Th(N, +) is decidable Presented by: Brian Lee Two Domains 1. We - - PowerPoint PPT Presentation

Th(N, +) is decidable

Presented by: Brian Lee Theorem 6.12

Two Domains

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

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

First Order Logic

• Treat the logical statements as strings and define a language consisting of

those statements are true.

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)

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}

Formulas

• Well formed strings over boolean alphabet(a set of strings part of a formal

language)

• It must have the requirements:

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

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

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

Example 6.10

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

Example 6.11

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.

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)

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

Th(N, +) is decidable

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

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:

Proof Idea

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

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

Proof

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

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

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

• ver

whenever phi(a1,...,ai) is true

Construct Ai from Ai+1

Construct Ai from Ai+1

• Every time Ai reads Fig. 1
• It nondeterministically guesses z and simulates Ai+1(Fig. 2)

Construct Ai from Ai+1

Construct Ai from Ai+1

• Ai nondeterministically guesses the leading bit of ai+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

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

Questions?