisms using probative and reductio proofs" (A7: 29a30–33). Aristotle reserved using the verb 'teleiousthai' specifically in relation to a process by which a valid statement, whose validity is not evident, is made axiomatic by performing a deduction during which a teleios sullogismos, one whose validity is obviously evident, is generated; this signals cogency in the deduction process from premisses to conclusion.
Deciding inconcludence: the method of contrasted instances
The method of assorted instances used in Prior Analytics A4–6 is the complement of the procedure of completion. The purpose of this method is to constitute which uncomplicated argument patterns are not panvalid. This proof-theoretic method is dissimilar from the method of counterargument, since (1) information technology treats patterns of premisses and statement patterns and not arguments and, thus, information technology establishes paninvalidity and not invalidity, and (2) it does non produce an argument in the aforementioned form as a given argument just with true premisses and a false conclusion. Rather, this method constructs ii arguments, each of whose premisses are truthful sentences plumbing fixtures the same premiss-pair pattern and whose conclusions also are true sentences, only in the one argument the conclusion is an a sentence, in the other an e sentence. This establishes that a given premiss pair pattern is inconcludent and that consequently its corresponding four statement patterns are paninvalid. No syllogism is possible in such a case. It is not possible to construct such arguments with a concludent premiss pair pattern: in that example every like construction that produces truthful sentences as premisses results in at least one false sentence amidst the conclusions. Thus, whatsoever two sentences of three terms fitting a given inconcludent premiss-pair pattern are shown never to result together in a valid argument. In this mode Aristotle was able to eliminate would-be syllogistic deduction rules.
Transforming patterns: assay
Analysis (analusis, analuein) is a proof-theoretic process that transforms i syllogistic pattern in whatever ane figure into some other syllogistic pattern of some other figure only if both patterns 'prove' the same problêma. Analyses are performed through conversion and premiss transposition. Assay is non directly concerned with making validity or panvalidity evident, non with a deduction process, nor with establishing whether a given syllogistic pattern is a redundant rule. Rather, Aristotle aimed to promote his students' facility with reasoning syllogistically to constitute
and to abnegate arguments past studying the logical relationships among their patterns.107 This is coordinating to how mod logicians accept studied the relationships among the rules of propositional logic. Aristotle'due south theorem concerning analysis is that 'the syllogisms in the different figures that bear witness the same problêma are analyzable into each' (run across A45: 50b5–seven).
Nils J. Nilsson , in Artificial Intelligence: A New Synthesis, 1998
thirteen.8.ane Language Distinctions
The propositional calculus is a formal linguistic communication that an artificial amanuensis uses to describe its globe. At that place is ever a possibility of confusing the informal languages of mathematics and of English (which I am using in this book to talk about the propositional calculus) with the formal language of the propositional calculus itself. When we say, for example, that {P, P ⊃ Q}
Q nosotros use the symbol That symbol is not a symbol in the language of the propositional calculus; information technology is a symbol in our language used to talk about the propositional calculus. The metalinguistic symbols and should never be confused with the symbol ⊃, for example.
Heimo H. Adelsberger , in Encyclopedia of Physical Scientific discipline and Engineering (Third Edition), 2003
IV.D Start-Society Predicate Calculus
Predicate calculus is a formal language for expressing statements that are built from atomic formulas. Atomic formulas can be constants, variables, functions, and predicates. Atomic formulas can be continued to form statements by using the following connectives: ∨ (logical or), ∧ (logical and), and ⇒ (implication). The symbol ∼ (not) is used to negate a formula. Atomic formulas and their negations are called literals. Variables in a statement tin be quantified by ∀ (the universal quantifier, "for all") and ∃ (the existential quantifier, "for at least ane"). The syntactically correct expressions of the predicate calculus are called well-formed formulas (wff's). Get-go-order predicate calculus is an important subset of predicate calculus. Statements are restricted in that quantification is not allowed over predicates or functions. For example, the sentence "Siblings are not married" can exist represented by the first-order predicate calculus statement
Paul A. Fishwick , in Emotions and Bear on in Homo Factors and Human-Computer Interaction, 2017
Collaborative roles, usability, and experience
Artful Computing begins with a formal language construct such every bit a number, data, model, or software. Then the challenge is to represent this construct through embodiment. We noted that "embodiment" can exist as simple every bit pure reification without representation of existing objects when we demonstrated the ability to catch concord of numbers and move them toward operators. However, reification tin can likewise propose object representation equally in Figs. 21.4–21.vii. I need to address the "who" and "why" aspects of aesthetic calculating.
Get-go, who is going to be creating these representations? In the case of collaboration, I recommend teams of humanist scholars, artists, and calculator scientists. Humanist scholars bring to bear different philosophies and theories which can help shape the resulting representation. The creative person has the creative perspective and tools to create the representation, and the computer scientist can serve two roles: to help construct tools used by the humanist and artist in the extraction of information and in enabling the interaction that ensues through externalizing apotheosis in the homo-estimator interface.
2d, who is going to use the representations? Students in my aesthetic computing form are often initially confused why 1 would construct anything but diagrams. This confusion is expected, simply we must be conscientious when defining usability: usable for whom and for what purpose? We demand to place (1) the goal of the representation, and (2) the end target users. Goals for the embodied representations are education, arts, and amusement (e.g., cinema, visual and performing arts, fiction). Target users may exist whatsoever form level in school or some segment of the general public. From a psychological perspective, a broad view of "usability" tin cover user goals including: increased valence, motivation, and attitudinal change, as well every bit improved brusque or long term memory. Mathematicians and estimator scientists are not the target, as these populations are adept at using existing notations. Aesthetic computing is less stressed on information extraction and more on the utilise of entertainment, arts, and humanities on formal languages with the largest practical effects being in education. Thus the target users are formal and informal learners of all elements of formal linguistic communication-based instruction (due east.chiliad., mathematics, computer science).
The roles of participants in aesthetic computing will likely be different given the interests of each political party. For the computer scientist, for instance, Fig. 21.5 serves as a pattern template for the creation of special effects and interactive games for the purpose of expressing elements of prime numbers and the factorization process into these numbers. The artist'due south work is a medium through which this aspect of formal language is creatively expressed. The goals of the artist and computer scientist are clearly unlike, but the means (i.e., representations of prime numbers) are common. This difference in ends, with similar means, plays out in the other examples. For instance, Perl poetry (i.e., poesy created using the programming language, Perl) may be an artful production to the writer—a valid end in itself. To the computer scientist, this product represents a medium in which to express a different finish—the formal language "message." Therefore, aesthetic computing past its organization of words comprising this phrase is focused on computing—the learning of formal languages. Nevertheless, aesthetic products play a key function in this learning action and allow for the creative person, scholar, and computer scientist to interact with different intentions and goals.
Other areas related to aesthetic calculating are data visualization (Carte du jour et al., 1999; Ward et al., 2010), and software visualization (Eades and Zhang, 1996; Stasko et al., 1998; Zhang, 2007; Diehl, 2010); however, the goals of these areas are generally quite different than for aesthetic computing. In information visualization, the goal is efficient communication of information and information, whereas for aesthetic computing, the goal is education through highly embodied, and interactive, artful products in the forms of art and entertainment. Every bit such, Aesthetic Computing fosters a deeper experience than building representations meant for immediate consumption (east.g., newspaper diagrams and maps). Readers volition observe that the use of metaphor is rich inside the high level interactions with computers. We are an interface culture (Johnson, 1997). However, the metaphors used on the "desktop," for instance, have not nevertheless made their fashion into the core of mathematics and computing. Efforts such as computational thinking (Wing, 2006) are a move in the right direction.
Laurel (1993) presciently captures a prerequisite for aesthetic computing in her "Calculating every bit Theatre." However, Laurel was mainly constructing a instance for human-calculator interaction as a circuitous theatrical production, involving many of the same elements found in theatre. The use of calculating, and its associated interaction phenomena, are similar theatre. However, what we find is that every bit we suspension open the lid of the black box containing the atomic elements of normally hidden data, formulas, code, and models is that computing is theatre all the fashion downward.
Computer-Generated Proofs of Mathematical Theorems
David 1000. Bressoud , in Encyclopedia of Physical Science and Technology (Tertiary Edition), 2003
I.A What Cannot Be Washed
Mathematics is frequently viewed as a formal linguistic communication with conspicuously established underlying assumptions or axioms and unambiguous rules for determining the truth of every argument couched in this linguistic communication. In the early decades of the twentieth century, works such as Russell and Whitehead'southward Principia Mathematica attempted to describe all mathematics in terms of the formal language of logic. Part of the reason for this undertaking was the hope that it would atomic number 82 to an algorithmic process for determining the truth of each mathematical statement. As the twentieth century progressed, this hope receded and finally vanished. In 1931, Kurt Gödel proved that no axiomatic system comparable to that of Russell and Whitehead could be used to decide the truth or falsehood of every mathematical statement. Every consistent organization of axioms is necessarily incomplete.
One wide grade of theorems deals with the existence of solutions of a particular course. Given the mathematical problem, the theorem either exhibits a solution of the desired blazon or states that no such solution exists. In 1900, every bit the tenth of his set of 20-iii problems, David Hilbert challenged the mathematical community: "Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a procedure co-ordinate to which it can be determined by a finite number of operations whether the equation is solvable in rational integers." A well-known example of such a Diophantine equation is the Pythagorean equation, x2 + y2 = z2, with the restriction that we just have integer solutions such as ten = 3, y = 4, and z = v. Another problem of this type is Fermat'southward Last Theorem. This theorem asserts that no such positive integer solutions be for the equation xn + ynorth = zn when n is an integer greater than or equal to 3. We know that the last exclamation is correct, thank you to Andrew Wiles.
For a Diophantine equation, if a solution exists and then it tin can be found in finite (though potentially very long) fourth dimension but by trying all possible combinations of integers, merely if no solution exists and then we cannot observe this fact only by trying possibilities. A proof that in that location is no solution is usually very difficult. In 1970, Yuri Matijasevič proved that Hilbert's algorithm could not exist. Information technology is impossible to construct an algorithm that, for every Diophantine equation, is able to determine whether it does or does not have a solution.
At that place have been other negative results. Let E be an expression that involves the rational numbers, π, ln 2, the variable x, the functions sine, exponential, and absolute value, and the operations of addition, multiplication, and composition. Does there exist a value of x where this expression is nil? As an example, is there a real ten for which
For this particular expression the answer is "yes" because sin(π ln ii) > 0, but in 1968, Daniel Richardson proved that information technology is impossible to construct an algorithm that would determine in finite time whether or non, for every such Eastward, there exists a solution to the equality E = 0.
John H. Heinrichs , ... Jeen S. Lim , in Encyclopedia of Data Systems, 2003
2.B.i. Codified vs Uncodified
Codified noesis tin can be represented in formal language such every bit mathematical, grammatical, digital, and symbolic codes. Codified involves the creation of perceptual and conceptual categories that facilitate the nomenclature of various phenomena. Thus, codified organizational cognition represents phenomena that have been classified into perceptual and conceptual categories meaningful to organizational members.
Notwithstanding, it is important to realize that the procedure of codified is not simple. Depending upon the complexity of the phenomena in question, codified can be fraught with potential problems. For example, codifying the sales performance of the shop'south salesperson could be viewed as a unproblematic task of associating the number of dollars and/or the number of product units sold by the salesperson. Withal, sales performance could too be assessed in boosted terms such as the amount of customer satisfaction generated with each sales transaction. Codifying customer satisfaction generated at the time of the sale would exist much more difficult and more complicated to achieve.
Thus, some knowledge is as well rich, ambiguous, complex, and personal to be articulated or codified. Such knowledge remains uncodified and is often referred to as "tacit." Uncodified or tacit knowledge tin take on two forms. Those forms are know-how and taken-for-granted beliefs. The know-how class of tacit knowledge is an embedded skill or ability caused from birth or over time from feel. For example, the knowledge of how to ride a cycle is probably tacit in most people who can ride a bicycle. These people can merely become on a bicycle and automatically begin riding it. However, for those who cannot ride a bike they demand to accept physical experience on a bike before they tin can identify the knowledge needed to learn. The taken-for-granted beliefs embody what is and what should be. Taken-for-granted beliefs or notions of what is and what should be represent noesis embedded in mental models and value systems that shape how one perceives and experiences the world. For example, i might take it for granted that everyone believes in God until he or she encounters an atheist for the first fourth dimension.
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