# Computational Models Lecture 8, Spring 2009

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 1 Computational Models Lecture 8, Spring 2009 Encoding of TMs Universal Turing Machines The Halting/Acceptance problem The Halting/Acceptance problems are undecidable Diagonalization Computable functions The busy beaver function is not computable (not in book) Reductions Reducing A to B by Mapping reductions Sipser s book, 4.1, 4.2, 5.1

2 Standard Encoding of Turing Machines (Slightly Modified) This formulation slightly simplifies last week s convention, without sacrifying generality. We assume that our TM, M, has one tape, Σ = {0,1} and Γ = {0,1,\$, }. For L {0,1}, this is not a restriction. The encoding M of a TM, M, will use a binary alphabet. Blocks of 0 s will be used as delimiters. A set Q with m states will be indicated by m in unary. By conventions states will be 1 through m. By convention, q 0 is indicated by state 1, q a = q 2 by 11, and q r = q 3 by 111 (& delimiters!). Finally, the transition function δ is encoded as a list of 5-tuples with correct size and no duplications in q, γ entries. Important (and Easy): An algorithm (TM) can check that a given string is legal encoding of (any other) TM. Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 2

3 Universal Turing Machine Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 3

4 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 4 Universal Turing Machines We now define the universal Turing machine, U. On input M,w, where M is a TM and a string w 1. Checks that M,w is a proper encoding of a TM, followed by a string from Σ. 2. Simulates M on input w (some details in next slide) 3. If M on input w enters its accept state, U accept, and if M on input w ever enters its reject state, U reject. Notice that as a consequence, if M on input w enters an infinite loop, so does U on input M,w.

5 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 5 Univeral Turing Machines: Some Simulation Details The simulated TM, M, has one tape, has Σ = {0, 1} and Γ = {0, 1, \$, }. To make life easier, the universal Turing machine, U, that will simulate M, will have several tapes (say five), and a larger work alphabet, Γ. On input M,w, where M is a TM and w {0, 1, } is a string Tape 2 of U is the "program tape". The contents of tape 2 will follow the contents of M s single tape, step by step. Tape 3 is the "simulation tape". Tape 4 is the "state tape". Tape 5 is the "scratch tape". U copies M to tape 2, and w to tape 3. It places the tape 3 head on the leftmost character of w.

6 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 6 Univeral Turing Machines: Some Simulation Details U copies both the the state q (a string of 1s) from tape 4, and the letter it sees on tape 3, a Γ, to tape 5 (with delimiters). U compares q,a to the entries in the transition function portion of tape 2. Once U finds a match, it updates the state tape (tape 4), writes a letter in the simulation tape (tape 3), and moves the head on tape 3 left/right accordingly. If M on input w enters its accept state q 2, U accepts M,w, and if M on input w ever enters its reject state q 3, U rejects M,w. Notice that as a consequence, if M on input w enters an infinite loop, so does U on input M,w.

7 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 7 Universal Turing Machines (4) The universal machine U uses some extra dotting and crossover marks (larger work alphabet) to facilitate comparisons, copying, and erasing. The universal machine U obviously has a fixed number of states (100 should do). Despite this, it can simulate machines M with many more states. Universal machines inspired the development of stored-program computers in the 40s and 50s. Most of you have seen a universal machine, and have even used one!

8 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 8 Universal Turing Machines (5) For example, Dr. Scheme (interpreter) is a universal Scheme machine. It accepts a two part input: Above the line the program (corresponding to M ), and below the line the input to run it on (corresponding to w).

9 In case You Forgot (or Repressed) Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 9

10 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 10 The Halting/Acceptance Problem One of the most philosophically important theorems of the theory of computation. Acceptance Problem: Does a Turing machine accept an input string? A TM = { M,w M is a TM that accepts w} Halting Problem: Does a Turing machine halt on an input string? H TM = { M,w M is a TM and M halts on input w} Theorem: Both A TM and H TM are undecidable.

11 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 11 The Acceptance Problem A TM = { M,w M is a TM that accepts w} Before approaching the proof of undecidability, we first notice Theorem: A TM is recursively enumerable (namely in RE). Proof: The universal machine accepts A TM.

12 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 12 The Halting Problem H TM = { M,w M is a TM that halts on the input string w} Before approaching the proof of undecidability, we first note Theorem: H TM is recursively enumerable (namely in RE). Proof: A slight modification of the universal machine, where the reject state is replaced by the accept state. The modified machine accepts H TM.

13 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 13 Acceptance, Again We are now able to prove the undecidability of A TM = { M,w M is a TM that accepts w}. Proof: By contradiction. Suppose a TM, H, is a decider for A TM. On input M,w, where M is a TM and w is a string, H halts and accepts if and only if M accepts w. Furthermore, H halts and rejects if M fails to accept w.

14 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 14 Acceptance (2) On input M,w, where M is a TM and w is a string, H halts and accepts if and only if M accepts w. Furthermore, H halts and rejects if M fails to accept w. H( M,w ) = { accept if M accepts w reject if M does not accept w

15 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 15 Acceptance (3) Now we construct a new TM, D, with H as a subroutine. D does the following Calls H to determine what TM, M, does when the input to M is its own description, M. When D determines this, it does the opposite. So D rejects if M accepts M, and accepts if M does not accept M.

16 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 16 Acceptance (4) More precisely, D does the following: Run H on input M, M. Output the opposite of what H outputs: If H accepts, reject, and If H rejects, accept.

17 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 17 Self Reference (4) Don t be confused by the notion of running a machine on its own description! Actually, you should get used to it. Notion of self-reference comes up again and again in diverse areas. Read Gödel, Escher, Bach, an Eternal Golden Braid, by Douglas Hofstadter. This notion of self-reference is the basic idea behind Gödel s revolutionary result. Compilers do this all the time....

18 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 18 The Punch Line So far we have, D( M ) = { reject if M accepts M accept if M does not accept M What happens if we run D on its own description? D( D ) = { reject if D accepts D accept if D does not accept D Oh, oh... Or, more accurately, a contradiction (to what?)

19 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 19 Once Again Assume that TM H decides A TM. Then use H to build a TM, D, that when given M, accepts exactly when M does not accept. Run D on its own description. D does: H accepts M,w when M accepts w. D rejects M exactly when M accepts M. D rejects D exactly when D accepts D. Last step leads to contradiction. Therefore neither TM D nor H can exist. So A TM is undecidable!

20 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 20 A Non-enumerable Language We already saw a non-decidable language: A TM. Can we do better (i.e., worse)? Mais, oui! We now display a language that isn t even recursively enumerable....

21 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 21 A Non-enumerable Language Earlier we saw Theorem: If L and L are both enumerable, then L is decidable. Corollary: If L is not decidable, then either L or L is not enumerable. Definition: A language is co-enumerable if it is the complement of an enumerable language. Reformulating theorem Theorem: A language is decidable if and only if it is both enumerable and co-enumerable.

22 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 22 A TM is not Enumerable Theorem: If L and L are both enumerable, then L is decidable. We proved that A TM is undecidable. On the other hand, we saw that the universal TM, U, accepts A TM. Therefore A TM is enumerable. If A TM were also enumerable, then by theorem A TM was decidable. Therefore A TM is not enumerable.

23 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 23 Languages??? co-enumerable A TM decidable A DFA enumerable A TM Question: Are there any languages in the area marked???? Answer: Yes, heaps (why?)

24 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 24 The Acceptance Problem (again) We saw A TM = { M,w M is a TM that accepts w} Our proof that A TM is undecidable was actually a diagonalization proof.

25 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 25 The Real Numbers Assume there is a correspondence between N and R. Write it down: n f(n) We now show that there is a number x not in this list.

26 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 26 Diagonalization Pick 0 x 1, so its significant digits follow decimal point. Will ensure x f(n) for all n. n f(n)

27 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 27 Diagonalization n f(n) First fractional digit of f(1) is 1, so pick first fractional digit of x to be something else (say, 2). Second fractional digit of f(2) is 5, so pick second fractional digit of x to be something else (say, 6). and so on... x =

28 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 28 Diagonalization A similar proof shows there are languages that are not enumerable. the set of Turing machines is countable, but the set of languages is uncountable! Ergo, there exist languages that are not enumerable (why?) indeed, most languages are not enumerable (explain)

29 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 29 Countably Many Turing Machines Claim: The set of strings, Σ, is countable. Proof: List strings of length 0, then length 1, then 2, and so on. This exhausts all of Σ. The union of countably many finite sets is countable.

30 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 30 Countably Many Turing Machines (2) Claim: The set of all Turing machines is countable. Proof: Each TM M has an encoding as a string M. Therefore there is a one-to-one mapping from the set of all TMs into (but not onto) Σ. Since Σ is countable, so is the set of all TMs.

31 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 31 The Set of All Languages is Uncountable Let B be the set of of infinite binary sequences. Claim B is uncountable. Proof Diagonalization argument, essentially identical to the proof that R is uncountable. (additional helpful clue: think of binary sequence as binary expansion!)

32 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 32 The Set of Languages is Uncountable (2) Let L be the set of all languages over alphabet Σ. Recall B is the set of of infinite binary sequences. We give a correspondence χ : L B called the language s characteristic sequence. Let Σ = {s 1,s 2,s 3,...} (in lexicographic order). Each language L L is associated with a unique sequence χ(l) B: the i-th bit of χ(l) is 1 if and only if s i L.

33 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 33 The Set of Languages is Uncountable (3) Each language L L has a unique sequence χ(l) B: the i-th bit of χ(l) is 1 if and only if s i L. Example: Σ {ε, 0, 1, 00, 01, 10, 11, } A { 0, 00, 01, } χ(a) {0, 1, 0 1, 1, 0, 0, 1...} The map χ : L B is one-to-one and onto (why?), and is hence a correspondence. It follows that L is uncountable.

34 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 34 TMs vs. Languages We saw that the set of all Turing machines is countable. We saw that the set L of all languages over alphabet Σ is uncountable. Therefore there are languages that are not accepted by any TM. This is an existential proof it does not explicitly show any such language.

35 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 35 Reflections on Diagonalization This proof that the acceptance problem is undecidable is actually diagonalization in transparent disguise. To unveil this, let s start by making a table. M 1 M 2 M 3 M 4... M 1 accept accept M 2 accept accept accept accept M 3 M 4 accept accept. Entry (i,j) is accept if M i accepts M j, and blank if M i rejects or loops on M j.

36 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 36 Diagonalization (2) M 1 M 2 M 3 M 4... M 1 accept accept M 2 accept accept accept accept M 3 M 4 accept accept.. Run H on on corresponding inputs. In new table, entry (i,j) states whether H accepts M i, M j. M 1 M 2 M 3 M 4... M 1 accept reject accept reject M 2 accept accept accept accept M 3 reject reject reject reject M 4 accept accept reject reject....

37 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 37 Diagonalization (3) Now we add D to the table. By assumption, H is a TM, and therefore so is D. D occurs on the list M 1,M 2,... of all TMs. D computes the opposite of the diagonal entries. At diagonal entry, D computes its own opposite! M 1 M 2 M 3... D M 1 accept reject accept M 2 accept accept accept M 3 reject reject reject M 4. accept accept. reject... D reject reject accept???.....

38 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 38 Halting vs Acceptance Problem We have already established that A TM is undecidable. Halting is a closely related problem. H TM = { M,w M is a TM and M halts on input w} Clarification: How does H TM differ from A TM?

39 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 39 Undecidable Problems H TM = { M,w M is a TM and M halts on input w} Theorem: H TM is undecidable. Proof idea: Again, proof by diagonalization. Will do this on the blackboard. Next week will prove this differently, by a reduction from A TM.

40 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 40 Computable Functions A TM computes a total function if the TM f : Σ Σ when starting with an input w, always halts with only f(w) written on tape. The definition can be extended to functions of more than one variable, where some special separator symbol indicates end of one variable and beginning of next. Sometimes we have a separate output tape where f(w) is written. This is more convenient, but otherwise makes no difference.

41 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 41 Computable Functions A TM computes a partial function if the TM f : Σ (Σ ) when starting with an input w, if f(w) is defined, TM halts with only f(w) on tape, if f(w) is undefined, TM does not halt. Computable functions are also called (total or partial) recursive functions.

42 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 42 Computable Functions Claim: All the usual arithmetic functions on integers are computable. These include addition, subtraction, multiplication, division (quotient and remainder), exponentiation, roots (to a specified precision), modular exponentiation, greatest common divisor. Even non-arithmetic functions, like logarithms and trigonometric functions, can be computed (to a specified precision), using Taylor expansion or other numeric mathematic techniques. Exercise: Design a TM that on input m,n, halts with m + n on tape.

43 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 43 Computable Functions A useful class of functions modifies TM descriptions. For example: On input w: if w = M for some TM, construct M, where L(M ) = L(M), but M never tries to move off LHS of tape. otherwise write ε and halt.

44 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 44 A Non-Computable Function: The Busy Beaver (taken from

45 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 45 A Non-Computable Function: The Busy Beaver We look at all one tape TMs with Σ = {0, 1} and Γ = {0, 1, \$, }. Consider the set S n of all such TMs that have n states and halt on the empty input, ε. The set S n is clearly finite. By definition, if M S n then M on ε runs for finitely many steps. Define BB(n) =maximum number of steps taken by machines in S n on the input ε. By the discussion above, BB(n) is a well defined, total function. Theorem: The busy beaver function is not computable. Proof: On board.

46 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 46 Reducibility Example: Finding your way around a new city reduces to... obtaining a city map.

47 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 47 Reducibility, In Our Context Always involves two problems, A and B. Desired Property: If A reduces to B, then any solution of B can be used to find a solution of A. Remark: This property says nothing about solving A by itself or B by itself.

48 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 48 Examples Reductions: Traveling from Boshton to Paris... reduces to buying plane ticket... which reduces to earning the money for that ticket... which reduces to finding a job (or getting the \$s from mom and dad... )

49 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 49 Examples Reductions: Measuring area of rectangle... reduces to measuring lengths of sides. Also: Solving a system of linear equations... reduces to inverting a matrix.

50 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 50 Reducibility If A is reducible to B, then A cannot be harder than B if B is decidable, so is A. if A is undecidable and reducible to B, then B is undecidable.

51 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 51 Additional Undecidable Problems We have already established that A TM is undecidable. Here is a related problem the original halting problem (of Shoshana and Uri :-). H TM = { M,w M is a TM and M halts on input w} Clarification: How does H TM differ from A TM?

52 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 52 Undecidable Problems H TM = { M,w M is a TM and M halts on input w} Theorem: H TM is undecidable. Proof idea: By contradiction. Assume H TM is decidable. Let R be a TM that decides H TM. Use R to construct S, a TM that decides A TM. So A TM is reduced to H TM. Since A TM is undecidable, so is H TM.

53 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 53 Undecidable Problems Theorem: H TM is undecidable. Proof: Assume, by way of contradiction, that TM R decides H TM. Define a new TM, S, as follows: On input M,w, run R on M,w. If R rejects, reject. If R accepts (meaning M halts on w), simulate M on w until it halts (namely run U on M,w ). If M accepted, accept; otherwise reject. TM S decides A TM, a contradiction

54 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 54 Undecidable Problems H TM = { M,w M is a TM and M halts on input w} Theorem: H TM is undecidable. What we actually did was a reduction from A TM to H TM. This will be formalized later.

55 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 55 Undecidable Problems (2) Does a TM accept any string at all? EMPTY TM = { M M is a TM and L(M) = } Theorem: EMPTY TM is undecidable. Proof structure: By contradiction. Assume EMPTY TM is decidable. Let R be a TM that decides EMPTY TM. Use R to construct S, a TM that decides A TM.

56 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 56 Undecidable Problems (2) EMPTY TM = { M M is a TM and L(M) = } First attempt: When S receives input M,w, it calls R with input M. If R accepts, then reject, because M does not accept any string, let alone w. But what if R rejects? Second attempt: Let s modify M.

57 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 57 Undecidable Problems (2) EMPTY TM = { M M is a TM and L(M) = } Define M 1 : on input x, 1. if x w, reject. 2. if x = w, run M on w and accept if M does. M 1 either accepts just w, or accepts nothing.

58 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 58 Undecidable Problems (2) Machine M 1 : on input x, 1. if x w, reject. 2. if x = w, run M on w and accept if M does. Question: Can a TM construct M 1 from M? Answer: Easily, because we need only hardwire w, and add a few extra states to perform the x = w? test.

59 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 59 Undecidable Problems (2) EMPTY TM = { M M is a TM and L(M) = }. Theorem: EMPTY TM is undecidable. Define S as follows: On input M,w, where M is a TM and w a string, Construct M 1 from M and w. Run R on input M 1, if R accepts, reject; if R rejects, accept. TM S decides A TM, a contradiction

60 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 60 Undecidable Problems (3) Does a TM accept a regular language? REG TM = { M M is a TM and L(M) is regular} Theorem: REG TM is undecidable. Skeleton of Proof: By contradiction. Assume REG TM is decidable. Let R be a TM that decides REG TM. Use R to construct S, a TM that decides A TM. But how?

61 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 61 Undecidable Problems (3) REG TM = { M M is a TM and L(M) is regular} Intuition: Modify M so that the resulting TM accepts a regular language if and only if M accepts w. Design M 2 so that if M does not accept w, then M 2 accepts {0 n 1 n n 0} (non-regular) if M accepts w, then M 2 accepts Σ (regular).

62 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 62 Undecidable Problems (3) Given M and w, construct M 2 : On input x, 1. If x has the form 0 n 1 n, accept it. 2. Otherwise, run M on input w and accept x if M accepts w. Claim: If M does not accept w, then M 2 accepts {0 n 1 n n 0}. If M accepts w, then M 2 accepts Σ. The function: On input M,w, output M 2, is computable.

63 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 63 Undecidable Problems (3) REG TM = { M M is a TM and L(M) is regular} Theorem: REG TM is undecidable. Define S: On input M,w, 1. Construct M 2 from M and w. 2. Run R on input M If R accepts, accept; if R rejects, reject. TM S decides A TM, a contradiction

64 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 64 Undecidable Problems (4) Are two TMs equivalent? EQ TM = { M 1,M 2 M 1,M 2 are TMs and L(M 1 ) = L(M 2 )} Theorem: EQ TM is undecidable. We are getting tired of reducing A TM to everything. Let s try instead a reduction from EMPTY TM to EQ TM.

65 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 65 Undecidable Problems (4) EQ TM = { M 1,M 2 M 1,M 2 are TMs and Theorem: EQ TM is undecidable. Idea: L(M 1 ) = L(M 2 )} EMPTY TM is the problem of testing whether a TM language is empty. EQ TM is the problem of testing whether two TM languages are the same. If one of these two TM languages happens to be empty, then we are back to EMPTY TM. So EMPTY TM is a special case of EQ TM. The rest is easy.

66 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 66 Undecidable Problems (4) EQ TM = { M 1,M 2 M 1,M 2 are TMs and L(M 1 ) = L(M 2 )} Theorem: EQ TM is undecidable. Let M NO be this TM: On input x, reject. Let R decide EQ TM. Let S be: On input M : 1. Run R on input M,M NO. 2. If R accepts, accept; if R rejects, reject. If R decides EQ TM, then S decides EMPTY TM.

67 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 67 Bucket of Undecidable Problems Same techniques prove undecidability of Does a TM accept a decidable language? Does a TM accept a context-free language? Does a TM accept a finite language? Does a TM halt on all inputs? Is there an input string that causes a TM to traverse all its states?

68 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 68 Reducibility So far, we have seen many examples of reductions from one language to another, but the notion was neither defined nor treated formally. Reductions play an important role in decidability theory (here and now) complexity theory (to come) Time to get formal.

69 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 69 Mapping Reductions Definition: Let A and B be two languages. We say that there is a mapping reduction from A to B, and denote A m B if there is a computable function such that, for every w, f : Σ Σ w A f(w) B. The function f is called the reduction from A to B.

70 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 70 Mapping Reductions Missing Figure Here A mapping reduction converts questions about membership in A to membership in B

71 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 71 Mapping Reductions Theorem: If A m B and B is decidable, then A is decidable. Proof: Let M be the decider for B, and f the reduction from A to B. Define N: On input w 1. compute f(w) 2. run M on input f(w) and output whatever M outputs.

72 Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 72 Mapping Reductions Corollary: If A m B and A is undecidable, then B is undecidable. In fact, this has been our principal tool for proving undecidability of languages other than A TM.

### FORMAL LANGUAGES, AUTOMATA AND COMPUTATION

FORMAL LANGUAGES, AUTOMATA AND COMPUTATION REDUCIBILITY ( LECTURE 16) SLIDES FOR 15-453 SPRING 2011 1 / 20 THE LANDSCAPE OF THE CHOMSKY HIERARCHY ( LECTURE 16) SLIDES FOR 15-453 SPRING 2011 2 / 20 REDUCIBILITY

### Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras

Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture No. # 31 Recursive Sets, Recursively Innumerable Sets, Encoding

### CSE 135: Introduction to Theory of Computation Decidability and Recognizability

CSE 135: Introduction to Theory of Computation Decidability and Recognizability Sungjin Im University of California, Merced 04-28, 30-2014 High-Level Descriptions of Computation Instead of giving a Turing

### CS 3719 (Theory of Computation and Algorithms) Lecture 4

CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a

### A TM. Recall the A TM language: A TM = { M, w M is a TM and M accepts w.} is undecidable. Theorem: The language. p.2/?

p.1/? Reducibility We say that a problem Q reduces to problem P if we can use P to solve Q. In the context of decidability we have the following templates : If A reduces to B and B is decidable, then so

### CS154. Turing Machines. Turing Machine. Turing Machines versus DFAs FINITE STATE CONTROL AI N P U T INFINITE TAPE. read write move.

CS54 Turing Machines Turing Machine q 0 AI N P U T IN TAPE read write move read write move Language = {0} q This Turing machine recognizes the language {0} Turing Machines versus DFAs TM can both write

### The Halting Problem is Undecidable

185 Corollary G = { M, w w L(M) } is not Turing-recognizable. Proof. = ERR, where ERR is the easy to decide language: ERR = { x { 0, 1 }* x does not have a prefix that is a valid code for a Turing machine

### (IALC, Chapters 8 and 9) Introduction to Turing s life, Turing machines, universal machines, unsolvable problems.

3130CIT: Theory of Computation Turing machines and undecidability (IALC, Chapters 8 and 9) Introduction to Turing s life, Turing machines, universal machines, unsolvable problems. An undecidable problem

### Computability Theory

CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2014 Computability Theory This section is partly inspired by the material in A Course in Mathematical Logic by Bell and Machover, Chap 6, sections 1-10.

### Turing Machines: An Introduction

CIT 596 Theory of Computation 1 We have seen several abstract models of computing devices: Deterministic Finite Automata, Nondeterministic Finite Automata, Nondeterministic Finite Automata with ɛ-transitions,

### Lecture 2: Universality

CS 710: Complexity Theory 1/21/2010 Lecture 2: Universality Instructor: Dieter van Melkebeek Scribe: Tyson Williams In this lecture, we introduce the notion of a universal machine, develop efficient universal

### Notes on Complexity Theory Last updated: August, 2011. Lecture 1

Notes on Complexity Theory Last updated: August, 2011 Jonathan Katz Lecture 1 1 Turing Machines I assume that most students have encountered Turing machines before. (Students who have not may want to look

### Chapter 7 Uncomputability

Chapter 7 Uncomputability 190 7.1 Introduction Undecidability of concrete problems. First undecidable problem obtained by diagonalisation. Other undecidable problems obtained by means of the reduction

### We give a basic overview of the mathematical background required for this course.

1 Background We give a basic overview of the mathematical background required for this course. 1.1 Set Theory We introduce some concepts from naive set theory (as opposed to axiomatic set theory). The

### Sets and Subsets. Countable and Uncountable

Sets and Subsets Countable and Uncountable Reading Appendix A Section A.6.8 Pages 788-792 BIG IDEAS Themes 1. There exist functions that cannot be computed in Java or any other computer language. 2. There

### Computing basics. Ruurd Kuiper

Computing basics Ruurd Kuiper October 29, 2009 Overview (cf Schaum Chapter 1) Basic computing science is about using computers to do things for us. These things amount to processing data. The way a computer

### 3515ICT Theory of Computation Turing Machines

Griffith University 3515ICT Theory of Computation Turing Machines (Based loosely on slides by Harald Søndergaard of The University of Melbourne) 9-0 Overview Turing machines: a general model of computation

### THE TURING DEGREES AND THEIR LACK OF LINEAR ORDER

THE TURING DEGREES AND THEIR LACK OF LINEAR ORDER JASPER DEANTONIO Abstract. This paper is a study of the Turing Degrees, which are levels of incomputability naturally arising from sets of natural numbers.

### CAs and Turing Machines. The Basis for Universal Computation

CAs and Turing Machines The Basis for Universal Computation What We Mean By Universal When we claim universal computation we mean that the CA is capable of calculating anything that could possibly be calculated*.

### 1 Definition of a Turing machine

Introduction to Algorithms Notes on Turing Machines CS 4820, Spring 2012 April 2-16, 2012 1 Definition of a Turing machine Turing machines are an abstract model of computation. They provide a precise,

### Outline. Database Theory. Perfect Query Optimization. Turing Machines. Question. Definition VU , SS Trakhtenbrot s Theorem

Database Theory Database Theory Outline Database Theory VU 181.140, SS 2016 4. Trakhtenbrot s Theorem Reinhard Pichler Institut für Informationssysteme Arbeitsbereich DBAI Technische Universität Wien 4.

### Automata and Computability. Solutions to Exercises

Automata and Computability Solutions to Exercises Fall 25 Alexis Maciel Department of Computer Science Clarkson University Copyright c 25 Alexis Maciel ii Contents Preface vii Introduction 2 Finite Automata

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### Finite Automata and Formal Languages

Finite Automata and Formal Languages TMV026/DIT321 LP4 2011 Lecture 14 May 19th 2011 Overview of today s lecture: Turing Machines Push-down Automata Overview of the Course Undecidable and Intractable Problems

### Turing Machines, Part I

Turing Machines, Part I Languages The \$64,000 Question What is a language? What is a class of languages? Computer Science Theory 2 1 Now our picture looks like Context Free Languages Deterministic Context

### Reading 13 : Finite State Automata and Regular Expressions

CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model

### Automata Theory CS F-15 Undecidiability

Automata Theory CS411-2015F-15 Undecidiability David Galles Department of Computer Science University of San Francisco 15-0: Universal TM Turing Machines are Hard Wired Addition machine only adds 0 n 1

### 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008

MIT OpenCourseWare http://ocw.mit.edu 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

### Automata and Languages

Automata and Languages Computational Models: An idealized mathematical model of a computer. A computational model may be accurate in some ways but not in others. We shall be defining increasingly powerful

### Theory of Computation Lecture Notes

Theory of Computation Lecture Notes Abhijat Vichare August 2005 Contents 1 Introduction 2 What is Computation? 3 The λ Calculus 3.1 Conversions: 3.2 The calculus in use 3.3 Few Important Theorems 3.4 Worked

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

### Introduction to Turing Machines

Automata Theory, Languages and Computation - Mírian Halfeld-Ferrari p. 1/2 Introduction to Turing Machines SITE : http://www.sir.blois.univ-tours.fr/ mirian/ Automata Theory, Languages and Computation

### 24 Uses of Turing Machines

Formal Language and Automata Theory: CS2004 24 Uses of Turing Machines 24 Introduction We have previously covered the application of Turing Machine as a recognizer and decider In this lecture we will discuss

### Theory of Computation Chapter 2: Turing Machines

Theory of Computation Chapter 2: Turing Machines Guan-Shieng Huang Feb. 24, 2003 Feb. 19, 2006 0-0 Turing Machine δ K 0111000a 01bb 1 Definition of TMs A Turing Machine is a quadruple M = (K, Σ, δ, s),

### Cardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set.

Section 2.5 Cardinality (another) Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted A = B, if and only if there is a bijection from A to B. If there is an injection

### Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### Practice Problems for Final Exam: Solutions CS 341: Foundations of Computer Science II Prof. Marvin K. Nakayama

Practice Problems for Final Exam: Solutions CS 341: Foundations of Computer Science II Prof. Marvin K. Nakayama 1. Short answers: (a) Define the following terms and concepts: i. Union, intersection, set

### Diagonalization. Ahto Buldas. Lecture 3 of Complexity Theory October 8, 2009. Slides based on S.Aurora, B.Barak. Complexity Theory: A Modern Approach.

Diagonalization Slides based on S.Aurora, B.Barak. Complexity Theory: A Modern Approach. Ahto Buldas Ahto.Buldas@ut.ee Background One basic goal in complexity theory is to separate interesting complexity

### Pushdown Automata. place the input head on the leftmost input symbol. while symbol read = b and pile contains discs advance head remove disc from pile

Pushdown Automata In the last section we found that restricting the computational power of computing devices produced solvable decision problems for the class of sets accepted by finite automata. But along

### CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

### 2 The Euclidean algorithm

2 The Euclidean algorithm Do you understand the number 5? 6? 7? At some point our level of comfort with individual numbers goes down as the numbers get large For some it may be at 43, for others, 4 In

### Mathematics for Computer Science

Mathematics for Computer Science Lecture 2: Functions and equinumerous sets Areces, Blackburn and Figueira TALARIS team INRIA Nancy Grand Est Contact: patrick.blackburn@loria.fr Course website: http://www.loria.fr/~blackbur/courses/math

### 4.6 The Primitive Recursive Functions

4.6. THE PRIMITIVE RECURSIVE FUNCTIONS 309 4.6 The Primitive Recursive Functions The class of primitive recursive functions is defined in terms of base functions and closure operations. Definition 4.6.1

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### Formal Languages and Automata Theory - Regular Expressions and Finite Automata -

Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March

### Turing Machine and the Conceptual Problems of Computational Theory

Research Inventy: International Journal of Engineering And Science Vol.4, Issue 4 (April 204), PP 53-60 Issn (e): 2278-472, Issn (p):239-6483, www.researchinventy.com Turing Machine and the Conceptual

### Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

### 6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010. Class 4 Nancy Lynch

6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010 Class 4 Nancy Lynch Today Two more models of computation: Nondeterministic Finite Automata (NFAs)

### 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008

MIT OpenCourseWare http://ocw.mit.edu 6.080 / 6.089 Great Ideas in Theoretical Computer Science Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

### Computer Science 281 Binary and Hexadecimal Review

Computer Science 281 Binary and Hexadecimal Review 1 The Binary Number System Computers store everything, both instructions and data, by using many, many transistors, each of which can be in one of two

### Course notes on Number Theory

Course notes on Number Theory In Number Theory, we make the decision to work entirely with whole numbers. There are many reasons for this besides just mathematical interest, not the least of which is that

### It is not immediately obvious that this should even give an integer. Since 1 < 1 5

Math 163 - Introductory Seminar Lehigh University Spring 8 Notes on Fibonacci numbers, binomial coefficients and mathematical induction These are mostly notes from a previous class and thus include some

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### the lemma. Keep in mind the following facts about regular languages:

CPS 2: Discrete Mathematics Instructor: Bruce Maggs Assignment Due: Wednesday September 2, 27 A Tool for Proving Irregularity (25 points) When proving that a language isn t regular, a tool that is often

### 3. Recurrence Recursive Definitions. To construct a recursively defined function:

3. RECURRENCE 10 3. Recurrence 3.1. Recursive Definitions. To construct a recursively defined function: 1. Initial Condition(s) (or basis): Prescribe initial value(s) of the function.. Recursion: Use a

### Universality in the theory of algorithms and computer science

Universality in the theory of algorithms and computer science Alexander Shen Computational models The notion of computable function was introduced in 1930ies. Simplifying (a rather interesting and puzzling)

### Lecture 1: Time Complexity

Computational Complexity Theory, Fall 2010 August 25 Lecture 1: Time Complexity Lecturer: Peter Bro Miltersen Scribe: Søren Valentin Haagerup 1 Introduction to the course The field of Computational Complexity

### 13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385

mcs-ftl 2010/9/8 0:40 page 379 #385 13 Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infinite

### Introduction to Automata Theory. Reading: Chapter 1

Introduction to Automata Theory Reading: Chapter 1 1 What is Automata Theory? Study of abstract computing devices, or machines Automaton = an abstract computing device Note: A device need not even be a

### Automata and Formal Languages

Automata and Formal Languages Winter 2009-2010 Yacov Hel-Or 1 What this course is all about This course is about mathematical models of computation We ll study different machine models (finite automata,

### A Formalization of the Turing Test Evgeny Chutchev

A Formalization of the Turing Test Evgeny Chutchev 1. Introduction The Turing test was described by A. Turing in his paper [4] as follows: An interrogator questions both Turing machine and second participant

### Announcements. CompSci 230 Discrete Math for Computer Science. Test 1

CompSci 230 Discrete Math for Computer Science Sep 26, 2013 Announcements Exam 1 is Tuesday, Oct. 1 No class, Oct 3, No recitation Oct 4-7 Prof. Rodger is out Sep 30-Oct 4 There is Recitation: Sept 27-30.

### Theory of Computation

Theory of Computation For Computer Science & Information Technology By www.thegateacademy.com Syllabus Syllabus for Theory of Computation Regular Expressions and Finite Automata, Context-Free Grammar s

### CS 341 Homework 9 Languages That Are and Are Not Regular

CS 341 Homework 9 Languages That Are and Are Not Regular 1. Show that the following are not regular. (a) L = {ww R : w {a, b}*} (b) L = {ww : w {a, b}*} (c) L = {ww' : w {a, b}*}, where w' stands for w

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### Mathematical Induction

Mathematical Induction Everybody do the wave! The Wave If done properly, everyone will eventually end up joining in. Why is that? Someone (me!) started everyone off. Once the person before you did the

### Prime Numbers. Chapter Primes and Composites

Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are

### ECS 120 Lesson 17 Church s Thesis, The Universal Turing Machine

ECS 120 Lesson 17 Church s Thesis, The Universal Turing Machine Oliver Kreylos Monday, May 7th, 2001 In the last lecture, we looked at the computation of Turing Machines, and also at some variants of Turing

### An example of a computable

An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

### Properties of Stabilizing Computations

Theory and Applications of Mathematics & Computer Science 5 (1) (2015) 71 93 Properties of Stabilizing Computations Mark Burgin a a University of California, Los Angeles 405 Hilgard Ave. Los Angeles, CA

### 1. R In this and the next section we are going to study the properties of sequences of real numbers.

+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

### Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

### Lecture 3: Finding integer solutions to systems of linear equations

Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture

### Classical Information and Bits

p. 1/24 Classical Information and Bits The simplest variable that can carry information is the bit, which can take only two values, 0 or 1. Any ensemble, description, or set of discrete values can be quantified

### Computing Functions with Turing Machines

CS 30 - Lecture 20 Combining Turing Machines and Turing s Thesis Fall 2008 Review Languages and Grammars Alphabets, strings, languages Regular Languages Deterministic Finite and Nondeterministic Automata

### NUMBER SYSTEMS CHAPTER 19-1

NUMBER SYSTEMS 19.1 The Decimal System 19. The Binary System 19.3 Converting between Binary and Decimal Integers Fractions 19.4 Hexadecimal Notation 19.5 Key Terms and Problems CHAPTER 19-1 19- CHAPTER

### GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY

GENERIC COMPUTABILITY, TURING DEGREES, AND ASYMPTOTIC DENSITY CARL G. JOCKUSCH, JR. AND PAUL E. SCHUPP Abstract. Generic decidability has been extensively studied in group theory, and we now study it in

### 6.080/6.089 GITCS Feb 12, 2008. Lecture 3

6.8/6.89 GITCS Feb 2, 28 Lecturer: Scott Aaronson Lecture 3 Scribe: Adam Rogal Administrivia. Scribe notes The purpose of scribe notes is to transcribe our lectures. Although I have formal notes of my

### APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

### Chapter 6. Number Theory. 6.1 The Division Algorithm

Chapter 6 Number Theory The material in this chapter offers a small glimpse of why a lot of facts that you ve probably nown and used for a long time are true. It also offers some exposure to generalization,

### Structural properties of oracle classes

Structural properties of oracle classes Stanislav Živný Computing Laboratory, University of Oxford, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK Abstract Denote by C the class of oracles relative

### (Refer Slide Time: 1:41)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 10 Sets Today we shall learn about sets. You must

### Fractions and Decimals

Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### NUMBER SYSTEMS. William Stallings

NUMBER SYSTEMS William Stallings The Decimal System... The Binary System...3 Converting between Binary and Decimal...3 Integers...4 Fractions...5 Hexadecimal Notation...6 This document available at WilliamStallings.com/StudentSupport.html

### Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

### MATHS 315 Mathematical Logic

MATHS 315 Mathematical Logic Second Semester, 2006 Contents 2 Formal Statement Logic 1 2.1 Post production systems................................. 1 2.2 The system L.......................................

University of Virginia - cs3102: Theory of Computation Spring 2010 Final Exam - Comments Problem 1: Definitions. (Average 7.2 / 10) For each term, provide a clear, concise, and precise definition from

### Cryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur

Cryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Module No. # 01 Lecture No. # 05 Classic Cryptosystems (Refer Slide Time: 00:42)

### Introduction Number Systems and Conversion

UNIT 1 Introduction Number Systems and Conversion Objectives 1. Introduction The first part of this unit introduces the material to be studied later. In addition to getting an overview of the material

### Continued fractions and good approximations.

Continued fractions and good approximations We will study how to find good approximations for important real life constants A good approximation must be both accurate and easy to use For instance, our

### Recall that in base 10, the digits of a number are just coefficients of powers of the base (10):

Binary Number System 1 Base 10 digits: 0 1 2 3 4 5 6 7 8 9 Base 2 digits: 0 1 Recall that in base 10, the digits of a number are just coefficients of powers of the base (10): 417 = 4 * 10 2 + 1 * 10 1

### This section demonstrates some different techniques of proving some general statements.

Section 4. Number Theory 4.. Introduction This section demonstrates some different techniques of proving some general statements. Examples: Prove that the sum of any two odd numbers is even. Firstly you

### Deterministic Finite Automata

1 Deterministic Finite Automata Definition: A deterministic finite automaton (DFA) consists of 1. a finite set of states (often denoted Q) 2. a finite set Σ of symbols (alphabet) 3. a transition function

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Binary Numbers. Bob Brown Information Technology Department Southern Polytechnic State University

Binary Numbers Bob Brown Information Technology Department Southern Polytechnic State University Positional Number Systems The idea of number is a mathematical abstraction. To use numbers, we must represent

### CHAPTER 7 GENERAL PROOF SYSTEMS

CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes