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

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

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)

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.

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

Construct A i from A i+1

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

Construct A i from A i+1

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

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?