Download Decidability of Monadic Theories

Decidability of Monadic Theories
Wolfgang Thomas
Edinburgh, October 2014
Some Undecidability Results
Wolfgang Thomas
Background: Theories of Arithmetic
The FO-theory of (N, +, ·, 0, 1) is undecidable.
Presburger arithmetic, the FO-theory of (N, +, 0, 1), is
decidable.
Skolem arithmetic, the FO-theory of (N, ·, 0, 1) is
decidable.
“Büchi’s arithmetic”, the MSO-theory of (N, +1, 0), is
decidable.
What about the MSO-theory of (N, +, 0, 1)?
We show that even the special case of x 7→ x + x gives
undecidability.
Wolfgang Thomas
Recall
Adding relation quantifiers to the FO-language of (N, +1, 0)
gives undecidability:
x + y = z iff each binary relation that contains (0, x) and is
closed under (m, n) 7→ (m + 1, n + 1) contains also (y, z)
x · y = z iff each binary relation that contains (0, 0) and is
closed under (m, n) 7→ (m + 1, n + x) contains also (y, z)
Wolfgang Thomas
Double Function
R.M. Robinson 1958:
The MSO-theory of (N, +1, 2x, 0) is undecidable.
We follow a proof idea of Elgot and Rabin (1966).
Idea: Code binary relations by sets and simulate relation
quantifiers by set quantifiers.
R = {(m0 , n0 ), (m1 , n1 ), . . .} 7→
MR = {m0′ < n0′ < m1′ < n1′ . . .}.
(m, n) ∈ R iff
in the MR -list there are successive codes m′ , n′ of m, n where
m′ sits on an odd position.
Wolfgang Thomas
Defining the Code
For each x we need an infinite set Cx of code numbers.
such that from z ∈ Cx the number x can be retrieved (by an
MSO-formula).
Set Cx := {2i · (2x + 1) | i ≥ 0}.
From a number z = 2i · (2x + 1) obtain x:
iteratively divide by 2
if this is no more possible subtract 1 and divide by 2
Obtain an MSO-formula for “z is a code of x”
Wolfgang Thomas
Details
Let ϕ2 (y, y′ ) := y′ is half of y
Then
“z is a code of x”: ∃s(ϕ∗2 (z, s) ∧ s = 2x + 1)
Translation of ∃ R( R( x, y) . . .):
∃ X (∃z∃z′ (z is code of x ∧ z′ is code of y
∧OddPos( X, z) ∧ Next( X, z, z′ ))
Wolfgang Thomas
A Sharper Result
Let f : N → N be
strictly increasing,
f − idN 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
The Infinite Grid
The infinite grid is the structure
G2 = (N × N, (0, 0), S1 , S2 )
where S1 (i, j) = (i + 1, j), S2 (i, j) = (i, j + 1)
Wolfgang Thomas
Undecidability of Monadic Grid-Theory
The monadic second-order theory of the infinite grid is
undecidable.
Wolfgang Thomas
Another (Direct) 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 G2 such that
M halts when started on the empty tape iff G2 |= 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
C0 , C1 , . . . , Cm .
We can arrange the configurations row by row in a
right-infinite rectangular array:
q0
a1
q0
a3
etc.
Wolfgang Thomas
a0
q1
a1
q2
a0
a0
a2
a2
a0
a0
a0
a0
a0
a0
a0
a0
a0
a0
a0
a0
a0
a0
a0
a0
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Describing an M -Run
The sentence ϕ M will express over G2 the existence of such an
array of configurations.
a0 , . . . , an are the tape symbols (a0 is the blank)
q0 , . . . , qk are the states of M , special halting state qs
We use set variables X0 , . . . , Xn , Y0 , . . . , Yk
Xi collects the grid positions where ai occurs,
Yi collects the grid positions where state qi occurs.
ϕ M : ∃ X0 , . . . , Xn , Y0 , . . . , Yk (Partition( X0 , . . . , Yk )
∧ “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
Use of Interpretations
An interpretation of a structure A = ( A, RA , . . .) in a structure
B is a description of A in B
Here we use MSO for the description and speak of
MSO-interpretations.
Assume A is MSO-interpretable in B .
Then:
MTh(A) undecidable implies MTh(B) undecidable.
MTh(B) decidable implies MTh(A) decidable.
Wolfgang Thomas
Interpretations Formally
An MSO-interpretation of a structure A = ( A, RA , . . .) in a
structure B is given by
a “domain formula” ϕ( x)
for each relation RA of A, say of arity m, an MSO-formula
ψ ( x1 , . . . , x m )
such that A is isomorphic to (ϕB , ψB , . . .)
Then there is a transformation of MSO-sentences χ (in the
signature of A) to sentences χ′ (in the signature of B ) such
that
A |= χ iff B |= χ′ .
Wolfgang Thomas
Adding the Equal Level Relation
Consider the binary tree with equal level relation E
E(u, v)
:⇔
|u| = |v|. Obtain ( T2 , E).
The MSO-theory of ( T2 , E) is undecidable.
Proof by describing the grid in ( T2 , E):
Wolfgang Thomas
A Hidden Grid
Consider the expansion of the tree T2 by the two
first-letter-adding functions:
p0 (w) = 0 · w,
p1 ( w ) = 1 · w
The MSO-theory of ( T2 , p0 , p1 ) is undecidable.
Proof: Give interpretation of G2 in ( T2 , p0 , p1 )
Domain formula, using σi (z, z′ ) : zi = z′ (i = 0, 1)
ϕ( x) : ∃y(σ0∗ (ε, y) ∧ σ1∗ (y, x))
ψ1 ( x, y) : p0 ( x) = y, ψ2 ( x, y) : x1 = y
Wolfgang Thomas
Decidability Results
Wolfgang Thomas
A First Example
Show Rabin’s Tree Theorem for T3 = ({0, 1, 2}∗ , S03 , S13 , S23 ).
Idea: Describe T3 in T2 :
Consider the T2 -vertices in (10 + 110 + 1110)∗ .
Wolfgang Thomas
Another Interpretation
The MSO-theory of (Q, <) is decidable. (Rabin 1969)
Work with the tree nodes w01
With the lexicographic order they give a countable dense
linear order.
This order is isomorphic to (Q, <) (Cantor)
So we have an MSO-interpretation of (Q, <) in T2 .
Much more difficult: MTh(R, <) is undecidable. (Shelah 1975)
Wolfgang Thomas
A Pushdown Graph
q0 ⊥
q1 ⊥
Wolfgang Thomas
a
b
b
q0 Z ⊥
q1 Z ⊥
a
b
b
q0 ZZ ⊥
q1 ZZ ⊥
a ...
b
b ...
Describing a pushdown graph in a tree
ε
⊥
Z⊥
ZZ ⊥
q0 ZZ ⊥
(q0 , Z, ZZ, q1 )
Wolfgang Thomas
ZZZ ⊥
q1 ZZZ ⊥
MSO-theory of a pushdown graph
Each pushdown graph can be obtained from a tree Tk by an
MSO-interpretation.
The MSO-theory of a pushdown graph is decidable.
(Muller Schupp 1985)
Wolfgang Thomas
Unfoldings
Given a graph (V , ( E a ) a∈Σ , (Pb )b∈Σ′ )
the unfolding of G from a given vertex v0 is the following tree
TG (v0 ) = (V ′ , ( E′a ) a∈Σ , (Pb′ )b∈Σ′ ):
V ′ consists of the vertices v0 a1 v1 . . . ar vr with
(vi−1 , vi ) ∈ E ai ,
E′a contains the pairs (v0 a1 v1 . . . ar vr , v0 a1 v1 . . . ar vr av)
with (vr , v) ∈ E a ,
Pb′ the vertices v0 a1 v1 . . . ar vr with vr ∈ Pb .
Wolfgang Thomas
Example
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Andrej Muchnik (1958 - 2007)
If the MSO-theory of a graph G is decidable and v0 an
MSO-definable vertex of G, then also the MSO-theory of the
unfolding of G from v0 is decidable.
Wolfgang Thomas
Proof Architecture (for Pushdown Graphs)
Given an unfolding T of a pushdown graph G.
T is finitely branching, with labels say in Σ inherited from G.
For each MSO-formula ϕ( X1 , . . . , Xn ) find a parity tree
automaton Aϕ such that
Aϕ accepts T (P1 , . . . , Pn ) iff T (P1 , . . . , Pn ) |= ϕ( X1 , . . . , Xn )
The construction of the Aϕ follows precisely the pattern of
Rabin’s equivalence theorem.
Essential: In the complementation step we use the finite
out-degree of G.
For sentences, one checks acceptance of the unlabelled
unfolding, which is done by invoking the MSO-theory of G
Wolfgang Thomas
Caucal’s Proposal
We have now two processes which preserve decidability of
MSO-theory:
interpretation (transforming a tree into a graph)
unfolding (transforming a graph into a tree)
Let us apply them in alternation!
We obtain the Caucal hierarchy or pushdown hierarchy.
Wolfgang Thomas
Definition
T0 = the class of finite trees
Gn = the class of graphs which are MSO-interpretable in a
tree of Tn
Tn+1 = the class of unfoldings of graphs in Gn
Each structure in the pushdown hierarchy has a decidable
MSO-theory.
Nontrivial fact:
The sequence G0 , G1 , G2 , . . . is strictly increasing.
Wolfgang Thomas
The First Levels
G0 is the class of finite graphs.
T1 contains the regular trees.
G1 contains the prefix-recognizable graphs.
Wolfgang Thomas
Towards Factorial Predicate
(N, Succ, Fac)
We start with the following pushdown graph,
obtained from a finite tree by an interpretation, an unfolding,
and another interpretation:
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Continuation: Unfolding and Interpretation
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···
Obtaining (N, +1, 0, Fac)
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On Structures (N, +1, 0, P ) with P ⊆ N
1. (N, +1, 0, P ) belongs to G1 iff P is ultimately periodic.
2. (N, +1, 0, P ) belongs to G2 iff P is morphic.
What about P √2 , Pπ , P?
Wolfgang Thomas
An Example
2
Pexp := set of iterated 2-powers 1, 2, 22 , 22 , etc.
1. (N, +1, 0, Pexp ) is outside the pushdown hierarchy.
2. MSO-Th(N, +1, 0, Pexp ) is decidable.
Method for 2:
The MSO-theory of (N, +1, 0, P ) is decidable iff the following
“acceptance problem for Büchi automata” is decidable:
Given a Büchi automaton A over {0, 1}, does A accept αP ?
(Here αP is the 0-1-sequence with α(i ) = 1 iff i ∈ P.)
Wolfgang Thomas
Outlook
The pushdown hierarchy is a very rich class of structures all of
which have a decidable MSO-theory.
Some open questions:
Understand which structures belong to the hierarchy.
Compute the smallest level on which a structure occurs.
Find ways to extend the hierarchy while keeping
decidability of the MSO-theory.
How to obtain a richer class of models when restricting to
a weaker logic, e.g. monadic transitive logic?
Wolfgang Thomas