Th(N, +) is decidable

Transcription

1 Theorem 6.12 Th(N, +) is decidable Presented by: Brian Lee

2 Two Domains 1. We can give an algorithm to decide truth 2. A problem is undecidable

3 First Order Logic Also known as First order predicate calculus Uses quantified variables over non-logical objects 1. Infinitely many primes exist. 2. Fermat s Last Theorem 3. Twin prime conjecture

4 First Order Logic Treat the logical statements as strings and define a language consisting of those statements are true.

5 Terminology Boolean operations, quantifiers, variables, and relations. Formula Atomic Formula Prenex Normal Form Sentences/Statement Free variables Universe Model Language of a Model theory of M - Th(M)

6 Boolean Operators Relations (R1,...,Rk) atomic formula: a string of the form R(x1,...,xn) arity: the number of arguments for relation R(x1,...,xn) is a relation taking variables as arguments x1,...,xn are arguments for the R and returns true/false R: x1,...xn -> {TRUE, FALSE}

7 Formulas Well formed strings over boolean alphabet(a set of strings part of a formal language) It must have the requirements:

8 Sentences A formula without any free variables Prenex Normal Form Quantifiers are in front of a sentence 1. x2 & x3 are free variables 2. x2 & x3 are free variables 3. Sentence

9 Variables and Relations First we need to specify 2 things Universe over where variables may take values Assignment of specific relations to relation symbols arity of a relation symbol must match that of its assigned relation

10 Model A universe with assignment of relations to relation symbols We say that a model M is a tuple of (U,P1,...,Pk) P1,...,Pk are relations assigned to relation symbols R1,...,Rk Language of a Model- collection of formulas that use only the relation symbols the model assigns that use each symbol with the correct arity Phi is a sentence in the language of a model has the value {TRUE,FALSE} if TRUE in model M, then M is a model of Phi

11 Example 6.10

12 Example 6.10 Summary M = (N,<=) R1(x,y) is (x <= y) Phi says All natural numbers x and y, there is x >= y or x <= y This sentence is true Using less than <, this sentence will not be in the model because if x == y, then R1 will not work

13 Example 6.11

14 Example 6.11 Universe is real numbers R1 is PLUS(a,b,c) = TRUE whenever a + b =c The sentence phi is true using N instead of R(real) universe would not work. We can represent functions like addition by relations.

15 Theory of M Written as Th(M): This is the collection of true sentences in the language of the model. So any phi that works for all values in the universe. In example 6.10: For Model M = (N, <=) [R1(x,y) or R1(y,x)] is inth(m)

16 Th(N, +) is decidable What this means theory of model (N,+) is decidable for the universe N. Is Decidable, proof will be shown in the next slides.

17 Proof Idea We use the finite automata that is capable of doing addition if the input is presented in a special form. The special form is 3 bit value that represents 8 different characters. Instead we use this form in general for i

18 Proof Idea Let Q s - are quantifiers Psi is a formula without quantifiers has variables: x1,...,xl each quantifier has a variable For each i from 0 to L define:

19 Proof Idea Also each Phi can take arguments a1,...,an from Natural Numbers writing and substitute them for their respective Xi s.

20 Proof Idea For each i from 0 to L, Create a finite automaton Ai that recognize strings that represent i-tuples of numbers that make phi(i) true. First construct A(L) directly, iterate i from L to 0 The construct A(i-1) using Ai Check A0 accepts empty string if it does, phi is true and algorithm accepts

21 Proof i is the size of the column [] is also a symbol ai represents a column

22 Algorithm to Decide Th(N,+) AL - phil = psi Addition is the atomic formula we are trying to prove is in N FA can be created to compute any of the individual relations use regular language operations to compute the boolean combinations of the atomic formulas

23 Algorithm to Decide Th(N,+) On input phi, where phi is a sentence Write phi, and define phi(i) for each i from 0 to L. For each i, create finite automaton Ai from phi(i) that accepts strings over whenever phi(a1,...,ai) is true

24 Construct A i from A i+1

25 Construct A i from A i+1 Every time Ai reads Fig. 1 It nondeterministically guesses z and simulates Ai+1(Fig. 2)

26 Construct A i from A i+1

27 Construct A i from A i+1 Ai nondeterministically guesses the leading bit of a i+1 corresponding to leading 0s in a1 through ai by branching with empty strings it s new start state to all states Ai+1 can reach from its start state with the input strings of the symbols

28 Final Step A0 accepts any input iff phi(0) is true Final Step: algorithm test if A0 accepts empty string if it does, phi is true and it accepts, else reject

29 Questions?