Decidability of logical theories

1
Decidability of logical theories
CSC 4170 Theory of Computation
Section 6.2
2
The language of arithmetic formulas
6.2.a
Giorgi Japaridze Theory of Computability
Formulas --- strings produced by the
following CFG
FORMULA ? ATOM
?(FORMULA) (FORMULA) ?
(FORMULA) (FORMULA) ?
(FORMULA) ? VARIABLE
(FORMULA) ATOM ? TERM TERM TERM ? VARIABLE
CONSTANT
(TERM) (TERM) (TERM) ?
(TERM) VARIABLE ? v VARIABLE CONSTANT ? 0
1 1CONSTANT
negation conjunction disjunction universal
quantifier



3
The language of arithmetic sentences
6.2.b
Giorgi Japaridze Theory of Computability
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 v0 ? (?v(v0)) ?
(?v(v0)) ?v(vv) ?v((v0)
?(vv)) (?v(v0))?(vv)
Is the following formula a sentence ?(010) v0
?v(vv) ?v(vv) ?v(v1 ??v(vv110))



4
Truth of arithmetic sentences
6.2.c
Giorgi Japaridze Theory of Computability

• 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
• ?xA(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)

101010?10 ?v(vvv?v) ?v (v?1v) ?v(?(vvv)) ?v
(v0??(vvv)) ?v?v(vvvv) ?v?v(v?(v1)(v
?v)v)


5
The undecidability of truth for arithmetic
sentences
6.2.d
Giorgi Japaridze Theory of Computability
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,
and 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.