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Computation in sets with atoms


Joanna Ochremiak

Universitat Politècnica de Catalunya


QuantLA, Krippen, 19th September 2016











Definable sets with atoms


Atoms = (N,=)

a,b,c, - denote atoms

Aut(Atoms) - all permutations


Examples:

  • Atoms
  • Atoms(2) - set of non-repeating pairs of atoms
  • a - a single atom

Set expressions


Examples:

  • {xxAtoms}
  • {(y1,y2)(y1,y2)Atoms2, y1y2}
  • x


A set expression can be:

  • a variable,
  • a formal tuple of set expressions,
  • a formal union of set expressions,
  • a set-builder expression of the form {e(ˉxˉy)ˉyAtomsn, ϕ(ˉxˉy)}.

{{y1}{y2}{y3}(y1,y2)Atoms2, y1y2y1y3y2y3}


Valuation:  y3a


{SAtoms|S|=3,aS}

Set expressions


Examples:

  • {xxAtoms}
  • {(y1,y2)(y1,y2)Atoms2, y1y2}
  • x


A set expression can be:

  • a variable,
  • a formal tuple of set expressions,
  • a formal union of set expressions,
  • a set-builder expression of the form {e(ˉxˉy)ˉyAtomsn, ϕ(ˉxˉy)}.

Orbit-finiteness


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:

  • Atoms - one orbit
  • Atoms2 - two orbits    (a,b)(c,d)  (a,a)(d,d)
  • P(Atoms) - infinitely many orbits (not definable)
  • Pfin(Atoms) - infinitely many orbits (not definable)

Context


  • Fraenkel-Mostowski set theory (1922)
  • Nominal sets [Gabbay and Pitts]


Alphabet: Atoms
States: {,}Atoms
Transitions: {(,a,a),(a,b,a),(a,a,)(a,b)Atoms2}

Definable automata

[Bojańczyk, Klin, Lasota]


Determinization fails


Minimisation works (syntactic automaton)


Emptiness is decidable (PSPACE-complete)


"some letter appears twice"



Powerset construction does not work.

Other models


Definable pushdown systems - reachability is decidable [Murawski, Ramsay, Tzevelekos].


Definable Petri nets - reachability is open.


Definability over different atoms.

TM with atoms


I ab bc ce A I ab bc R ab bc ce B B ... ab bc ea ce ad B B ... language of "paths"

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.

Standard alphabets


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.

{(a,c,e),(a,d,f),(b,c,f),(b,d,e)}


Standard alphabets


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.

Quest for logic for PTime


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?

Alphabets 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.

The power of LFP+C


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.

Choiceless polynomial time


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)}

Constraint satisfaction problem


Problem: CSP(B)

Input: structure A (over the same signature)

Decide: Is there a homomorphism from A to B?









3-colorability


3-colorability


3-colorability


Linear equations mod 2


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))


3-SAT


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)}

¬xyz

A=({x,y,z},R100(x,y,z))


Definable CSP


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.

Definable 3-colorability


Definable linear equations mod 2



xab+xba=1, where a and b are distincxab+xbc+xca=0,where ab and c are distinct



Definable CSP


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.

Complexity


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?