Nathan Brunelle Department of Computer Science University of Virginia Theory of Computation CS3102 – Spring 2014 A tale of computers, math, problem solving, life, love and tragic death

Rice’s Theorem Any nontrivial property about the language of a Turing machine is undecidable. A property is trivial if it either applies to ALL languages of Turing Machines, or NONE of them. Henry Gordon Rice, 1951 M w x

Which of these are Undecidable? Does TM M accept any strings? Does TM M accept “Hello”? Does TM M take more than 1000 steps to process input w ? Does TM M accept a finite number of strings? Does TM M accept a Decidable Language? Does TM M accept a Recognizable Language? Decidable Undecidable Decidable Warning: Rice’s theorem applies to properties (i.e., sets of languages), not (directly to) TM’s or other object types!

Undecidability + Rice + Church-Turing Undecidability: undecidable languages that cannot be decided by any Turing Machine Rice’s Theorem: all nontrivial properties about the language of a TM are undecidable Church-Turing Thesis: any mechanical computation can be done by some TM Conclusion: any nontrivial property about general mechanical computations cannot be decided! Language and Problems: any problem can be restated as a language recognition problem

Some “useful” undecidable Problems Malware detection Virus Scanning Bug detection Program equivalence Type checking Solutions to polynomials N-body problems Fluid dynamics Compressibility Enscheidungsproblem M w x

Recognizing vs. Deciding Turing-decidable: A language L is “decidable” if there exists a TM M such that for all strings w: If w  L, M enters q accept. If w  L, M enters q reject. Turing-recognizable: A language L is “Turing-recognizable” if there exists a TM M such that for all strings w: If w  L eventually M enters q accept If w  L either M enters q reject or M never terminates “Computable” “Non-Computable”

Recognizable Languages Revised Hierarchy Decidable Languages Regular Languages CFLs finite Languages Deterministic CFLs Computable Non- Computable

Example of a Recognizable Language The Halting Problem, or A TM Algorithm: Universal Turing Machine. Universal Turing Machine M w Output Tape for running TM- M starting on tape w

Closure Properties of Recognizable Languages Theorem: The recognizable languages are closed under . 1.Copy input 2.Run machine 1 on copy of input (never cross $) 1.If reject then move into reject state 3.Erase everything to right of $ 4.Run machine 2 abba$abba input copy M1 M2

Closure Properties of Recognizable Languages Theorem: The recognizable languages are closed under union. 1.Copy input 2.Run machine 1 on copy of input (never cross $) 1.If accept then move into accept state 3.Erase everything to right of $ 4.Run machine 2 Problem: What if the machine runs forever?

Closure Properties of Recognizable Languages Theorem: The recognizable languages are closed under union. 1.Copy input to second tape 2.Run machine 1 on tape 1, machine 2 on tape 2 “in parallel”, “time-multiplexed” 1.If either accept then move into accept state M1 M2 Time Accept!

Closure Properties of Recognizable Languages Theorem: The recognizable languages are closed under concat. Using non-determinism: Tape 1: input Tape 2: tape for Machine 1 Tape 3: tape for Machine 2 Copy some characters from the first tape to tape 2 Nondeterminstically decide when to copy the rest onto tape 3 abba ab ba “choose” Non- deterministically!

Closure Properties of Recognizable Languages Theorem: The recognizable languages are closed under Kleene star. Similar to concatenation, but now we also try all possible numbers of concatenations. Non-deterministically insert $’s into the input. The machine must accept each of these substrings. a$bb$a Inserted Non- deterministically! Each substring then used as input to the machine

Closure Properties of Recognizable Languages Theorem: The recognizable languages are not closed under complement. Hint: reduction from halting problem. This one takes some new tools

Co-Recognizable Languages Turing-recognizable: A language L is “Turing-recognizable” if there exists a TM M such that for all strings w: If w  L eventually M enters q accept If w  L either M enters q reject or M never terminates Co-Turing-recognizable: A language L is “Co-Turing-recognizable” if there exists a TM M such that for all strings w: If w  L eventually M enters q accept If w  L either M enters q reject or M never terminates

Closure Properties of Recognizable Languages Theorem: If a language is both Recognizable and Co-Recognizable then it is decidable. Proof: Let M1 be a recognizer for L, M2 be a Co-Recognizer for L. We build MD, a Decider for L: Simulate M1 and M2 “in parallel” (time multiplexed)on input w. If M1 accepts then accept, else if M2 accepts, then reject. Every string will be accepted by exactly one machine in finite time. M1 M2 Time Reject!

Closure Properties of Recognizable Languages

Oracles Originated in Turing’s Ph.D. thesis Named after the “Oracle of Apollo” at Delphi, ancient Greece Black-box subroutine / language Can compute arbitrary functions Instant computations “for free” Can greatly increase computation power of basic TMs E.g., oracle for halting problem

The “Oracle of Omaha”

The “Oracle” of the Matrix

A special case of “hyper-computation” Allows “what if” analysis: assumes certain undecidable languages can be recognized An oracle can profoundly impact the decidability & tractability of a language Any language / problem can be “relativized” WRT an arbitrary oracle Undecidability / intractability exists even for oracle machines! Turing Machines with Oracles Theorem [Turing]: Some problems are still not computable, even by Turing machines with an oracle for the halting problem!

Reaching a Contradiction What happens if we run D on its own description, ? Define D ( ) = Construct a TM that: Outputs the opposite of the result of simulating H on input (M, ) Assume there exists some TM H that decides A TM. If M accepts its own description, D( ) rejects. If M rejects its own description, D( ) accepts. If D accepts its own description, D( ) rejects. If D rejects its own description, D( ) accepts. substituting D for M… M* is Relativized to O H* is Relativized to O D* is Relativized to O * * * * * * * * * * * * * * * * * * * * *

Ø Turing (1937); studied by Post (1944) and Kleene (1954) Quantifies the non-computability (i.e., algorithmic unsolvability) of (decision) problems and languages Some problems are “more unsolvable” than others! Turing Degrees Emil Post Alan Turing Stephen Kleene Students of Alonzo Church: H H H*H* Turing degree 0 Turing degree 1 Turing degree 2 Defines computation “relative” to an oracle. “Relativized computation” - an infinite hierarchy! A “relativity theory of computation”! Georg Cantor Diagonalization

Turing degree of a set X is the set of all Turing-equivalent (i.e., mutually-reducible) sets: an equivalence class [X] Turing degrees form a partial order / join-semilattice [0]: the unique Turing degree containing all computable sets For set X, the “Turing jump” operator X’ is the set of indices of oracle TMs which halt when using X as an oracle [0’]: Turing degree of the halting problem H; [0’’]: Turing degree of the halting problem H* for TMs with oracle H. Turing Degrees Emil Post Alan Turing Stephen Kleene Students of Alonzo Church: Turing jump Turing jump

Turing Degrees Emil Post Alan Turing Stephen Kleene Students of Alonzo Church: Turing jump Turing jump Each Turing degree is countably infinite (has exactly  0 sets) There are uncountably many (2  0 ) Turing degrees A Turing degree X is strictly smaller than its Turing jump X’ For a Turing degree X, the set of degrees smaller than X is countable; set of degrees larger than X is uncountable (2  0 ) For every Turing degree X there is an incomparable degree (i.e., neither X  Y nor Y  X holds). There are 2  0 pairwise incomparable Turing degrees For every degree X, there is a degree D strictly between X and X’ so that X < D < X’ (there are actually  0 of them) The structure of the Turing degrees semilattice is extremely complex!

PSPACE-complete QBF The Extended Chomsky Hierarchy Finite {a,b} Regular a* Det. CF a n b n Context-free ww R PanbncnPanbncn NP PSPACE EXPSPACE Recognizable Not Recognizable H H Decidable Presburger arithmetic NP-complete SAT Not finitely describable  ** EXPTIME EXPTIME-complete Go EXPSPACE-complete =RE  Turing Context sensitive LBA H* degrees