Undecidability Results Wolfgang Thomas Francqui Lecture, Mons, - PowerPoint PPT Presentation

Undecidability Results Wolfgang Thomas Francqui Lecture, Mons, April 2013

Fighting the Undecidable Wolfgang Thomas

Overview 1. Undecidability? 2. The grid 3. Defining addition and multiplication 4. Undecidability in weak arithmetics 5. Conclusion Wolfgang Thomas

Undecidability? Wolfgang Thomas

Example: Hilbert’s 10th Problem (1900) Given a Diophantine equation with any number of unknowns and with rational integral numerical coefficients: To devise a process (“Verfahren”) according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers. Wolfgang Thomas

Axel Thue (1863-1922) Wolfgang Thomas

The “First Tree” Wolfgang Thomas

Thue’s Problem (1910) Given two terms s , t and a set of xioms in the form of equations u ( x 1 , . . . , x n ) = v ( x 1 , . . . , x n ) decide whether from s one can obtain t in finitely many steps by applications of axioms. Thue’s suspicion: Eine L¨ osung dieser Aufgabe im allgemeinsten Falle d¨ urfte vielleicht mit un¨ uberwindlichen Schwierigkeiten verbunden sein. (A solution of this problem in the general case might perhaps be connected with insurmountable difficulties.) Wolfgang Thomas

Wolfgang Thomas

The Grid Wolfgang Thomas

The Infinite Grid The infinite grid is the structure G 2 = ( N × N , ( 0, 0 ) , S 1 , S 2 ) where S 1 ( i , j ) = ( i + 1, j ) , S 2 ( i , j ) = ( i , j + 1 ) Wolfgang Thomas

Undecidability of Monadic Grid-Theory The monadic second-order theory of the infinite grid is undecidable. Proof by reduction of the halting problem for Turing machines: For any TM M construct a sentence ϕ M of the monadic second-order language of G 2 such that M halts when started on the empty tape iff G 2 | = ϕ M . Wolfgang Thomas

Configurations of M Assume that M works on a left-bounded tape. A halting computation of M can be coded by a finite sequence of configuration words C 0 , C 1 , . . . , C m . We can arrange the configurations row by row in a right-infinite rectangular array: q 0 a 0 a 0 a 0 a 0 a 0 a 0 . . . a 1 q 1 a 0 a 0 a 0 a 0 a 0 . . . q 0 a 1 a 2 a 0 a 0 a 0 a 0 . . . a 3 q 2 a 2 a 0 a 0 a 0 a 0 . . . etc. Wolfgang Thomas

Describing an M -Run The sentence ϕ M will express over G 2 the existence of such an array of configurations. a 0 , . . . , a n are the tape symbols ( a 0 is the blank) q 0 , . . . , q k are the states of M , special halting state q s We use set variables X 0 , . . . , X n , Y 0 , . . . , Y k X i collects the grid positions where a i occurs, Y i collects the grid positions where state q i occurs. ϕ M : ∃ X 0 , . . . , X n , Y 0 , . . . , Y k ( Partition ( X 0 , . . . , Y k ) ∧ “the first row is the initial M -configuration” ∧ “a successor row is the successor configuration of the preceding one” ∧ “at some position the halting state is reached” ) Wolfgang Thomas

A Hidden Grid Consider the expansion of the tree T 2 by the two first-letter-adding functions: p 0 ( w ) = 0 · w , p 1 ( w ) = 1 · w The MSO-theory of ( T 2 , p 0 , p 1 ) is undecidable. Proof: Define the grid on the domain 0 ∗ 1 ∗ . Wolfgang Thomas

Another Hidden Grid Consider the binary tree with Equal-Level Predicate E E ( u , v ) : ⇔ | u | = | v | Obtain ( T 2 , E ) . The MSO-theory of ( T 2 , E ) is undecidable. Proof: Use E to define again the grid 0 ∗ 1 ∗ . Wolfgang Thomas

Path Logic over the Grid In path logic we have first-order quantifiers and set quantifiers ranging only over paths. The finite-path theory of G 2 is undecidable. [W. Th. Path logics with synchronization, in K. Lodaya et al., Perspectives in Concurrency Theory, IARCS, Universities Press, India, 2009] Idea: Transform 2-counter machine M into a finite-path sentence ϕ M such that M stops when started with counters ( 0, 0 ) iff G 2 | = ϕ M M -configuration: (instruction label, value of counter 1, value of counter 2) Wolfgang Thomas

A 2-Counter Machine M 1. if X 2 = 0 goto 5 2. decr ( X 2 ) 3. incr ( X 1 ) 4. goto 1 5. stop Configurations: ( 1, 3, 2 ) , ( 2, 3, 2 ) , ( 3, 3, 1 ) , ( 4, 4, 1 ) , . . . , , ( 5, 5, 0 ) Wolfgang Thomas

Desribing an M -Configuration over G 2 We use three paths Y , X 1 , X 2 ❄ ✲ ✲ ✲ ✲ ✲ ✲ ❄ ❄ ❄ S m S ℓ ❄ ❄ ❄ S n ❄ ❄ ❄ ✲ ❄ ❄ ❄ ✲ ❄ ❄ ✲ Coding configuration ( ℓ , m , n ) = ( 4, 2, 5 ) . Wolfgang Thomas

Update of Configuration Wolfgang Thomas

An Intermediate Summary MTh ( T 2 ) is decidable MTh ( T 2 , E ) is undecidable MTh ( G 2 ) is undecidable. PathTh ( G 2 ) is undecidable. We now show: ChainTh ( T 2 , E ) is decidable. [W. Th., Infinite trees and automaton definable relations over ω -words, TCS 103 (1992)] Wolfgang Thomas

Back to Tree with Equal-Level Predicate We consider a “path logic” over T 2 , or even over any regular tree equipped with the equal-level predicate. We call chain logic the fragment of MSO logic where all set quantifications are restricted to subsets of paths (“chains”). Wolfgang Thomas

Chain Logic over Regular Trees The chain theory of a regular (binary) tree with equal level predicate is decidable. Idea: Reduction to the MSO-theory of ( N , + 1 ) Code a chain C in ( T 2 , E , P ) by a pair ( α C , β C ) of ω -words over { 0, 1 } : α C is the sequence d 0 d 1 d 2 . . . of “directions” β C ( i ) = 1 iff d 0 . . . d i − 1 ∈ C A third sequence γ C signals membership of the reached vertices in P This result gives decidability of CTL ∗ -model-checking even when the “synchronization” via E is added. Wolfgang Thomas

Defining Addition and Multiplication Wolfgang Thomas

Quantification over Binary Relations By the results of G¨ odel, Tarski, Turing we know: The first-order theory of ( N , + , · , 0, 1 ) is undecidable. Already G¨ odel remarked in 1931: In the second-order language (with quantifiers over elements and relations) one can define define + and · in ( N , + 1 ) . Consequence: The second-order theory of ( N + 1 ) is undecidable. x + y = z iff ∀ R ([ R ( 0, x ) ∧ ∀ s , t ( R ( s , t ) → R ( s + 1, t + 1 ))] → R ( y , z )) Wolfgang Thomas

Adding Double Function to ( N , + 1 ) double ( x ) : = 2 x . Robinson 1958: The (weak) MSO-theory of ( N , + 1, double ) is undecidable. We follow a proof idea of Elgot and Rabin [JSL 31 (1966)]. Code a relation R = { ( m 1 , n 1 ) , . . . , ( m k , n k ) } by a set M R = { m ′ 1 < n ′ 1 < . . . < m ′ k < n ′ k } For each n we need an infinite set of code numbers. Take as codes of n all numbers 2 i · ( double ( n ) + 1 ) Wolfgang Thomas

Example R = { ( 2, 1 ) , ( 0, 2 ) } A code set M R contains 1 · 5 < 2 · 3 < 8 · 1 < 2 · 5 Wolfgang Thomas

A Remark There is an MSO-formula OddPos ( X , x ) that expresses X ( x ) in the < -listing of X -elements, x occurs on an odd position. Use ψ ( X , z , z ′ ) : X ( z ) ∧ X ( z ′ ) ∧ there is precisely one y between z , z ′ with X ( y ) OddPos ( X , x ) : ψ ∗ ( X , min ( X ) , x ) Next ( X , x , y ) says “in X , y is the next element after x Wolfgang Thomas

Definability of Decoding Let ϕ 2 ( z , z ′ ) : = double ( z ) = z ′ Then “ s is a code of x ”: ∃ y ( double ( x ) + 1 = y ∧ ϕ ∗ 2 ( y , s )) Translation of ∃ R ( R ( x , y ) . . . ) : ∃ X ( ∃ s ∃ t ( s is code of x ∧ t is code of y ∧ OddPos ( X , s ) ∧ Next ( X , s , t )) Wolfgang Thomas

A Sharper Result Let f : N → N be strictly increasing, f − id N be monotone and unbounded. Then MTh ( N , + 1, 0, f ) is undecidable. [W. Th., A note on undecidable extensions of monadic second order arithmetic, Arch math. Logik 17 (1975)] Wolfgang Thomas

Undecidability of Weak Arithmetics Wolfgang Thomas

Successor Structure + Unary Predicate Consider ( N , + 1, P ) χ P is the characteristic function of P χ P = 0 0 1 1 0 1 0 1 0 0 . . . Consequence of B¨ uchi’s analysis of MTh ( N , + 1 ) : For each monadic formula ϕ ( X ) one can construct a B¨ uchi (or Muller) automaton A ϕ such that ( N , + 1 ) | = ϕ [ P ] iff A ϕ accepts χ P . Acceptance Problem Acc ( P ) : uchi autoamaton A , does A accept χ P ? Given a B¨ Then MTh ( N , + 1, P ) is decidable iff Acc ( P ) is decidable. Wolfgang Thomas

The Prime Predicate P Can we decide for any B¨ uchi automaton A whether A accepts χ P = 0 0 1 1 0 1 0 1 . . . ? Wolfgang Thomas

Prime Numbers Decidability of MTh ( N , + 1, P ) (and even of FOTh ( N , + 1, < , P ) ) is open. Twin prime hypothesis TPH: � � ∀ x ∃ y x < y ∧ P ( y ) ∧ P ( y + 1 + 1 ) Dirchlet’s Theorem: Let A m , n : = { m + i · n | i ≥ 0 } If m , n are relatively prime, then | A m , n ∩ P | = ∞ For fixed m , n , this claim is expressible in MTh ( N , + 1, P ) Wolfgang Thomas

More on Arithmetical Progressions An arithmetic progression of length k in P is a sequence m , m + d , . . . , m + ( k − 1 ) · d of successive prime numbers B. Green, T. Tao (2006): For each k there are infinitely many arithmetical progressions of length k in P . Illustration (Frind, Underwood, Jobling (2004)): m = 56211383760397 , d = 44546738095860 , k = 22 Wolfgang Thomas