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1 CS 3261 Computability Course Summary Zeph Grunschlag

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2 Announcements Last hw due now Look out for a final exam practice problems coming out over the weekend I will hold final review session on Tuesday 12/11, 3-5 pm, 833 Mudd. Pick-up final hws. I will hold daily OHs next week and Monday 12/17, 12:00-1:30 except Thursday, 12/13 Final exam: Tuesday 12/18, 9-12, 833 Mudd

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3 Computability Concepts AIM: Reduce Computer Science to its bare theoretical essentials. APPROACH: Algorithmic Problems Formal Languages

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4 Formal Languages Fundamental insight of Theoretical CS: By understanding how formal languages can be computed, will understand how any algorithmic problem can be solved. Algorithmic input/output problems involve creating procedures for procuring outputs from given inputs. Can be turned into a formal languages by re-writing as yes/no questions. EG: Find the shortest path… becomes Is there a path shorter than…

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5 Computability Concepts AIM: Reduce Computer Science to its bare theoretical essentials. Algorithmic Problems Formal Languages Computers Graph based machine models Questions to investigate: 1) What sorts of problems can be solved by each computer model? 2) What languages does each model accept? 3) What are the practical limits on what a computer can do?

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6 Abstract Machine Models DFAs DFAs model computers with strictly bounded memory. 1 3 2 a b b a a,b

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7 Abstract Machine Models DFAs Q: Whats the accepted language? 1 3 2 a b b a a,b

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8 Abstract Machine Models DFAs A: a*b + 1 3 2 a b b a a,b

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9 Abstract Machine Models NFAs Nondeterminism is a powerful concept. Often 1 st view of a problem is nondeterministic. 1 3 2 a a b a a,b

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10 Abstract Machine Models NFAs Q: Whats the accepted language? 1 3 2 a a b a a,b

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11 Abstract Machine Models NFAs A: a + b* 1 3 2 a a b a a,b

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12 Abstract Machine Models PDAs By allowing a pushdown stack, increase flexibility and accept more languages. 12 3 a, X b,X 0 $ $

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13 Abstract Machine Models PDAs Q: Whats the accepted language? 12 3 a, X b,X 0 $ $

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14 Abstract Machine Models PDAs A: {a n b n | n 0} 12 3 a, X b,X 0 $ $

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15 Abstract Machine Models TMs By allowing a read-write tape, amazingly get most general possible computer model! 12 X R 0 1 $,R L acc 34 X L $ L 1 L $ R 1 X,R 1 R X R X L 5 1 L 1|X L

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16 Abstract Machine Models TMs Q: Whats the accepted language? 12 X R 0 1 $,R L acc 34 X L $ L 1 L $ R 1 X,R 1 R X R X L 5 1 L 1|X L

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17 Abstract Machine Models TMs A: Unary powers of 2. 12 X R 0 1 $,R L acc 34 X L $ L 1 L $ R 1 X,R 1 R X R X L 5 1 L 1|X L

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18 I/O Versions Each class of languages has its own I/O version. Regular: Finite State Transducers More powerful models exist (e.g. with s) Context free: (didnt study any) Compilers: Input is a string, output is a parse-tree (or even executable code) Turing Machines: I/O TMs

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19 Robust Formal Language Classes Turns out these models are very robust Many equivalent ways to generate same classes: Regular languages FAs, NFAs, Regular Expressions, Right-Linear Grammars Context Free Languages PDAs, Context Free Grammars Recognizable languages –Church-Turing thesis TMs, k-tape machines, k-track machines NTMs, Queue Machines, 2-Stack PDAs, RAMs, Unrestricted Grammars Complexity classes P and NP For NP: Poly. NTMs, Poly. Verifiers, Poly. Proofs We learned algorithms for converting between most of the different views Language classes closed under natural operations.

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20 System Design Often computer system creation involves designing a formal language to describe system communication. Components receive communication streams and have to effect actions based on these. Computability theory can help drive design at a high level. EG: Might come up with a communication stream thats seems like regular language. Could then show that it isnt using pumping lemma. With this knowledge, final design tweaks original to obtain a regular language and therefore DFA based ultra-fast and super-reliable system components!

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21 Negative Examples As above, it is very important to be able to tell when particular languages cannot be accepted by a certain model of computation. We have several tools at our disposal: Irregularity: pumping lemma (PL) Non-Context-Freeness: CFPL Undecidability: Reductions from undecidable languages Intractability: Poly-time reduction from NP- hard languages

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22 System Design Learned useful concepts that can help modularization when designing systems. Often can express a language as a union, intersection, negation, concatenation or Kleene-* of simpler languages. More complex language may be put together by using simple components along with off the shelf reconstruction techniques:

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Language Design Class Negate Concat. Kleene- * DFAs Cartesian Product Accept Non-accepts NFAs Parallel Cartesian Product SerialLoop PDAs Parallel SerialLoop Deciders Run in parallel Accept Non-accepts Break string up Recursive algorithm Recognizers Run in parallel

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24 Language Class Hierarchy All REC = accepted by TM DEC = decided by TM Context Free Deterministic Context Free Regular = accepted by FAs Finite languages

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25 Known Complexity Hierarchy Get the following RAM hierarchy diagram: REC DEC P TIME(n) CFL TIME(n 3 ) REG EQ REX

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26 Unknown Complexity Hierarchy Decidable NP NP but not NP-hard P Finite languages Does anything exist here?

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27 Conjectured Hierarchy Inside of DEC most conjecture: DEC NP P co-NP NP complete PRIME SAT

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28 Follow-ups to Computability Related Electives: Analysis of Algorithms 4231 (fall) Computational Complexity 4236 (spring) Cryptography –generic course no. 4995 (this Spring with Michael Rabin!!!) Courses Requiring Computability: Programming Languages and Translators 4115 (every semester) Compilers 4117 (this Spring with Al Aho!!!) Portions of several other courses

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29 Final Remarks With the horrors at the beginning of the semester….

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30 Final Remarks Thanks for putting in the effort and helping make this my best semester thus far!

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