Exploring Universal Turing Machines and Undecidability in Computation

1. CSE 599 Lecture 2 • In the previous lecture, we discussed: • What is Computation? • History of Computing • Theoretical Foundations of Computing • Abstract Models of Computation • Finite Automata • Languages • The Chomsky Hierarchy • Turing Machines

2. Overview of Today’s Lecture • Universal Turing Machines • The Halting Problem: Some problems cannot be solved! • Nondeterministic Machines • Time and Space Complexity • Classes P and NP • NP-completeness: Some problems (probably) cannot be solved fast (on classical sequential computers)!

3. Recall from last class: Turing Machines • Can be defined as a “program”: • (current state, current symbol) to (next state, new symbol to write, direction of head movement), or • A list of quintuples • (q0, s0, q1, s1, d0) • (q1, s1, q2, s2, d1) • (q2, s2, q3, s3, d2) • (q3, s3, q4, s4, d3) • (q4, s4, q5, s5, d4) • etc. • Includes an initial state q0 and • A set of halting states (for example, {q3, q5})

4. Turing Machines • Example: (q0, 1, q1, 0, R)

5. Universal Turing Machines (UTMs) • A UTM is a “programmable” Turing machine that can simulate any TM - it is like an interpreter running a program on a given input • Takes as input a description or “program” (list of quintuples) of a TM and the TM’s input, and executes that program • Uses a fixed program (like all TMs) but this program specifies how to execute arbitrary programs • Analogous to a digital computer running a program; in fact, Von Neumann’s stored program concept was partly motivated by UTMs

6. Universal Turing Machines – How They Work • A UTM U receives as input a binary description of • an arbitrary Turing machine T (list of quintuples) and • contents of T’s tape (a binary string t). • U simulates T on t using a fixed program as follows: • INITIALIZE: copy T’s initial state and T’s initial input symbol to a fixed “machine condition” (or workspace) area of the tape • LOCATE: Given T’s current state q and current symbol s: Find the quintuple (q, s, q’, s’, d) in the description of T; If q not found, then halt (q is a halt state); else • COPY: Write q’ in workspace and s’ in T’s simulated tape region; Move T’s simulated head according to d Read new symbol s’’ and write it next to q’ in workspace • REPEAT: Go to 2.

7. Universal Turing Machines – Diagram • See Feynman text, Figure 3.23 • This machine uses 8 symbols and 23 different states • Exercise: Identify which portions of the machine executes which subroutine (LOCATE, COPY, etc.)

8. Now we are ready for the big question… • Are there computational problems that no algorithm (or Turing Machine) can solve? • Surprising answer – YES! • Proof relies on self-reference: UTMs trying to solve problems about themselves. • Related to the paradox: “This sentence is false” • Can you prove the above statement true or false? • Related to Cantor’s proof by “diagonalization” that there are more real numbers than natural numbers

9. Proof by diagonalization: An example • Show that there are more real numbers than natural numbers • Proof: Form a 1-1 mapping from natural numbers to reals, and form a new real number by changing the ith digit of the ith real number: For example, if 1-1 map is given by: New Real Number = 0.2404……. (Add 1 to diagonal) This number is different from all the ones listed

10. The Halting Problem and Undecidability • Question: Are there problems that no algorithm can solve? • Consider the Halting Problem: Is there a general algorithm that can tell us whether a Turing machine T with tape t will halt, for any given T and input t? • Answer: No! • Proof: By contradiction. Suppose such an algorithm exists. Let D be the corresponding Turing machine. Note that D is just like a UTM except that it is guaranteed to halt: • D halts with a “Yes” if T halts on input t • D halts with a “No” if T does not halt on input t

11. Diagram of Halting Problem Solver D • Input to D: description dT of a TM T and its input data t

12. Diagram of New Machine E • Define E as the TM that takes as input dT, makes a second copy in the adjacent part of the tape, and then runs D:

13. Diagram of final machine Z (the “diagonalizer”) • Consider the machine Z obtained by modifying E as follows: • On input dT: if E halts with a “No” answer, then halt if E halts with a “Yes” answer, then loop forever • What happens when Z is given dZ as input?

14. The machine D cannot exist! • By construction, Z halts on dT if and only if the machine T does not halt on dT • Therefore, on input dZ: • Z halts on dZ if and only if Z does not halt on dZ • A contradiction! • We constructed Z and E legally from D. So, D cannot exist. The halting problem is therefore undecidable. • Conclusion: There exist computational problems (such as the halting problem) that cannot be solved by any Turing machine or algorithm.

15. Computability • We can now make a distinction between two types of computability: • Decidable (or recursive) • Turing Computable (or partial recursive/recursively enumerable) • A language is decidable if there is a TM that accepts every string in that language and halts, and rejects every string not in the language and halts. • A language is Turing computable if there is a TM that accepts every string in that language (and no strings that are not) • Are all decidable languages Turing computable? • Are all Turing computable languages decidable?

16. Examples: Decidable Languages • The language 0n1n is decidable: • If the tape is empty, ACCEPT • Otherwise, if the input does not look like 0* 1*, REJECT • Cross off the first 0. Then move right until the first 1 (not crossed off) • Cross off that 1, then move left until the first 0. • Repeat the above 2 steps until we run out of 0’s or 1’s • If there are more 0’s than 1’s REJECT • If there are more 1’s than 0’s REJECT • If there are an equal number, ACCEPT • L = { 1n | n is a composite number } is decidable • All decidable languages are Turing computable. Are there Turing computable languages that are not decidable?

17. Examples: Turing Computable Languages • The Halting Problem is Turing Computable: • HALT = { <dT,t > | dT is a description of TM T, and T halts on input t} • Proof Sketch: The following UTM H accepts HALT • Simulate T on input t. • If T halts, then ACCEPT • Crucial Point: H may not halt in some cases because T doesn’t, but if T does halt, so does H. So L(H) = HALT. • Hilbert’s 10th problem: Given a polynomial equation (e.g. 7x2-5xy-3y2+2x-11=0, or x3+y3=z3), give an algorithm that says whether the equation has at least one integer solution. • Try all possible tuples of integers. • If one of the tuples is a solution, ACCEPT

18. Beyond Turing Computability • Are there problems that are even harder than HALT i.e. that are not even Turing Computable? • Consider the language DOESN’T-HALT = {<dT,t> | T does not halt on input t} • Result: DOESN’T HALT is not Turing Computable! • Proof: Suppose DOESN’T-HALT was Turing computable and a Turing Machine DH accepts it. Let H be our UTM that accepts HALT. • Define another TM D as follows: On input <dT, t>: • Run H and DH simultaneously (alternate step by step) • If H accepts, ACCEPT • If DH accepts, REJECT • Then D decides HALT! A contradiction, therefore DH does not exist and DOESN’T-HALT is not Turing Computable.

19. Discussion • THERE ARE PROBLEMS WE JUST CAN’T SOLVE!!! • Software verification is impossible without restrictions • You can’t get around this, but you can reduce the pathological cases • Be careful-- these pathological cases won’t go away • There are mathematical facts we can’t prove: • Gödels Theorem: Any arithmetic system large enough to contain Q (a subset of number theory) will contain unprovable statements • Based on constructing the statement: “This statement is unprovable” • Uses numerical encodings of statements (called Gödel numbers) like those for a TM • Any sufficiently complex system will have holes in it.

20. 5-minute break… Next: Nondeterministic Machines, Time and Space Efficiency of Algorithms

21. Nondeterminism • We have seen the limitations of sequential machines that transition from one state to another unique state at each time step. • Consider a new model of computation where at each step, the machine may have a choice of more than one state. • For the same input, the machine may follow different computational paths when run at different times • If there exists any path that leads to an accept state, the machine is said to accept the input • Simple Example: NFA (Nondeterministic Finite Automata)

22. NFA Example • Here’s one example: • There is only one nondeterministic transition in this machine • What strings does this machine accept? • Are NFAs more powerful than DFAs?

23. NFAs and DFAs are equivalent! • The models are computationally equivalent (the DFA has exponentially more states though in this sketch) • Proof sketch: • each state of the DFA represents a subset of states in the NFA • What about Nondeterministic Turing machines?

24. Nodeterministic Turing Machines (NTMs) • NTMs may have multiple successor states, e.g. • (q,s,q1,s1,d1) and (q,s,q2,s2,d2) • NTMs are equivalent in computational power to deterministic TMs! • Proof sketch: • Design a machine based on a Universal TM U to simulate NTM M • When there is a nondeterministic transition: (q,s,q1,s1,d1) and (q,s,q2,s2,d2), • U alternates between simulating one computation path and the other (similar to “breadth first search”) • If one of the paths halts, the U halts (with M’s output on its tape) • Nondeterminism useful for time and space complexity issues

25. Time and Space Efficiency • The fact that a problem is decidable doesn’t mean it is easy to solve. • We are interested in answering: what problems can/cannot be efficiently solved? • Time complexity (worst case run time) is a major concern • Space complexity (maximum memory utilized) is a second concern • We first need to define how to measure the time and space complexity of an algorithm or a TM

26. Time and Space Complexity of an Algorithm • Example Problem DUP: Given an array A of n positive integers, are there any duplicates? • For example, A: 34, 9, 40, 87, 223, 109, 58, 9, 71, 8 • An easy algorithm for DUP: • for i = 1 to n-1 • for j = i+1 to n • if A[i] = A[j] • Output i and j • Halt • else continue • Space complexity: n + 2 • Time complexity: How many steps in the worst case?

27. Time and Space Complexity of an Algorithm • An easy algorithm for DUP: • for i = 1 to n-1 • for j = i+1 to n • if A[i] = A[j] • Output i and j • Halt • else continue • Time complexity: How many steps in the worst case? • Worst case = last two numbers are duplicates • Total time steps = • [1 + 3(n-1)] + [1 + 3(n-2)] + … upto n-1 terms • = approximately n2 • = O(n2) (on the order of n2)

28. “Big O” notation for expressing complexity • A function f(n) is big O of a function g(n), i.e. f(n) = O(g(n)), if there exists an integer N and constant C such that • f(n)  c·g(n) for all n  N • Thus, our algorithm uses O(n) space and O(n2) time at worst • Exercise: Design an algorithm for DUP that uses O(n) space and O(n log n) time

29. Time versus Space Tradeoffs • New Algorithm for DUP: • Idea: Use A[i] as index into new array B initialized to 0’s • for i = 1 to n • If B[A[i]] = 1 • Output A[i] • Halt else B [A[i]]  1 • Similar to detecting collisions in hashing • Worst Case Time complexity = O(n) • Worst Case Space complexity = O(2m) where m is the number of bits required to represent numbers that can potentially occur in A.

30. Polynomial Time • DTIME(t(n)) = All languages decided by a deterministic TM in time O(t(n)) • P =  k1 DTIME (nk) • Importance of P: It corresponds to our notion of the class of problems that can be solved efficiently in time (runs in polynomial number of steps with respect to size of input) • Example: DUP is in P; so is sorting. • P for a TM  P in most other models • Multitape TM, different alphabet, the RAM model • DNA Computing • Not necessarily for nondeterministic TMs • Not necessarily for Quantum Computers

31. The Satisfiability Problem (SAT) • SAT = {Boolean formula f | f is an AND of many ORs and there is an assignment of 0’s and 1’s that makes f true} • Example: f = (x1 + NOT(x2) + x3)(NOT(x1) + x2 + x3) • f is satisfiable: x1 = 0, x2 = 0, x3 = 0 (or x1 = 0, x2 = x3 = 1) • Very hard for large formulas – exhaustive search of all assignments of n variables • Best known algorithm runs in exponential time in the number of variables • BUT: once you guess an assignment, very easy to check • Nondeterminism might help!

32. The Class NP • NTIME (t(n)) = All languages decided by a nondeterministic TM in time O( t(n)) • NP = k1 NTIME (nk) • NP stands for Nondeterministic Polynomial Time • NTM can answer NP problems in polytime • Another definition uses the idea of a verifier • A verifier takes a string allegedly in the language, along with some piece of evidence. • Using this evidence, it verifies that the string is indeed in the language. • If the string is not in the language, or if the evidence isn’t right, it REJECTS

33. Verifiers • A verifier for a language L is an algorithm V, where L={ w | V accepts < w,c > for some string c that is “evidence” of w’s membership in L} • We measure the time of the verifier in terms of w -- not c, the evidence • The language L is polynomially verifiable if it has a polytime verifier • NP is the class of languages that have polynomial time verifiers • SAT = { Boolean formulas f | f has a satisfying assignment} • Evidence c is an assignment of variables that makes f true

34. NTMs can solve NP problems using verifiers • Say language L is in NP. • Let V be a polytime verifier for L. • Define Nondeterministic Turing Machine N as follows • N: On input w of length n: • Nondeterministically choose an “evidence” string c of polynomial length • Run V on < w, c > • If V accepts, ACCEPT • Else, REJECT • Note: One of the choices for c will be the correct evidence if w is in L.

35. Exercise • Show that the following problems are in NP: • VERTEX-COVER = { < G, k > | G is an undirected graph that has a k-node vertex cover i.e. all edges are “covered” by at least one vertex from a set of k vertices} • TSP = {<C,b> | there is a “tour” of all the cities in C with total length no more than b} • COMPOSITE = { numbers n | n is composite }

36. Reductions • Basic Idea: Use one problem to solve another • Problem A is reducible to problem B if you can transform any instance of problem A to an instance of problem B and solve problem A by solving problem B • Example: • Language ACC = { <dT ,t> | T is a TM that accepts input t } • HALT = { <dT,t> | T halts on input t} (Let H decide HALT) • ACC is reducible to HALT: On input < dT, t >, • Run H on input < dT, t >. • If H rejects (T does not halt on t), then REJECT • Else, simulate T on input t. • If T accepts, ACCEPT • If T rejects, REJECT • We have solved ACC using an algorithm H for HALT

37. NP Hard, NP Complete • A problem is NP hard if solving it means we can solve every problem in NP • Specifically, there is a polynomial time reduction from every problem in NP to the NP hard problem • Note: By this definition, a problem is NP-hard if there is a polynomial time reduction from a known NP-hard problem to the given problem (easier to show) • A problem is NP complete if it is NP hard and in NP • These problems epitomize the class NP (i.e. they are the hardest problems in NP)

38. Importance of NP completeness • A large number of problems in optimization, engineering, computer science and mathematics are known to be NP complete, including problems in compiler optimization, scheduling, etc. • No one has found an efficient (polynomial time) algorithm for any NP complete problem • If someone finds a polynomial time algorithm for any one NP complete problem, then we can solve all NP complete problems (and all problems in NP) efficiently in polynomial time.

39. The Graph of NP Complete Problems

40. Cook-Levin Theorem: SAT is NP complete • Cook and Levin independently proved that SAT is NP complete • Proof involves constructing a very large Boolean formula that captures the operation of a nondeterministic TM N that runs in polynomial time and solves a problem A in NP • The large formula takes into account: • Basic facts such as: N can be in only one state q at any time t, a tape cell can only contain 1 symbol, read/write head is scanning 1 cell etc. • e.g. [S(t,q)  ~S(t,q’)] for all q  q’ and for t = 0, 1, …, nk • Initial and final conditions after nk steps have been executed • N’s program i.e. list of quintuples

41. Cook-Levin Theorem: SAT is NP complete • Crucial facts: • It takes only a polynomial amount of time to generate the Boolean formula for any NTM • The Boolean formula limits the NTM to behaving just as it should • Thus, the constructed formula is satisfiable if and only if the NTM halts in nk time steps and outputs a “Yes” on its tape (which means the original NP problem has the answer “Yes” for the given input) • We have thus shown that any NP problem is polynomial time reducible to SAT i.e. SAT is NP complete • Now, suppose you have a new problem you suspect is NP complete: to show that it is, just reduce SAT to the problem!

42. Proving NP completeness via Reductions • VERTEX-COVER = { < G, k > | G is an undirected graph that has a k-node vertex cover} • Vertex cover is a subset of nodes such that every edge in the graph touches at least one node in the cover • Show that VERTEX-COVER is NP complete • Proof: • Show that VERTEX-COVER is in NP • Show that SAT is polytime reducible to VERTEX-COVER

43. PSPACE • DSPACE(s(n)) is the set of languages that can be decided using no more than O(s(n)) cells of the tape. • PSPACE =  k1 DSPACE (nk) • We can reuse space, but not time. • Is PSPACE as big as NP? (Homework problem) • We are asking if NP problems can be solved by PSPACE machines (Hint: Try all possibilities for a solution (exhaustive search) and figure out how much space you really need for simulating the NP machine).

44. General Issues: Applicability • Issues to think about as we examine other models • How do the time and space bounds apply? • DNA computing is massively parallel - but only so big • same with neural systems • Quantum Computers? • Do the decidability results really apply? • Approximate solutions may suffice for many ill defined questions in • vision • speech understanding, speech production • learning • navigation/ movement

45. General Issues: Representation • Does the TM model apply to neural computing? • Neurons compute using distributed signals and stochastic pulses • Is thinking about symbol processing the wrong way to think about neural systems? • Could some other model (e.g. probabilistic computing) provide us with a way to describe neural processing? • How useful is the TM model in capturing the abstract computations involved in DNA computing and Quantum computing? • Keep these questions in mind as we explore alternative computing paradigms…

46. Summary and where we are headed… • We asked: What functions are computable? • Are there functions that no algorithm (or Turing machine) can ever compute? (Yes) • We asked: What functions are tractable? • Formalized the notion of P as the class of problems with time-efficient solutions • Lots of problems are NP complete with no fast algorithms known • Big question: Is P = NP? • We are now ready to explore: • How digital computers embody the theory we have discussed • How problems that are hard to solve on digital computers may be solved more efficiently using alternative computing methods such as DNA, neural, or Quantum Computing.

47. Next Week: Digital Computing • We will see how a hierarchical approach allows us to build a general purpose digital computer (a finite state automaton): • Transistors  switches  gates  combinational and sequential logic  finite-state behavior  register-transfer behavior  … • The physical basis is silicon integrated-circuit technology • Guest Lecture by Chris Diorio on IC technology (first hour or so) • We will discuss the theory and practice of digital computing, and end by examining their future: Moore’s law and semiconductor scaling in the years to come.

48. Things to do this week… Finish Homework Assignment # 1 (due next class 1/18) Read the handouts and Feynman chapters 1, 2, and 7… Have a great weekend!