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71-431: The machine operates on an infinite memory tape divided into discrete cells, each of which can hold a single symbol drawn from a finite set of symbols called the alphabet of the machine. It has a "head" that, at any point in the machine's operation, is positioned over one of these cells, and a "state" selected from a finite set of states. At each step of its operation, the head reads the symbol in its cell. Then, based on

142-770: A recurrence relation or difference equation . Difference equations are similar to differential equations , but replace differentiation by taking the difference between adjacent terms; they can be used to approximate differential equations or (more often) studied in their own right. Many questions and methods concerning differential equations have counterparts for difference equations. For instance, where there are integral transforms in harmonic analysis for studying continuous functions or analogue signals, there are discrete transforms for discrete functions or digital signals. As well as discrete metric spaces , there are more general discrete topological spaces , finite metric spaces , finite topological spaces . The time scale calculus

213-526: A 0th symbol S 0 = "erase" or "blank", etc. However, he did not allow for non-printing, so every instruction-line includes "print symbol S k " or "erase" (cf. footnote 12 in Post (1947), The Undecidable , p. 300). The abbreviations are Turing's ( The Undecidable , p. 119). Subsequent to Turing's original paper in 1936–1937, machine-models have allowed all nine possible types of five-tuples: Any Turing table (list of instructions) can be constructed from

284-429: A Turing machine, programming languages themselves do not necessarily have this limitation. Kirner et al., 2009 have shown that among the general-purpose programming languages some are Turing complete while others are not. For example, ANSI C is not Turing complete, as all instantiations of ANSI C (different instantiations are possible as the standard deliberately leaves certain behaviour undefined for legacy reasons) imply

355-432: A computer, with the canonical machine using sequential memory to store data. Typically, the sequential memory is represented as a tape of infinite length on which the machine can perform read and write operations. In the context of formal language theory, a Turing machine ( automaton ) is capable of enumerating some arbitrary subset of valid strings of an alphabet . A set of strings which can be enumerated in this manner

426-407: A course that is basically intended to develop mathematical maturity in first-year students; therefore, it is nowadays a prerequisite for mathematics majors in some universities as well. Some high-school-level discrete mathematics textbooks have appeared as well. At this level, discrete mathematics is sometimes seen as a preparatory course, like precalculus in this respect. The Fulkerson Prize

497-415: A desultory manner"). More explicitly, a Turing machine consists of: In the 4-tuple models, erasing or writing a symbol (a j1 ) and moving the head left or right (d k ) are specified as separate instructions. The table tells the machine to (ia) erase or write a symbol or (ib) move the head left or right, and then (ii) assume the same or a new state as prescribed, but not both actions (ia) and (ib) in

568-412: A different class. A method is formally called effective for a class of problems when it satisfies these criteria: Optionally, it may also be required that the method never returns a result as if it were an answer when the method is applied to a problem from outside its class. Adding this requirement reduces the set of classes for which there is an effective method. An effective method for calculating

639-535: A different convention, with new state q m listed immediately after the scanned symbol S j : For the remainder of this article "definition 1" (the Turing/Davis convention) will be used. In the following table, Turing's original model allowed only the first three lines that he called N1, N2, N3 (cf. Turing in The Undecidable , p. 126). He allowed for erasure of the "scanned square" by naming

710-414: A drawing. Whether a drawing represents an improvement on its table must be decided by the reader for the particular context. The reader should again be cautioned that such diagrams represent a snapshot of their table frozen in time, not the course ("trajectory") of a computation through time and space. While every time the busy beaver machine "runs" it will always follow the same state-trajectory, this

781-479: A finite-space memory. This is because the size of memory reference data types, called pointers , is accessible inside the language. However, other programming languages like Pascal do not have this feature, which allows them to be Turing complete in principle. It is just Turing complete in principle, as memory allocation in a programming language is allowed to fail, which means the programming language can be Turing complete when ignoring failed memory allocations, but

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852-669: A mathematical description of a very simple device capable of arbitrary computations, he was able to prove properties of computation in general—and in particular, the uncomputability of the Entscheidungsproblem ('decision problem'). Turing machines proved the existence of fundamental limitations on the power of mechanical computation. While they can express arbitrary computations, their minimalist design makes them too slow for computation in practice: real-world computers are based on different designs that, unlike Turing machines, use random-access memory . Turing completeness

923-453: A model through which one can reason about an algorithm or "mechanical procedure" in a mathematically precise way without being tied to any particular formalism. Studying the abstract properties of Turing machines has yielded many insights into computer science , computability theory , and complexity theory . In his 1948 essay, "Intelligent Machinery", Turing wrote that his machine consists of: ...an unlimited memory capacity obtained in

994-405: A number of challenging problems which have focused attention within areas of the field. In graph theory, much research was motivated by attempts to prove the four color theorem , first stated in 1852, but not proved until 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computer assistance). In logic , the second problem on David Hilbert 's list of open problems presented in 1900

1065-402: A part of number theory and analysis , partition theory is now considered a part of combinatorics or an independent field. Order theory is the study of partially ordered sets , both finite and infinite. Graph theory, the study of graphs and networks , is often considered part of combinatorics, but has grown large enough and distinct enough, with its own kind of problems, to be regarded as

1136-538: A point, or as the spectrum Spec ⁡ K [ x ] ( x − c ) {\displaystyle \operatorname {Spec} K[x]_{(x-c)}} of the local ring at (x-c) , a point together with a neighborhood around it. Algebraic varieties also have a well-defined notion of tangent space called the Zariski tangent space , making many features of calculus applicable even in finite settings. In applied mathematics , discrete modelling

1207-425: A real computer program, it is possible for a Turing machine to go into an infinite loop which will never halt. The Turing machine was invented in 1936 by Alan Turing , who called it an "a-machine" (automatic machine). It was Turing's doctoral advisor, Alonzo Church , who later coined the term "Turing machine" in a review. With this model, Turing was able to answer two questions in the negative: Thus by providing

1278-449: A single conclusion). The truth values of logical formulas usually form a finite set, generally restricted to two values: true and false , but logic can also be continuous-valued, e.g., fuzzy logic . Concepts such as infinite proof trees or infinite derivation trees have also been studied, e.g. infinitary logic . Set theory is the branch of mathematics that studies sets , which are collections of objects, such as {blue, white, red} or

1349-419: A source of confusion, as it can mean two things. Most commentators after Turing have used "state" to mean the name/designator of the current instruction to be performed—i.e. the contents of the state register. But Turing (1936) made a strong distinction between a record of what he called the machine's "m-configuration", and the machine's (or person's) "state of progress" through the computation—the current state of

1420-693: A subject in its own right. Graphs are one of the prime objects of study in discrete mathematics. They are among the most ubiquitous models of both natural and human-made structures. They can model many types of relations and process dynamics in physical, biological and social systems. In computer science, they can represent networks of communication, data organization, computational devices, the flow of computation, etc. In mathematics, they are useful in geometry and certain parts of topology , e.g. knot theory . Algebraic graph theory has close links with group theory and topological graph theory has close links to topology . There are also continuous graphs ; however, for

1491-453: A third element of the set of directions { L , R } {\displaystyle \{L,R\}} . The 7-tuple for the 3-state busy beaver looks like this (see more about this busy beaver at Turing machine examples ): Initially all tape cells are marked with 0 {\displaystyle 0} . In the words of van Emde Boas (1990), p. 6: "The set-theoretical object [his formal seven-tuple description similar to

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1562-491: A unified framework for counting permutations , combinations and partitions . Analytic combinatorics concerns the enumeration (i.e., determining the number) of combinatorial structures using tools from complex analysis and probability theory . In contrast with enumerative combinatorics which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae . Topological combinatorics concerns

1633-497: A universal machine). Another mathematical formalism, lambda calculus , with a similar "universal" nature was introduced by Alonzo Church . Church's work intertwined with Turing's to form the basis for the Church–Turing thesis . This thesis states that Turing machines, lambda calculus, and other similar formalisms of computation do indeed capture the informal notion of effective methods in logic and mathematics and thus provide

1704-407: Is tiling of the plane . In algebraic geometry , the concept of a curve can be extended to discrete geometries by taking the spectra of polynomial rings over finite fields to be models of the affine spaces over that field, and letting subvarieties or spectra of other rings provide the curves that lie in that space. Although the space in which the curves appear has a finite number of points,

1775-484: Is a procedure for solving a problem by any intuitively 'effective' means from a specific class. An effective method is sometimes also called a mechanical method or procedure. The definition of an effective method involves more than the method itself. In order for a method to be called effective, it must be considered with respect to a class of problems. Because of this, one method may be effective with respect to one class of problems and not be effective with respect to

1846-467: Is a theorem. For classical logic, it can be easily verified with a truth table . The study of mathematical proof is particularly important in logic, and has accumulated to automated theorem proving and formal verification of software. Logical formulas are discrete structures, as are proofs , which form finite trees or, more generally, directed acyclic graph structures (with each inference step combining one or more premise branches to give

1917-457: Is a unification of the theory of difference equations with that of differential equations , which has applications to fields requiring simultaneous modelling of discrete and continuous data. Another way of modeling such a situation is the notion of hybrid dynamical systems . Discrete geometry and combinatorial geometry are about combinatorial properties of discrete collections of geometrical objects. A long-standing topic in discrete geometry

1988-418: Is awarded for outstanding papers in discrete mathematics. Theoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on graph theory and mathematical logic . Included within theoretical computer science is the study of algorithms and data structures. Computability studies what can be computed in principle, and has close ties to logic, while complexity studies

2059-418: Is called a recursively enumerable language . The Turing machine can equivalently be defined as a model that recognises valid input strings, rather than enumerating output strings. Given a Turing machine M and an arbitrary string s , it is generally not possible to decide whether M will eventually produce s . This is due to the fact that the halting problem is unsolvable, which has major implications for

2130-469: Is equivalent to a single-stack pushdown automaton (PDA) that has been made more flexible and concise by relaxing the last-in-first-out (LIFO) requirement of its stack. In addition, a Turing machine is also equivalent to a two-stack PDA with standard LIFO semantics, by using one stack to model the tape left of the head and the other stack for the tape to the right. At the other extreme, some very simple models turn out to be Turing-equivalent , i.e. to have

2201-495: Is not true for the "copy" machine that can be provided with variable input "parameters". The diagram "progress of the computation" shows the three-state busy beaver's "state" (instruction) progress through its computation from start to finish. On the far right is the Turing "complete configuration" (Kleene "situation", Hopcroft–Ullman "instantaneous description") at each step. If the machine were to be stopped and cleared to blank both

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2272-474: Is supposed to not to appear elsewhere) and then by the note of instructions. This expression is called the "state formula". Earlier in his paper Turing carried this even further: he gives an example where he placed a symbol of the current "m-configuration"—the instruction's label—beneath the scanned square, together with all the symbols on the tape ( The Undecidable , p. 121); this he calls "the complete configuration " ( The Undecidable , p. 118). To print

2343-436: Is the P = NP problem , which involves the relationship between the complexity classes P and NP . The Clay Mathematics Institute has offered a $ 1 million USD prize for the first correct proof, along with prizes for six other mathematical problems . Effective method In logic , mathematics and computer science , especially metalogic and computability theory , an effective method or effective procedure

2414-449: Is the ability for a computational model or a system of instructions to simulate a Turing machine. A programming language that is Turing complete is theoretically capable of expressing all tasks accomplishable by computers; nearly all programming languages are Turing complete if the limitations of finite memory are ignored. A Turing machine is an idealised model of a central processing unit (CPU) that controls all data manipulation done by

2485-481: Is the discrete analogue of continuous modelling . In discrete modelling, discrete formulae are fit to data . A common method in this form of modelling is to use recurrence relation . Discretization concerns the process of transferring continuous models and equations into discrete counterparts, often for the purposes of making calculations easier by using approximations. Numerical analysis provides an important example. The history of discrete mathematics has involved

2556-499: Is the unlimited amount of tape and runtime that gives it an unbounded amount of storage space . Following Hopcroft & Ullman (1979 , p. 148), a (one-tape) Turing machine can be formally defined as a 7- tuple M = ⟨ Q , Γ , b , Σ , δ , q 0 , F ⟩ {\displaystyle M=\langle Q,\Gamma ,b,\Sigma ,\delta ,q_{0},F\rangle } where A variant allows "no shift", say N, as

2627-450: The NFA to DFA conversion algorithm). For practical and didactic intentions, the equivalent register machine can be used as a usual assembly programming language . A relevant question is whether or not the computation model represented by concrete programming languages is Turing equivalent. While the computation of a real computer is based on finite states and thus not capable to simulate

2698-403: The calculus of finite differences , a function defined on an interval of the integers is usually called a sequence . A sequence could be a finite sequence from a data source or an infinite sequence from a discrete dynamical system . Such a discrete function could be defined explicitly by a list (if its domain is finite), or by a formula for its general term, or it could be given implicitly by

2769-420: The computer graphics incorporated into modern video games and computer-aided design tools. Several fields of discrete mathematics, particularly theoretical computer science, graph theory, and combinatorics , are important in addressing the challenging bioinformatics problems associated with understanding the tree of life . Currently, one of the most famous open problems in theoretical computer science

2840-735: The first programmable digital electronic computer being developed at England's Bletchley Park with the guidance of Alan Turing and his seminal work, On Computable Numbers. The Cold War meant that cryptography remained important, with fundamental advances such as public-key cryptography being developed in the following decades. The telecommunications industry has also motivated advances in discrete mathematics, particularly in graph theory and information theory . Formal verification of statements in logic has been necessary for software development of safety-critical systems , and advances in automated theorem proving have been driven by this need. Computational geometry has been an important part of

2911-425: The right of the scanned square. But Kleene refers to "q 4 " itself as "the machine state" (Kleene, p. 374–375). Hopcroft and Ullman call this composite the "instantaneous description" and follow the Turing convention of putting the "current state" (instruction-label, m-configuration) to the left of the scanned symbol (p. 149), that is, the instantaneous description is the composite of non-blank symbols to

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2982-601: The "complete configuration" on one line, he places the state-label/m-configuration to the left of the scanned symbol. A variant of this is seen in Kleene (1952) where Kleene shows how to write the Gödel number of a machine's "situation": he places the "m-configuration" symbol q 4 over the scanned square in roughly the center of the 6 non-blank squares on the tape (see the Turing-tape figure in this article) and puts it to

3053-742: The "state register" and entire tape, these "configurations" could be used to rekindle a computation anywhere in its progress (cf. Turing (1936) The Undecidable , pp. 139–140). Many machines that might be thought to have more computational capability than a simple universal Turing machine can be shown to have no more power (Hopcroft and Ullman p. 159, cf. Minsky (1967)). They might compute faster, perhaps, or use less memory, or their instruction set might be smaller, but they cannot compute more powerfully (i.e. more mathematical functions). (The Church–Turing thesis hypothesises this to be true for any kind of machine: that anything that can be "computed" can be computed by some Turing machine.) A Turing machine

3124-456: The (infinite) set of all prime numbers . Partially ordered sets and sets with other relations have applications in several areas. In discrete mathematics, countable sets (including finite sets ) are the main focus. The beginning of set theory as a branch of mathematics is usually marked by Georg Cantor 's work distinguishing between different kinds of infinite set , motivated by the study of trigonometric series, and further development of

3195-478: The above nine 5-tuples. For technical reasons, the three non-printing or "N" instructions (4, 5, 6) can usually be dispensed with. For examples see Turing machine examples . Less frequently the use of 4-tuples are encountered: these represent a further atomization of the Turing instructions (cf. Post (1947), Boolos & Jeffrey (1974, 1999), Davis-Sigal-Weyuker (1994)); also see more at Post–Turing machine . The word "state" used in context of Turing machines can be

3266-587: The above] provides only partial information on how the machine will behave and what its computations will look like." For instance, Definitions in literature sometimes differ slightly, to make arguments or proofs easier or clearer, but this is always done in such a way that the resulting machine has the same computational power. For example, the set could be changed from { L , R } {\displaystyle \{L,R\}} to { L , R , N } {\displaystyle \{L,R,N\}} , where N ("None" or "No-operation") would allow

3337-506: The branch of mathematics dealing with countable sets (finite sets or sets with the same cardinality as the natural numbers). However, there is no exact definition of the term "discrete mathematics". The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business. Research in discrete mathematics increased in

3408-718: The compiled programs executable on a real computer cannot. Discrete mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables , having a bijection with the set of natural numbers ) rather than "continuous" (analogously to continuous functions ). Objects studied in discrete mathematics include integers , graphs , and statements in logic . By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers , calculus or Euclidean geometry . Discrete objects can often be enumerated by integers ; more formally, discrete mathematics has been characterized as

3479-645: The curves are not so much sets of points as analogues of curves in continuous settings. For example, every point of the form V ( x − c ) ⊂ Spec ⁡ K [ x ] = A 1 {\displaystyle V(x-c)\subset \operatorname {Spec} K[x]=\mathbb {A} ^{1}} for K {\displaystyle K} a field can be studied either as Spec ⁡ K [ x ] / ( x − c ) ≅ Spec ⁡ K {\displaystyle \operatorname {Spec} K[x]/(x-c)\cong \operatorname {Spec} K} ,

3550-412: The form of an infinite tape marked out into squares, on each of which a symbol could be printed. At any moment there is one symbol in the machine; it is called the scanned symbol. The machine can alter the scanned symbol, and its behavior is in part determined by that symbol, but the symbols on the tape elsewhere do not affect the behavior of the machine. However, the tape can be moved back and forth through

3621-566: The latter half of the twentieth century partly due to the development of digital computers which operate in "discrete" steps and store data in "discrete" bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science , such as computer algorithms , programming languages , cryptography , automated theorem proving , and software development . Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems. Although

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3692-561: The left of the scanned symbol or to the right of the scanned symbol. Turing's biographer Andrew Hodges (1983: 107) has noted and discussed this confusion. To the right: the above table as expressed as a "state transition" diagram. Usually large tables are better left as tables (Booth, p. 74). They are more readily simulated by computer in tabular form (Booth, p. 74). However, certain concepts—e.g. machines with "reset" states and machines with repeating patterns (cf. Hill and Peterson p. 244ff)—can be more readily seen when viewed as

3763-420: The left, state of the machine, the current symbol scanned by the head, and the non-blank symbols to the right. Example: total state of 3-state 2-symbol busy beaver after 3 "moves" (taken from example "run" in the figure below): This means: after three moves the tape has ... 000110000 ... on it, the head is scanning the right-most 1, and the state is A . Blanks (in this case represented by "0"s) can be part of

3834-477: The machine to stay on the same tape cell instead of moving left or right. This would not increase the machine's computational power. The most common convention represents each "Turing instruction" in a "Turing table" by one of nine 5-tuples, per the convention of Turing/Davis (Turing (1936) in The Undecidable , p. 126–127 and Davis (2000) p. 152): Other authors (Minsky (1967) p. 119, Hopcroft and Ullman (1979) p. 158, Stone (1972) p. 9) adopt

3905-424: The machine, this being one of the elementary operations of the machine. Any symbol on the tape may therefore eventually have an innings. The Turing machine mathematically models a machine that mechanically operates on a tape. On this tape are symbols, which the machine can read and write, one at a time, using a tape head. Operation is fully determined by a finite set of elementary instructions such as "in state 42, if

3976-409: The main objects of study in discrete mathematics are discrete objects, analytic methods from "continuous" mathematics are often employed as well. In university curricula, discrete mathematics appeared in the 1980s, initially as a computer science support course; its contents were somewhat haphazard at the time. The curriculum has thereafter developed in conjunction with efforts by ACM and MAA into

4047-1050: The most part, research in graph theory falls within the domain of discrete mathematics. Number theory is concerned with the properties of numbers in general, particularly integers . It has applications to cryptography and cryptanalysis , particularly with regard to modular arithmetic , diophantine equations , linear and quadratic congruences, prime numbers and primality testing . Other discrete aspects of number theory include geometry of numbers . In analytic number theory , techniques from continuous mathematics are also used. Topics that go beyond discrete objects include transcendental numbers , diophantine approximation , p-adic analysis and function fields . Algebraic structures occur as both discrete examples and continuous examples. Discrete algebras include: Boolean algebra used in logic gates and programming; relational algebra used in databases ; discrete and finite versions of groups , rings and fields are important in algebraic coding theory ; discrete semigroups and monoids appear in

4118-489: The same computational power as the Turing machine model. Common equivalent models are the multi-tape Turing machine , multi-track Turing machine , machines with input and output, and the non-deterministic Turing machine (NDTM) as opposed to the deterministic Turing machine (DTM) for which the action table has at most one entry for each combination of symbol and state. Read-only, right-moving Turing machines are equivalent to DFAs (as well as NFAs by conversion using

4189-403: The same instruction. In some models, if there is no entry in the table for the current combination of symbol and state, then the machine will halt; other models require all entries to be filled. Every part of the machine (i.e. its state, symbol-collections, and used tape at any given time) and its actions (such as printing, erasing and tape motion) is finite , discrete and distinguishable ; it

4260-605: The study of various continuous computational topics. Information theory involves the quantification of information . Closely related is coding theory which is used to design efficient and reliable data transmission and storage methods. Information theory also includes continuous topics such as: analog signals , analog coding , analog encryption . Logic is the study of the principles of valid reasoning and inference , as well as of consistency , soundness , and completeness . For example, in most systems of logic (but not in intuitionistic logic ) Peirce's law ((( P → Q )→ P )→ P )

4331-407: The symbol and the machine's own present state, the machine writes a symbol into the same cell, and moves the head one step to the left or the right, or halts the computation. The choice of which replacement symbol to write, which direction to move the head, and whether to halt is based on a finite table that specifies what to do for each combination of the current state and the symbol that is read. Like

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4402-522: The symbol seen is 0, write a 1; if the symbol seen is 1, change into state 17; in state 17, if the symbol seen is 0, write a 1 and change to state 6;" etc. In the original article (" On Computable Numbers, with an Application to the Entscheidungsproblem ", see also references below ), Turing imagines not a mechanism, but a person whom he calls the "computer", who executes these deterministic mechanical rules slavishly (or as Turing puts it, "in

4473-405: The theoretical limits of computing. The Turing machine is capable of processing an unrestricted grammar , which further implies that it is capable of robustly evaluating first-order logic in an infinite number of ways. This is famously demonstrated through lambda calculus . A Turing machine that is able to simulate any other Turing machine is called a universal Turing machine (UTM, or simply

4544-534: The theory of formal languages . There are many concepts and theories in continuous mathematics which have discrete versions, such as discrete calculus , discrete Fourier transforms , discrete geometry , discrete logarithms , discrete differential geometry , discrete exterior calculus , discrete Morse theory , discrete optimization , discrete probability theory , discrete probability distribution , difference equations , discrete dynamical systems , and discrete vector measures . In discrete calculus and

4615-411: The theory of infinite sets is outside the scope of discrete mathematics. Indeed, contemporary work in descriptive set theory makes extensive use of traditional continuous mathematics. Combinatorics studies the ways in which discrete structures can be combined or arranged. Enumerative combinatorics concentrates on counting the number of certain combinatorial objects - e.g. the twelvefold way provides

4686-536: The time, space, and other resources taken by computations. Automata theory and formal language theory are closely related to computability. Petri nets and process algebras are used to model computer systems, and methods from discrete mathematics are used in analyzing VLSI electronic circuits. Computational geometry applies algorithms to geometrical problems and representations of geometrical objects, while computer image analysis applies them to representations of images. Theoretical computer science also includes

4757-423: The total state as shown here: B 01; the tape has a single 1 on it, but the head is scanning the 0 ("blank") to its left and the state is B . "State" in the context of Turing machines should be clarified as to which is being described: the current instruction, or the list of symbols on the tape together with the current instruction, or the list of symbols on the tape together with the current instruction placed to

4828-428: The total system. What Turing called "the state formula" includes both the current instruction and all the symbols on the tape: Thus the state of progress of the computation at any stage is completely determined by the note of instructions and the symbols on the tape. That is, the state of the system may be described by a single expression (sequence of symbols) consisting of the symbols on the tape followed by Δ (which

4899-436: The use of techniques from topology and algebraic topology / combinatorial topology in combinatorics . Design theory is a study of combinatorial designs , which are collections of subsets with certain intersection properties. Partition theory studies various enumeration and asymptotic problems related to integer partitions , and is closely related to q-series , special functions and orthogonal polynomials . Originally

4970-415: The values of a function is an algorithm . Functions for which an effective method exists are sometimes called effectively calculable . Several independent efforts to give a formal characterization of effective calculability led to a variety of proposed definitions ( general recursive functions , Turing machines , λ-calculus ) that later were shown to be equivalent. The notion captured by these definitions

5041-595: Was to prove that the axioms of arithmetic are consistent . Gödel's second incompleteness theorem , proved in 1931, showed that this was not possible – at least not within arithmetic itself. Hilbert's tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. In 1970, Yuri Matiyasevich proved that this could not be done . The need to break German codes in World War II led to advances in cryptography and theoretical computer science , with

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