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68-434: Formally, for a formal language L to belong to L/poly, there must exist an advice function f that maps an integer n to a string of length polynomial in n , and a Turing machine M with two read-only input tapes and one read-write tape of size logarithmic in the input size, such that an input x of length n belongs to L if and only if machine M accepts the input x ,  f ( n ) . Alternatively and more simply, L

136-634: A deductive apparatus (also called a deductive system ). The deductive apparatus may consist of a set of transformation rules , which may be interpreted as valid rules of inference, or a set of axioms , or have both. A formal system is used to derive one expression from one or more other expressions. Although a formal language can be identified with its formulas, a formal system cannot be likewise identified by its theorems. Two formal systems F S {\displaystyle {\mathcal {FS}}} and F S ′ {\displaystyle {\mathcal {FS'}}} may have all

204-399: A formal grammar may be closer to the intuitive concept of a "language", one described by syntactic rules. By an abuse of the definition, a particular formal language is often thought of as being accompanied with a formal grammar that describes it. The following rules describe a formal language  L over the alphabet Σ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, +, =}: Under these rules,

272-493: A formal grammar . The alphabet of a formal language consists of symbols, letters, or tokens that concatenate into strings called words. Words that belong to a particular formal language are sometimes called well-formed words or well-formed formulas . A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar , which consists of its formation rules . In computer science, formal languages are used, among others, as

340-400: A "formal language of pure language." In the first half of the 20th century, several developments were made with relevance to formal languages. Axel Thue published four papers relating to words and language between 1906 and 1914. The last of these introduced what Emil Post later termed 'Thue Systems', and gave an early example of an undecidable problem . Post would later use this paper as

408-409: A belief that became obsessional. Gödel may have failed to appreciate the magnitude of the task facing the editors of Leibniz's manuscripts, given that Leibniz left about 15,000 letters and 40,000 pages of other manuscripts. Even now, most of this huge Nachlass remains unpublished. Others in the 17th century, such as George Dalgarno , attempted similar philosophical and linguistic projects, some under

476-467: A compendium of all human knowledge. Many Leibniz scholars writing in English seem to agree that he intended his characteristica universalis or "universal character" to be a form of pasigraphy , or ideographic language . This was to be based on a rationalised version of the 'principles' of Chinese characters , as Europeans understood these characters in the seventeenth century. From this perspective it

544-461: A finite number of elements, and many results apply only to them. It often makes sense to use an alphabet in the usual sense of the word, or more generally any finite character encoding such as ASCII or Unicode . A word over an alphabet can be any finite sequence (i.e., string ) of letters. The set of all words over an alphabet Σ is usually denoted by Σ (using the Kleene star ). The length of

612-692: A good compromise between expressivity and ease of parsing , and are widely used in practical applications. Certain operations on languages are common. This includes the standard set operations, such as union, intersection, and complement. Another class of operation is the element-wise application of string operations. Examples: suppose L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} are languages over some common alphabet Σ {\displaystyle \Sigma } . Such string operations are used to investigate closure properties of classes of languages. A class of languages

680-578: A high level programming language, following his work in the creation of FORTRAN . Peter Naur was the secretary/editor for the ALGOL60 Report in which he used Backus–Naur form to describe the Formal part of ALGOL60. An alphabet , in the context of formal languages, can be any set ; its elements are called letters . An alphabet may contain an infinite number of elements; however, most definitions in formal language theory specify alphabets with

748-454: A lack of specifics in both English language translations and modern English language interpretations of Leibniz's writings render a clear exposition difficult. As with Leibniz's calculus ratiocinator two different schools of philosophical thought have come to emphasise two different aspects that can be found in Leibniz's writing. The first point of view emphasizes logic and language , and

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816-438: A language or characteristic which includes at once both the arts of discovery and judgement, that is, one whose signs and characters serve the same purpose that arithmetical signs serve for numbers, and algebraic signs for quantities taken abstractly. Yet it does seem that since God has bestowed these two sciences on mankind, he has sought to notify us that a far greater secret lies hidden in our understanding, of which these are but

884-461: A language usable within the framework of a universal logical calculation or calculus ratiocinator . The characteristica universalis is a recurring concept in the writings of Leibniz. When writing in French, he sometimes employed the phrase spécieuse générale to the same effect. The concept is sometimes paired with his notion of a calculus ratiocinator and with his plans for an encyclopaedia as

952-498: A language  L as just L  = {a, b, ab, cba}. The degenerate case of this construction is the empty language , which contains no words at all ( L  =  ∅ ). However, even over a finite (non-empty) alphabet such as Σ = {a, b} there are an infinite number of finite-length words that can potentially be expressed: "a", "abb", "ababba", "aaababbbbaab", .... Therefore, formal languages are typically infinite, and describing an infinite formal language

1020-611: A man who is neither a prophet nor a prince can ever undertake any thing of greater good to mankind of more fitting for divine glory" (Loemker 1969: 225). But later in life, a more sober note emerged. In a March 1706 letter to the Electress Sophia of Hanover , the spouse of his patron, he wrote: It is true that in the past I planned a new way of calculating suitable for matters which have nothing in common with mathematics, and if this kind of logic were put into practice, every reasoning, even probabilistic ones, would be like that of

1088-400: A meaning to each of the formulas—usually, a truth value . The study of interpretations of formal languages is called formal semantics . In mathematical logic, this is often done in terms of model theory . In model theory, the terms that occur in a formula are interpreted as objects within mathematical structures , and fixed compositional interpretation rules determine how the truth value of

1156-439: A part of my characteristic, a task which is not easy, especially in my present condition and without the advantage of discussions with men who could stimulate and help me in work of this nature. Eventually, by discovering binary digits again from Chinese works, which was now from the I Ching , Leibniz arrived at what he felt was a discovery of a link that would thereby create his characteristica universalis. It eventually created

1224-417: A point; for numbers, points; for the relations of one entity with another, lines; for the variation of angles and of extremities in lines, kinds of relations. If these are correctly and ingeniously established, this universal writing will be as easy as it is common, and will be capable of being read without any dictionary; at the same time, a fundamental knowledge of all things will be obtained. The whole of such

1292-522: A polynomial advice tape may be simulated by a branching program the states of which represent the combination of the configuration of the writable tape and the position of the Turing machine heads on the other two tapes. In 1979, Aleliunas et al. showed that symmetric logspace is contained in L/poly. However, this result was superseded by Omer Reingold 's result that SL collapses to uniform logspace. BPL

1360-516: A system of notations and symbols intended to facilitate the description of machines"). Heinz Zemanek rated it as an equivalent to a programming language for the numerical control of machine tools. Noam Chomsky devised an abstract representation of formal and natural languages, known as the Chomsky hierarchy . In 1959 John Backus developed the Backus-Naur form to describe the syntax of

1428-413: A tool like lex , identifies the tokens of the programming language grammar, e.g. identifiers or keywords , numeric and string literals, punctuation and operator symbols, which are themselves specified by a simpler formal language, usually by means of regular expressions . At the most basic conceptual level, a parser , sometimes generated by a parser generator like yacc , attempts to decide if

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1496-466: A universal science ( scientia universalis ) for coordinating all human knowledge into a systematic whole comprises two parts: (1) a universal notation ( characteristica universalis ) by use of which any item of information whatever can be recorded in a natural and systematic way, and (2) a means of manipulating the knowledge thus recorded in a computational fashion, so as to reveal its logical interrelations and consequences (the calculus ratiocinator ). Near

1564-600: A way of understanding the syntactic regularities of natural languages . In the 17th century, Gottfried Leibniz imagined and described the characteristica universalis , a universal and formal language which utilised pictographs . Later, Carl Friedrich Gauss investigated the problem of Gauss codes . Gottlob Frege attempted to realize Leibniz's ideas, through a notational system first outlined in Begriffsschrift (1879) and more fully developed in his 2-volume Grundgesetze der Arithmetik (1893/1903). This described

1632-418: A word is the number of letters it is composed of. For any alphabet, there is only one word of length 0, the empty word , which is often denoted by e, ε, λ or even Λ. By concatenation one can combine two words to form a new word, whose length is the sum of the lengths of the original words. The result of concatenating a word with the empty word is the original word. In some applications, especially in logic ,

1700-471: A writing will be made of geometrical figures, as it were, and of a kind of pictures — just as the ancient Egyptians did, and the Chinese do today. Their pictures, however, are not reduced to a fixed alphabet... with the result that a tremendous strain on the memory is necessary, which is the contrary of what we propose. Nicholas Rescher , reviewing Cohen's 1954 article, wrote that: Leibniz's program of

1768-465: Is associated with analytic philosophy and rationalism . The second point of view is more in tune with Couturat's views as expressed above, which emphasize science and engineering . This point of view is associated with synthetic philosophy and empiricism . Either or both of these aspects Leibniz hoped would guide human reasoning like Ariadne's thread and thereby suggest solutions to many of humanity's urgent problems. Because Leibniz never described

1836-570: Is closed under a particular operation when the operation, applied to languages in the class, always produces a language in the same class again. For instance, the context-free languages are known to be closed under union, concatenation, and intersection with regular languages , but not closed under intersection or complement. The theory of trios and abstract families of languages studies the most common closure properties of language families in their own right. A compiler usually has two distinct components. A lexical analyzer , sometimes generated by

1904-519: Is common to find the characteristica universalis associated with contemporary universal language projects like Esperanto , auxiliary languages like Interlingua , and formal logic projects like Frege 's Begriffsschrift . The global expansion of European commerce in Leibniz's time provided mercantilist motivations for a universal language of trade so that traders could communicate with any natural language. Others, such as Jaenecke, for example, have observed that Leibniz also had other intentions for

1972-429: Is contained in L/poly, which is a variant of Adleman's theorem . This theoretical computer science –related article is a stub . You can help Misplaced Pages by expanding it . Formal language In logic , mathematics , computer science , and linguistics , a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules called

2040-412: Is in L/poly if and only if it can be recognized by branching programs of polynomial size. One direction of the proof that these two models of computation are equivalent in power is the observation that, if a branching program of polynomial size exists, it can be specified by the advice function and simulated by the Turing machine. In the other direction, a Turing machine with logarithmic writable space and

2108-482: Is not as simple as writing L  = {a, b, ab, cba}. Here are some examples of formal languages: Formal languages are used as tools in multiple disciplines. However, formal language theory rarely concerns itself with particular languages (except as examples), but is mainly concerned with the study of various types of formalisms to describe languages. For instance, a language can be given as Typical questions asked about such formalisms include: Surprisingly often,

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2176-412: Is not the same as that used in statistical analysis. In 1918, Clarence Irving Lewis , the first English-speaking logician to translate and discuss some of Leibniz's logical writings, elaborated on "logistic" as follows: Logistic may be defined as the science which deals with types of order as such. It is not so much a subject as a method. Although most logistic is either founded upon or makes large use of

2244-532: Is reproduced in several texts including Saemtliche Schriften und Briefe ( Saemtliche Schriften und Briefe , Reihe VI, Band 1: 166, Loemker 1969: 83, 366, Karl Popp and Erwin Stein 2000: 33). Leibniz rightly saw that creating the characteristica would be difficult, fixing the time required for devising it as follows: "I think that some selected men could finish the matter in five years" (Loemker 1969: 224), later remarking: "And so I repeat, what I have often said, that

2312-423: Is responsible for the distorted picture of Leibniz's work found in the literature. As Louis Couturat wrote, Leibniz criticized the linguistic systems of George Dalgarno and John Wilkins for this reason since they focused on ...practical uses rather than scientific utility, that is, for being chiefly artificial languages intended for international communication and not philosophical languages that would express

2380-495: Is still not mature enough to lay claim to the advantages which this method could provide. In another 1714 letter to Nicholas Remond, he wrote: I have spoken to the Marquis de l'Hôpital and others about my general algebra, but they have paid no more attention to it than if I had told them about a dream of mine. I should have to support it too by some obvious application, but to achieve this it would be necessary to work out at least

2448-583: The characteristica to fulfill those functions. In the domain of science, Leibniz aimed for his characteristica to form diagrams or pictures, depicting any system at any scale, and understood by all regardless of native language. Leibniz wrote: And although learned men have long since thought of some kind of language or universal characteristic by which all concepts and things can be put into beautiful order, and with whose help different nations might communicate their thoughts and each read in his own language what another has written in his, yet no one has attempted

2516-463: The characteristica universalis in operational detail, many philosophers have deemed it an absurd fantasy. In this vein, Parkinson wrote: Leibniz's views about the systematic character of all knowledge are linked with his plans for a universal symbolism, a Characteristica Universalis . This was to be a calculus which would cover all thought, and replace controversy by calculation. The ideal now seems absurdly optimistic..." The logician Kurt Gödel , on

2584-430: The characteristica universalis , and these aspects appear to be a source of the aforementioned vagueness and inconsistency in modern interpretations. According to Jaenecke, the Leibniz project is not a matter of logic but rather one of knowledge representation, a field largely unexploited in today's logic-oriented epistemology and philosophy of science. It is precisely this one-sided orientation of these disciplines, which

2652-461: The foundations of mathematics , formal languages are used to represent the syntax of axiomatic systems , and mathematical formalism is the philosophy that all of mathematics can be reduced to the syntactic manipulation of formal languages in this way. The field of formal language theory studies primarily the purely syntactic aspects of such languages—that is, their internal structural patterns. Formal language theory sprang out of linguistics, as

2720-531: The " Invisible College " and now seen as forerunners of the Royal Society . A wide variety of constructed languages have emerged over the past 150 years which may be seen as supporting some of Leibniz's intuitions. On Leibniz's lifelong interest in the characteristica and the like, see the following texts in Loemker (1969): 165–66, 192–95, 221–28, 248–50, and 654–66. On

2788-492: The adjective "formal" is often omitted as redundant. While formal language theory usually concerns itself with formal languages that are described by some syntactic rules, the actual definition of the concept "formal language" is only as above: a (possibly infinite) set of finite-length strings composed from a given alphabet, no more and no less. In practice, there are many languages that can be described by rules, such as regular languages or context-free languages . The notion of

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2856-530: The alphabet is also known as the vocabulary and words are known as formulas or sentences ; this breaks the letter/word metaphor and replaces it by a word/sentence metaphor. A formal language L over an alphabet Σ is a subset of Σ , that is, a set of words over that alphabet. Sometimes the sets of words are grouped into expressions, whereas rules and constraints may be formulated for the creation of 'well-formed expressions'. In computer science and mathematics, which do not usually deal with natural languages ,

2924-491: The answer to these decision problems is "it cannot be done at all", or "it is extremely expensive" (with a characterization of how expensive). Therefore, formal language theory is a major application area of computability theory and complexity theory . Formal languages may be classified in the Chomsky hierarchy based on the expressive power of their generative grammar as well as the complexity of their recognizing automaton . Context-free grammars and regular grammars provide

2992-401: The art of complication of the sciences, i.e., of inventive logic... But when the tables of categories of our art of complication have been formed, something greater will emerge. For let the first terms, of the combination of which all others consist, be designated by signs; these signs will be a kind of alphabet. It will be convenient for the signs to be as natural as possible—e.g., for one,

3060-484: The basis for a 1947 proof "that the word problem for semigroups was recursively insoluble", and later devised the canonical system for the creation of formal languages. In 1907, Leonardo Torres Quevedo introduced a formal language for the description of mechanical drawings (mechanical devices), in Vienna . He published "Sobre un sistema de notaciones y símbolos destinados a facilitar la descripción de las máquinas" ("On

3128-478: The basis for defining the grammar of programming languages and formalized versions of subsets of natural languages, in which the words of the language represent concepts that are associated with meanings or semantics . In computational complexity theory , decision problems are typically defined as formal languages, and complexity classes are defined as the sets of the formal languages that can be parsed by machines with limited computational power. In logic and

3196-530: The characters and the words themselves would direct the mind, and the errors — excepting those of fact — would only be calculation mistakes. It would be very difficult to form or invent this language or characteristic, but very easy to learn it without any dictionaries. The universal "representation" of knowledge would therefore combine lines and points with "a kind of pictures" ( pictographs or logograms ) to be manipulated by means of his calculus ratiocinator . He hoped his pictorial algebra would advance

3264-410: The compiler to eventually generate an executable containing machine code that runs directly on the hardware, or some intermediate code that requires a virtual machine to execute. In mathematical logic , a formal theory is a set of sentences expressed in a formal language. A formal system (also called a logical calculus , or a logical system ) consists of a formal language together with

3332-427: The end of his life, Leibniz wrote that combining metaphysics with mathematics and science through a universal character would require creating what he called: ... a kind of general algebra in which all truths of reason would be reduced to a kind of calculus. At the same time, this would be a kind of universal language or writing, though infinitely different from all such languages which have thus far been proposed; for

3400-597: The epistemic synthesis of empirical science, mathematics, pictographs and metaphysics in the way Leibniz described. Hence scholars have had difficulty in showing how projects such as the Begriffsschrift and Esperanto embody the full vision Leibniz had for his characteristica . The writings of Alexander Gode suggested that Leibniz' characteristica had a metaphysical bias which prevented it from reflecting reality faithfully. Gode emphasized that Leibniz established certain goals or functions first, and then developed

3468-504: The formula can be derived from the interpretation of its terms; a model for a formula is an interpretation of terms such that the formula becomes true. Characteristica universalis The Latin term characteristica universalis , commonly interpreted as universal characteristic , or universal character in English, is a universal and formal language imagined by Gottfried Leibniz able to express mathematical, scientific, and metaphysical concepts. Leibniz thus hoped to create

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3536-418: The foundations of symbolic logic and modern philosophy, specifically the predicate-based Analytic Philosophy and Boolean Logic . C. J. Cohen (1954) set out three criteria which any project for a philosophical language would need to meet before it could be considered a version of the characteristica universalis . In setting out these criteria, Cohen made reference to the concept of "logistic". This concept

3604-517: The frontispiece to his 1666 De Arte Combinatoria (On the Art of Combinations), represents the Aristotelian theory of how all material things are formed from combinations of the elements earth, water, air, and fire. These four elements make up the four corners of a diamond (see picture). Opposing pairs of these are joined by a bar labeled "contraries" (earth-air, fire-water). At the four corners of

3672-737: The heading of mathesis universalis . A notable example was John Wilkins , the author of An Essay towards a Real Character and a Philosophical Language , who wrote a thesaurus as a first step towards a universal language. He intended to add to his thesaurus an alphabet of human thought (an organisational scheme, similar to a thesaurus or the Dewey decimal system ), and an "algebra of thought", allowing rule-based manipulation. The philosophers and linguists who undertook such projects often belonged to pansophical (universal knowledge) and scientific knowledge groups in London and Oxford, collectively known as

3740-548: The logical relations of concepts. He favors, and opposes to them, the true "real characteristic", which would express the composition of concepts by the combination of signs representing their simple elements, such that the correspondence between composite ideas and their symbols would be natural and no longer conventional. Leibniz said that his goal was an alphabet of human thought , a universal symbolic language (characteristic) for science , mathematics , and metaphysics . According to Couturat, "In May 1676, he once again identified

3808-446: The mathematician: if need be, the lesser minds which had application and good will could, if not accompany the greatest minds, then at least follow them. For one could always say: let us calculate, and judge correctly through this, as much as the data and reason can provide us with the means for it. But I do not know if I will ever be in a position to carry out such a project, which requires more than one hand; and it even seems that mankind

3876-525: The moon). It should be noted that Leibniz sometimes employs planetary signs in place of letters in his algebraic calculations Hartley Rogers emphasised the metaphysical aspect of the characteristica universalis by relating it to the "elementary theory of the ordering of the reals," defining it as "a precisely definable system for making statements of science" (Rogers 1963: 934). Universal language projects like Esperanto, and formal logic projects like Frege 's Begriffsschrift are not commonly concerned with

3944-471: The other hand, believed that the characteristica universalis was feasible, and that its development would revolutionize mathematical practice (Dawson 1997). He noticed, however, that a detailed treatment of the characteristica was conspicuously absent from Leibniz's publications. It appears that Gödel assembled all of Leibniz's texts mentioning the characteristica , and convinced himself that some sort of systematic and conspiratorial censoring had taken place,

4012-440: The principles of symbolic logic, still a science of order in general does not necessarily presuppose or begin with symbolic logic. Following from this Cohen stipulated that the universal character would have to serve as: These criteria together with the notion of logistic reveal that Cohen and Lewis both associated the characteristica with the methods and objectives of general systems theory . Inconsistency , vagueness , and

4080-436: The same theorems and yet differ in some significant proof-theoretic way (a formula A may be a syntactic consequence of a formula B in one but not another for instance). A formal proof or derivation is a finite sequence of well-formed formulas (which may be interpreted as sentences, or propositions ) each of which is an axiom or follows from the preceding formulas in the sequence by a rule of inference . The last sentence in

4148-501: The scientific treatment of qualitative phenomena, thereby constituting "that science in which are treated the forms or formulas of things in general, that is, quality in general" ( On Universal Synthesis and Analysis , 1679, in Loemker 1969: 233). Since the characteristica universalis is diagrammatic and employs pictograms (see picture), the diagrams in Leibniz's work warrant close study. On at least two occasions, Leibniz illustrated his philosophical reasoning with diagrams. One diagram,

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4216-411: The sequence is a theorem of a formal system. Formal proofs are useful because their theorems can be interpreted as true propositions. Formal languages are entirely syntactic in nature, but may be given semantics that give meaning to the elements of the language. For instance, in mathematical logic , the set of possible formulas of a particular logic is a formal language, and an interpretation assigns

4284-407: The shadows. P. P. Weiner raised an example of a large scale application of Leibniz's characteristica to climatic science. A weather-forecaster invented by Athanasius Kircher "interested Leibniz in connection with his own attempts to invent a universal language" (1940). Leibniz talked about his dream of a universal scientific language at the very dawn of his career, as follows: We have spoken of

4352-417: The source program is syntactically valid, that is if it is well formed with respect to the programming language grammar for which the compiler was built. Of course, compilers do more than just parse the source code – they usually translate it into some executable format. Because of this, a parser usually outputs more than a yes/no answer, typically an abstract syntax tree . This is used by subsequent stages of

4420-517: The string "23+4=555" is in  L , but the string "=234=+" is not. This formal language expresses natural numbers , well-formed additions, and well-formed addition equalities, but it expresses only what they look like (their syntax ), not what they mean ( semantics ). For instance, nowhere in these rules is there any indication that "0" means the number zero, "+" means addition, "23+4=555" is false, etc. For finite languages, one can explicitly enumerate all well-formed words. For example, we can describe

4488-400: The superimposed square are the four qualities defining the elements. Each adjacent pair of these is joined by a bar labeled "possible combination"; the diagonals joining them are labeled "impossible combination". Starting from the top, fire is formed from the combination of dryness and heat; air from wetness and heat; water from coldness and wetness; earth from coldness and dryness. This diagram

4556-408: The universal language with the characteristic and dreamed of a language that would also be a calculus—a sort of algebra of thought" (1901, chp 3.). This characteristic was a universalisation of the various "real characteristics". Couturat wrote that Leibniz gave Egyptian and Chinese hieroglyphics and chemical signs as examples of real characteristics writing: This shows that the real characteristic

4624-409: Was for him an ideography, that is, a system of signs that directly represent things (or, rather, ideas) and not words, in such a way that each nation could read them and translate them into its own language. In a footnote, Couturat added: Elsewhere Leibniz even includes among the types of signs musical notes and astronomical signs (the signs of the zodiac and those of the planets, including the sun and

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