Word equations and EDT0L languages: from logic to automata Dedicated to Manfred Kudlek (1940 – 2012) Volker Diekert 1 Universit¨ at Stuttgart AutoMathA 2015 Leipzig, May 6 - 9, 2015 1 Based on joint work with: Laura Ciobanu and Murray Elder (ICALP 2015); and with Artur Je˙ z and Wojciech Plandowski (CSR 2014). The corresponding papers are on the arXiv
Manfred Kudlek (1940 – 2012) The main result in this paper will be presented at ICALP 2015. The 42nd International Colloquium on Automata, Languages, and Programming (ICALP 2015) will take place in the period 6-10 July 2015 in Kyoto, Japan. Professor Kudlek has the distinction of being the only person to have attended all ICALP conferences during his lifetime. He worked on Lindenmayer systems, visited Kyoto several times, and taught the speaker that bikes are the best means of transport inside Kyoto. 1
PART I 2
MORE THAN 1700 YEARS OF WORD EQUATIONS Diophantus of Alexandria. Greek mathematician before 364 AD. 1900: Hilbert’s address at the International Congress of Mathematicians in Paris leading 1901 to a list 23 published problems. The Tenth Problem is: “Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.” 3
History 1960’s: WordEquations is a special instance of Hilbert 10. 1970 Matiyasevich: Hilbert 10 is undecidable based on previous work by Davis, Putnam, and Robinson. 1977 Makanin: WordEquations is decidable. 1982/84 Makanin: Existential and positive theories of free groups are decidable. 1987 Razborov: Description of all solutions for an equation in a free group via “Makanin-Razborov” diagrams. 1998 – 2006: Tarski’s conjectures: Kharlampovich and Myasnikov and independent work of Sela 4
Complexity of Makanin’s algorithms WordEquations. Complexity (first published estimation): 2 2 222poly( n ) � � DTIME GroupEquations. The scheme of Makanin is not primitive recursive. (Ko´ scielski/Pacholski 1990) 1999 Plandowski: WordEquations is in PSPACE. 2000 Guti´ errez: GroupEquations is in PSPACE. 2001 D., Guti´ errez, Hagenah: GroupEquations with rational constraints is PSPACE-complete. 5
From Lempel-Ziv Compression to recompression ICALP 1998. Plandowski and Rytter: Application of Lempel-Ziv Encodings to the Solution of Word Equations. New conjecture: WordEquations is NP-complete. Compression became a main tool in solving equations. STACS 2013. Artur Je˙ z applied recompression to WordEquations and simplified all known proofs (!!!) for decidability. Consequence. A full proof can lectureed in about 2 hours. 6
PART II All Solutions 7
From logic to automata Let Φ be some formula, say written in MSO, FO, LTL, . . . Models are sets in some domain, say words, trees, graphs, . . . Question. How to check satisfiability of Φ ? AutoMathA. Construct a trim nondeterministic finite automaton A such that L ( A ) = { w | w | = Φ } . The NFA A tells us: The formula Φ is not satisfiable if L ( A ) is empty. The formula Φ has only finitely many models if L ( A ) is finite. The formula Φ has infinitely many models if L ( A ) is infinite. 8
The existential theory of equations for a monoid M Let A be a finite set of generators and π : A ∗ → M be the canonical representation. Consider a first order sentence over atomic formulas U = V , where U, V ∈ ( A ∪ Ω) ∗ are words over the constants A and variables Ω . The interpretation is in M . This leads to the existential theory of equations in M , positive theory of equations in M , theory of equations in M . free monoids free groups Existential theory Makanin 1977 Makanin 1982 Positive theory undecidable Makanin/Razborov 1984 Theory undecidable Kharlampovich/Myasnikov 2004 9
The existential theory with rational constraints We allow additional atomic formulas of the form X ∈ L , where L ⊆ M is rational. For simplicity, M = A ∗ a free monoid with involution. a = a and xy = y x. So in this case being rational means we can specify L by some NFA A with L = L ( A ) ⊆ A ∗ and the truth value of X ∈ L for σ : Ω → A ∗ becomes σ ( X ) ∈ ( L ( A )) . 10
NFAs and rational subsets Let M be any monoid, eg. either M = F ( A ) or M = C ∗ or M = End( C ∗ ) . A nondeterministic finite automaton (NFA) over M is a finite directed graph A with initial and final states where the arcs are labeled with elements of M . Reading the labels of paths from initial to final states defines the accepted language L ( A ) ⊆ M . Definition L ⊆ M is rational if L = L ( A ) for some NFA. Rational = regular for f.g. free monoids. In general, rational sets are not closed under intersection. Benois (1969): Rational sets in free groups form a Boolean algebra. 11
Automata theoretical approach Consider a Boolean formula Φ of equations with rational constraints. The models of Φ are σ : Ω → A ∗ such that σ (Φ) is true. (For a free group we wish that σ ( X ) is a (freely) reduced word.) We have the logic, now we need an NFA! This leads us to EDT0L and Lindenmayer systems. a a Lindenmayer (1925 – 1989) 12
Ciobanu, D., Elder (ICALP 2015). The result is an easy to understand algorithm for the following problem. Input. An equation U = V in variables X 1 , . . . , X k with rational constraints in free groups (resp. free monoids with involution). Ouput. An NFA A such that A accepts a rational language R of endomorphisms over C ∗ . A ⊆ C . The alphabet C is of linear size in the input. The set of all solutions σ in reduced words is { ( σ ( X 1 ) , . . . , σ ( X k )) ∈ A ∗ × · · · × A ∗ | σ ( U ) = σ ( V ) } = { ( h ( c 1 ) , . . . , h ( c k )) ∈ C ∗ × · · · × C ∗ | h ∈ R } where c 1 , . . . , c k ∈ C are letters. Hence, { ( σ ( X 1 )# · · · # σ ( X k )) ∈ ( A ∪ #) ∗ | σ ( U ) = σ ( V ) } is an EDT0L language of reduced words. This was previously known only for quadratic word equations by Fert´ e, Marin, S´ enizergues (2014). 13
The finite monoid N keeping the words reduced Define N = { 1 , 0 } ∪ A × A to “remember first and last letters” � 0 if b = c ( a, b ) · ( c, d ) = ( a, d ) b � = c. The monoid N has an involution by ( a, b ) = ( b, a ) . Fix the morphism µ 0 : A ∗ → N given by µ 0 (#) = 0 and µ 0 ( a ) = ( a, a ) otherwise. µ 0 respects the involution. µ 0 ( w ) = 0 if and only if either w is not reduced or contains # . 14
A single word Starting point: Define # = # . Replace U = V by a single word – almost a palindrome. W init = # X 1 · · · # X k # U # V # U # V # X k # · · · X 1 # . ∈ � a ∈ A A ∗ aaA ∗ via a Introduce rational constraints σ ( X ) / morphism µ : A ∗ → N where N is the finite monoid above with zero 0 = µ (#) . This ensures that solutions are in reduced words and do not use # . Definition A solution of a word W ∈ ( A ∪ Ω) ∗ is a morphism σ : Ω → A ∗ such that σ ( W ) = σ ( W ) . That is a palindrome! µσ ( X ) � = 0 for all X ∈ Ω , ie. σ ( X ) is reduced without # . 15
Preludium The guiding example is a linear Diophantine system Solve AX = b, where A ∈ Z n × n , X = ( X 1 , . . . , X n ) T , b ∈ Z n × 1 . The X i are variables over natural numbers. We measure the complexity w.r.t to: � � n, � b � 1 = | b i | , � A � 1 = | a ij | . i i,j 16
How to solve linear Diophantine systems by compression? We let � V � 1 be the 1 -norm for vectors and matrices V . In particular, � � � b � 1 = | b i | , � A � 1 = | a ij | . i i,j Without restriction we assume that � b � 1 ≤ � A � 1 . We define the compression method with respect to a given solution x ∈ N n × 1 . Attention: Of course the algorithm does not know the solution. We do nondeterministic guesses! Think of a finite graph where the vertices are all linear Diophantine systems satisfying a certain a priori space bound. The arcs transform the these systems. The space bound will depend on the system, but not on the solution. If there is no solution then the algorithm cannot terminate. This is soundness. Fix any given solution x ∈ N X . If x = 0 we do nothing. Otherwise we repeat the following rounds until x = 0 . Then we stop. Each round consists of the following steps. 17
While x � = 0 do 1 For all i define x ′ i = x i − 1 if x i is odd and x ′ i = x i otherwise. Thus, all x ′ i are even. Rewrite the system with a new vector b ′ such that Ax ′ = b ′ . Note that � b ′ � 1 ≤ � b � 1 + � A � 1 . 2 Now, all b ′ i must be even. Otherwise we made a mistake and x was not a solution. 3 Define b ′′ i = b ′ i / 2 and x ′′ i = x ′ i / 2 . We obtain a new system AX = b ′′ with solution Ax ′′ = b ′′ . 4 The clou: since � b � 1 ≤ � A � 1 we obtain � � b ′′ � � � b ′ � 1 = 1 / 2 ≤ � b � 1 / 2 + � A � 1 / 2 ≤ � A � 1 . � � 5 Rename b ′′ and x ′′ as b and x . 18
Why does it work? The compression method defines a path in the finite set of systems b + AX = Y where A is fixed and � b � 1 ≤ � A � 1 . There are at most � A � n 1 such systems. 19
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