l10 - Lecture 10 Presburger arithmetic David Dill...

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture 10: Presburger arithmetic David Dill Department of Computer Science 1
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Outline A little more on Rice’s theorem Presburger arithmetic 2
Image of page 2
Statement of Rice’s theorem Thm: Every non-trivial property P of the R.E. languages is undecidable. (a). Property : A set of languages (in this case, a set of Turing Machines). (b). Trivial : Either every language has it or no language has it. 3
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
WRONG Applications of Rice’s theorem Rice’s theorem does not apply to these Whether a TM has less than 7 states (not a language property). Whether a TM has a final state (not a language property). Whether a TM has a start state (not a language property). Whether the language is RE (trivial – all RE languages). Whether the language is L d (trivial – no RE languages). 4
Image of page 4
Logical decision procedures Being able to solve logic problems automatically would be extremely valuable to computer science (and many other fields). Lots of problems can be reduced to logic problems, so general-purpose logic solvers can be extremely helpful for solving hard problems. Example: Program verification: Is a program correct? Unfortunately, we’ll say later that many such problems are provably not decidable using computers. Example: Validity of first-order logic formulas: x, y z : ( ¬ P ( x, y ) ( P ( x, z ) P ( z, y ))) Example: Non-linear arithmetic over the integers. n : n > 2 ⇒ ¬∃ x, y, z : ( x n + y n = z n ) There is an inherent conflict between expressive power , which we want so we can encode more problems, and computational complexity (including decidability), which is helpful if we want a computer to solve it. 5
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Presburger arithmetic Presburger arithmetic is the quantified theory of linear inequalities over the integers.
Image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern