Atoms = (N,=)
a,b,c,… - denote atoms
Aut(Atoms) - all permutations
Examples:
Examples:
A set expression can be:
{{y1}∪{y2}∪{y3}∣(y1,y2)∈Atoms2, y1≠y2∧y1≠y3∧y2≠y3}
Valuation: y3↦a
{S⊆Atoms∣|S|=3,a∈S}
Examples:
A set expression can be:
Aut(Atoms) acts on a set with atoms by atom renaming.
Fact. Every definable set with atoms has finitely many orbits under the action of Aut(Atoms).
Examples:
Alphabet: Atoms |
States: {⊤,⊥}∪Atoms |
Transitions: {(⊥,a,a),(a,b,a),(a,a,⊤)∣(a,b)∈Atoms2} |
[Bojańczyk, Klin, Lasota]
Determinization fails
Minimisation works (syntactic automaton)
Emptiness is decidable (PSPACE-complete)
"some letter appears twice"
Powerset construction does not work.
Definable pushdown systems - reachability is decidable [Murawski, Ramsay, Tzevelekos].
Definable Petri nets - reachability is open.
Alphabet: Atoms(2)
States: {I,A,R}∪Atoms(2)
Turing machine with atoms is a Turing machine whose alphabet, state space, and transition relation are definable sets with atoms.
An alphabet A is standard iff every language L over A recognisable by a nondeterministic TMA is recognisable by some deterministic TMA.
Theorem [Bojańczyk, Klin, Lasota, Toruńczyk]. There exists a non-standard alphabet.
Backtracking does not work.
No definable function
from {{a,b}∣(a,b)∈Atoms2} to Atoms.
Theorem [Klin, Lasota, O., Toruńczyk]. It is decidable if an alphabet A is standard.
An alphabet A is standard iff every language L over A recognisable by a nondeterministic TMA is recognisable by some deterministic TMA.
Theorem [Bojańczyk, Klin, Lasota, Toruńczyk]. There exists a non-standard alphabet.
Backtracking does not work.
No definable function
from {{a,b}∣(a,b)∈Atoms2} to Atoms.
Theorem [Klin, Lasota, O., Toruńczyk]. It is decidable if an alphabet A is standard.
Logic | Expresses | Fails to express |
FO | "there is a loop" | "there is a path from p to q" |
Datalog | "there is a path from p to q" | 2-colorability |
LFP | 2-colorability | evenness |
LFP+counting | evenness | ? |
Question: Is there a logic that expresses exactly PTime properties of graphs?
A={({a,b,c,d},{(a,c),(b,d)})∣(a,b,c,d)∈Atoms4, ϕ(a,b,c,d)}
Each alphabet of graphs A gives rise to a computational problem IA - isomorphism of "patched" graphs.
Theorem [Furst, Hopcroft, Luks]. Each problem IA is solvable in PTime (bounded color classes).
Theorem [Klin, Lasota, O., Toruńczyk]. The problem IA is expressible in LFP+C iff A is standard.
Corollary [Cai, Fürer, Immerman]. LFP+C does not capture PTime.
Key observation: Over "patched" graphs LFP+C = polynomial time (deterministic) TMAs.
Question: Does CPT express all PTime properties of graphs?
"Easier" question: Does it express the isomorphism problem on graphs with bounded colour class size, i.e., IA for any A?
(Over "patched" graphs LFP+C = polynomial time deterministic TMAs.)
Idea: Allow some kind of nondeterminism in TMA to capture CPT over "patched" graphs.
Vertices: Atoms(2) |
Edges:{{(a,b),(b,c)}∣(a,b,c)∈Atoms3,ϕ(a,b,c)} |
Problem: CSP(B)
Input: structure A (over the same signature)
Decide: Is there a homomorphism from A to B?
B=({0,1},R1,R0)
R0={(x,y,z)∈{0,1}3 | x+y+z=0 mod 2}
R1={(x,y)∈{0,1}2 | x+y=1 mod 2}
x+y+z=0x+y=1
A=({x,y,z},R0(x,y,z),R1(x,y))
B=({0,1},R000,R100,R110,R111)
R000={0,1}3∖{(0,0,0)}
R100={0,1}3∖{(1,0,0)}
R110={0,1}3∖{(1,1,0)}
R111={0,1}3∖{(1,1,1)}
¬x∧y∧z
A=({x,y,z},R100(x,y,z))
Problem: CSPinf(B)
Input: definable structure A (over the same signature)
Decide: Is there a homomorphism from A to B?
Theorem [Klin, Lasota, O., Toruńczyk]. There exists a definable structure B for which the problem Hom(B) is undecidable.
xab+xba=1, where a and b are distincxab+xbc+xca=0,where a, b and c are distinct
Problem: CSPinf(B)
Input: definable structure A (over the same signature)
Decide: Is there a homomorphism from A to B?
Theorem [Klin, Lasota, O., Toruńczyk]. There exists a definable structure B for which the problem CSPinf(B) is undecidable.
Theorem [Klin, Kopczyński, O., Toruńczyk]. For a finite structure B, if CSP(B) is C-complete, then CSPinf(B) is Exp(C)-complete.
Corollary. 3-colorability of definable graphs is NEXP-complete.
Open problem: Is it decidable, given two definable relational structures, whether they are isomorphic?