1 / 9

Trachtenbrot’s Theorem

Trachtenbrot’s Theorem. The following problem is undecidable . Given: A sentence of first order logic Question: Is satisfiable in a finite structure?. Theorem of Matyasevich. The following problem is undecidable

olive
Télécharger la présentation

Trachtenbrot’s Theorem

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Trachtenbrot’s Theorem

  2. The following problem is undecidable Given: A sentence of first order logic Question: Is satisfiable in a finite structure?

  3. Theorem of Matyasevich The following problem is undecidable Given: A polynomial of multiple natural varibales and integer coefficients Ques: Does have a zero, i.e., do there exist natural numbers s Hilbert’s 10th Problem

  4. Proof: Reduction of Hilbert’s 10th problem to the problem is a given first order sentence is satisfiable in the finite. The form of Hilbert’s problem: Given: Two multivariate polynomials of degree with all coefficients equal 1. Question: D have a natural solution?

  5. Reduction • For each variable in the equation we create a unary relation symbol

  6. For each monomial we use: • A constant • An -ary relation symbol

  7. Acorresponding to

  8. That sentence enforces And additionally the relation contains only tuples beginning with

  9. We add a -ary relation symbol and a sentence " is a 1-1 mapping From the set of all tuples in relations r Onto the set of all tuples in relations r

More Related