Word equations and EDT0L languages: from logic to automata - - PowerPoint PPT Presentation
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
Dedicated to Manfred Kudlek (1940 – 2012)
Universit¨ at Stuttgart
AutoMathA 2015 Leipzig, May 6 - 9, 2015
1Based 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
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
the speaker that bikes are the best means of transport inside Kyoto.
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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
“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.”
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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
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DTIME
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.
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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.
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All Solutions
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Let Φ be some formula, say written in MSO, FO, LTL, . . . Models are sets in some domain, say words, trees, graphs, . . .
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.
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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
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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)).
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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.
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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
aLindenmayer (1925 – 1989)
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easy to understand algorithm for the following problem.
constraints in free groups (resp. free monoids with involution).
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(X1), . . . , σ(Xk)) ∈ A∗ × · · · × A∗ | σ(U) = σ(V ) } = { (h(c1), . . . , h(ck)) ∈ C∗ × · · · × C∗ | h ∈ R } where c1, . . . , ck ∈ C are letters. Hence, { (σ(X1)# · · · #σ(Xk)) ∈ (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).
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Define N = {1, 0} ∪ A × A to “remember first and last letters” (a, b) · (c, d) = if b = c (a, d) b = c. The monoid N has an involution by (a, b) = (b, a). Fix the morphism µ0 : A∗ → N given by µ0(#) = 0 and µ0(a) = (a, a) otherwise. µ0 respects the involution. µ0(w) = 0 if and only if either w is not reduced or contains #.
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Starting point: Define # = #. Replace U = V by a single word – almost a palindrome. Winit = #X1 · · · #Xk#U#V #U#V #Xk # · · · X1 #. Introduce rational constraints σ(X) / ∈
a∈A A∗aaA∗ via a
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 #.
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The guiding example is a
Solve AX = b, where A ∈ Zn×n, X = (X1, . . . , Xn)T , b ∈ Zn×1. The Xi are variables over natural numbers. We measure the complexity w.r.t to: n, b1 =
|bi|, A1 =
|aij|.
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We let V 1 be the 1-norm for vectors and matrices V . In particular, b1 =
|bi|, A1 =
|aij|. Without restriction we assume that b1 ≤ A1. We define the compression method with respect to a given solution x ∈ Nn×1. Attention: Of course the algorithm does not know the
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
Fix any given solution x ∈ NX. 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.
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1 For all i define x′
i = xi − 1 if xi is odd and x′ i = xi otherwise.
Thus, all x′
i are even. Rewrite the system with a new vector b′
such that Ax′ = b′. Note that b′1 ≤ b1 + A1.
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 b1 ≤ A1 we obtain
5 Rename b′′ and x′′ as b and x.
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The compression method defines a path in the finite set of systems b + AX = Y where A is fixed and b1 ≤ A1 . There are at most An
1 such systems.
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Equation (U, V ). Solution σ(U) = σ(V ) with m huge.
X Y X P
b2 a b b2 a b2 a b4 b b2m+3 a b5 a
Z Q
Maximal visible blocks from left to right: b4, b3, b2, b2m+8. Replace all b in these blocks by c.
X Y X P
c2 a c c2 a c2 a c4 c c2m+3 a b5 a
Z Q
Split Y = Y ′Y , P = P ′P, Q = Q′Q. Type θ(X) = θ(Y ′) = θ(P ′) = θ(Q′) = c.
X Y ′ Y X P ′ P
c2 a c c2 a c2 a c4 c c2m+3 a b5 a
Y Z Q′ Q
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X Y ′ Y X P ′ P
c2 a c c2 a c2 a c4 c c2m+3 a b5 a
Y Z Q′ Q
Pop c2 from typed variables. Thus, Y ′ vanishes.
X Y X P ′ P
c2 a c c2 a c2 a c2 c2 c3 c2m+1 a b5 a
Z Q′ Q
Use Y = a and let c4,c3,c2,cµ fresh letters.
X X P ′ P
c2 a c3 c2 a c2 c a cµ c2 c4 c2m+1 a b5 a
Q′ Q
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X X P ′ P
c2 a c3 c2 a c2 c a cµ c2 c4 c2m+1 a b5 a
Q′ Q
Pop P ′ = cP ′. Compress c4c → c4, c2c → c2, cµc → cµ.
X X P ′ P
c2 a c3 c2 a c2 a cµ c2 c4 c2m a b5 a
Q′ Q
Compress c2 → c. This halves the exponents!
X X P ′ P
c a c3 c a c2 a cµ c c2 cm a b5 a
Q′ Q
Repeat until the red c vanish . . .
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1 Forget the involution and the rational constraints. 2 Construct the NFA A using simple rules. 3 Prove soundness. 4 Prove completeness using (a modified) Je˙
z compression.
5 Modification uses the compression method for linear
Diophantine system for “block compression”.
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Let C be a fixed extended alphabet with A ⊆ C and |C| ∈ O(|Winit|). A ⊆ B ⊆ C and X ⊆ Ω. A type is a partial mapping θ : (B ∪ X) \ A → B which defines a free partially commutative monoid M(B ∪ X, θ) = (B ∪ X)∗/ { θ(x)x = xθ(x) | x ∈ B ∪ X } . M(B) denotes the submonoid of M(B ∪ X, θ) generated by B. We have A∗ ⊆ M(B) since θ(a) is not defined for a ∈ A. M(B) and M(B ∪ X, θ) are trace monoids in the sense of Mazurkiewicz.
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States are (W, B, X, θ) W = equation, the solution σ is a “palindrome” σ(W) = σ(W). B = constants with # ∈ A ⊆ B = B ⊆ C. X = variables in W. θ = partial commutation Arcs change these parameters. Following arcs allows to find all solutions.
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1 τ(X) = 1, remove X from W. Potentially removes partial
commutation.
2 τ(X) = aX, where a is a constant. 3 τ(X) = Y X, split X as Y X and define a type θ(Y ) = a,
where a is a constant. After that Y commutes with a. This commuting relation is used for compressing blocks aℓ into a single fresh letter aℓ.
4 There are symmetric rules for the right side.
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Two rules, only.
1 Rename a as b. 2 Compress ab into a single letter c. This includes compressions
ab → a, ab → b, aa → a. Arcs (h(W ′), B, X, θ)
h
− → (W ′, B′, X, θ′) change the constants. Compression ab → c yields label h ∈ End(C∗) with h(c) = ab which induces a morphism h : M(B′) → M(B).
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Definition A state of A is a tuple P = (W, B, X, θ) such that: W ∈ M(B ∪ X, θ). |W| ∈ O(|Winit|). W is called the equation at P. Initial states (Winit, A, Ω, ∅) Final states (W, B, ∅, ∅) with W = W and #c1 · · · #ck is a prefix of W.
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Soundness of A Let h1 · · · ht be the labels of a path from an initial state P0 = (Winit, A, Ω, ∅) to a final state (W, B, ∅, ∅). Then σ(Xi) = h1 · · · ht(ci) defines a solution at P0.
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Theorem The graph A can be constructed deterministically in singly exponential time via some NSPACE(n log n) algorithm which
final vertices. The NFA A satisfies the soundness property, i.e., the corresponding EDT0L language is a subset of solutions in reduced words. Proof. The complexity statement is trivial by standard methods. Soundness was stated above.
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By soundness of the NFA A it remains to prove the following purely existential statement Theorem Let (idA∗, σ) be a solution at an initial vertex (Winit, A, Ω, ∅). Then there exists a path inside A to a some final vertex. Proof. Iterate block compression and pair compression based on the method of Je˙ z presented at STACS 2013. Details are on arXiv.
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By the soundness of the NFA A it remains to prove the following purely existential statement: Theorem Let (idA∗, σ) be a solution at an initial vertex (Winit, A, Ω, ∅). Then there exists a path inside A to some final vertex. Proof. Iterate block compression and pair compression based on the method of Je˙ z presented at STACS 2013. Details are on arXiv.
This is the end. Thank you.
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