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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

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Trachtenbrot’s Theorem

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  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

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