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Cousot_MIT_2005_Course_03_4-1

Cousot_MIT_2005_Course_03_4-1 - Abstract Program Invariance...

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« Abstract Program Invariance and Termination Proofs » Patrick Cousot École normale supérieure 45 rue d’Ulm, 75230 Paris cedex 05, France [email protected] www.di.ens.fr/ ~ cousot Course 16.399: “Abstract interpretation” — MIT — Tuesday, February 22, 2005 Course 16.399: “Abstract interpretation”, Tuesday, February 22, 2005 1 ľ P. Cousot , 2005 Undecidability Course 16.399: “Abstract interpretation”, Tuesday, February 22, 2005 2 ľ P. Cousot , 2005 The fundamental limitation: undecidability – In 1921, David Hilbert put forward the so-called Hilbert’s Program , calling for a formalization of all of mathe- matics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent, to be carried out using only “finitary” methods. Kurt Gödel ’s incompleteness theorems [1] essentially show that Hilbert’s Program cannot be carried out. Reference [1] Gödel, K., “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I”, Monatshefte für Mathematik und Physik , vol. 38 (1931), pp. 173–198. Course 16.399: “Abstract interpretation”, Tuesday, February 22, 2005 3 ľ P. Cousot , 2005 David Hilbert Kurt Gödel Course 16.399: “Abstract interpretation”, Tuesday, February 22, 2005 4 ľ P. Cousot , 2005
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Decision Problems – A decision problem is a computational problem where the answer is always YES/NO: solve_problem ( data ) 7!f YES, NO g – The complement : P of a decision problem P is one where all the YES and NO answers are exchanged. Course 16.399: “Abstract interpretation”, Tuesday, February 22, 2005 5 ľ P. Cousot , 2005 Decidability/Semidecidable/Undecidability A decision problem is: – “ Decidable ” if and only if there exists an algorithm to solve the problem in finite time; – “ Undecidable ” if and only if there exists no algorithm to solve the problem in finite time; – “ Semidecidable ” if and only if there exists an algorithm to solve the problem in finite time when the answer is YES but which may not terminate when the answer is NO ; Course 16.399: “Abstract interpretation”, Tuesday, February 22, 2005 6 ľ P. Cousot , 2005 The termination problem – “ Termination problem ”: given a sequential program and its input data, will execution of this program on these data ever terminate? – The complement is the “ Nontermination problem ”: given a sequential program and its input data, will execution of this program on these data never terminate? – Termination is undecidable (but semidecidable); – Nontermination is undecidable (and not semidecid- able). Course 16.399: “Abstract interpretation”, Tuesday, February 22, 2005 7 ľ P. Cousot , 2005 Interpretor – We let Ff be the text file (of type text ) containing the text of a program encoding a function f of type text -> bool ; – We let Fd be the text file (of type text ) containing the text encoding the data d – An interpreter I : text * text -> bool is a program which execution I(Ff, Fd) is the result f ( d ) of the evaluation of function f on the data d .
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