# l10 - Lecture 10 Presburger arithmetic David Dill...

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Lecture 10: Presburger arithmetic David Dill Department of Computer Science 1

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Outline A little more on Rice’s theorem Presburger arithmetic 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

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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
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

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Presburger arithmetic Presburger arithmetic is the quantified theory of linear inequalities over the integers.
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