Giorgi Japaridze Theory of Computability Decidability of logical
Giorgi Japaridze Theory of Computability Decidability of logical theories Section 6. 2
6. 2. a Giorgi Japaridze Theory of Computability The language of arithmetic: formulas Formulas --- strings produced by the following CFG: FORMULA ATOM | (FORMULA) | (FORMULA) | VARIABLE (FORMULA) ATOM TERM = TERM VARIABLE | CONSTANT | (TERM) + (TERM) VARIABLE v | VARIABLE’ CONSTANT 0 | 1 CONSTANT negation conjunction disjunction universal quantifier
6. 2. b Giorgi Japaridze Theory of Computability The language of arithmetic: sentences An occurrence of variable x is bound in formula F, if it is in the scope of x, i. e. F=. . . x( … x …) … Otherwise it is free. A sentence is a formula without free occurrences of variables Is v’ free or bound in: Is the following formula a sentence: v’=0 (0=10) ( v’(v’=0)) v=0 ( v’’(v’=0)) v(v=v) v’’(v’=v’’) v(v=v’) v’((v’=0) (v’=v)) v(v=1 v’(v+v’=110)) ( v’(v’=0)) (v’=v)
6. 2. c Giorgi Japaridze Theory of Computability Truth of arithmetic sentences • An atomic sentence is true iff it is true under the standard interpretation of constants and +, , =. • A is true iff A is false • A B is true iff both A and B are true • A B is true iff either A or B (or both) are true • x. A(x) is true iff for all constants c, A(c) is true A(c) --- the result of substituting all free occurrences of x by c in A(x) 10+10=10 10 v(v+v=v v) v (v 1=v) v( (v+v=v)) v(v=0 (v+v=v)) v v’(v+v’=v’+v) v v’(v (v’+1)=(v’ v)+v)
6. 2. d Giorgi Japaridze Theory of Computability The undecidability of truth for arithmetic sentences Let Th(N, +, ) = {A | A is a true arithmetical sentence} Theorem 6. 13: Th(N, +, ) is undecidable. Corollary: Th(N, +, ) is not Turing recognizable, either. Proof: Suppose a TM M recognizes Th(N, +, ). Construct a TM D: D = “On input A, an arithmetic sentence, 1. Run M on both A and A in parallel. 2. If M accepts A, accept; if M accepts A, reject” Obviously D decides Th(N, +, ) , which is in contradiction with Theorem 6. 11. Let Th(N, +) = {A | A is a true arithmetical sentence not containing } Theorem 6. 12: Th(N, +) is decidable.
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