Choiceless Polynomial Time, Counting, and the CaiFurerImmerman Graphs

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Choiceless Polynomial Time, Counting, and the
Cai-Furer-Immerman Graphs

• Anuj Dawar, David Richerby, Benjamin Rossman

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Ordered vs Unordered Structures

• Mathematicians often work with properties of
  abstract first-order structures such as
  (unlabeled) graphs, which have no intrinsic
  ordering of elements.
• Complexity theory does not deal with abstract
  structures as such, but with encodings of
  structures ultimately, as inputs to Turing
  machines (bit strings). Such encodings generally
  impose an ordering on input structures, which may
  be exploited by the machine. Thus, an algorithm
  (on unlabeled graphs) may speak of the first
  vertex. Equivalently, the algorithm chooses an
  arbitrary vertex.

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Ordered vs Unordered Structures

• A Turing machine whose inputs are encodings of
  finite unlabeled graphs, in order to describe an
  algorithm, must produce the same output for
  different encodings of the same graph.
• The question whether a machine is
  encoding-invariant is undecidable.

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Background Descriptive Complexity

• Fagins Theorem NP ?SO (existential
  second-order logic).
• Immerman-Vardi PTIME LFP (first-order logic
  plus least-fixed-point operator) on ordered
  structures.
• Conjecture (Gurevich) No logic captures PTIME on
  unordered structures. (A logic is something
  with a recursive syntax and an effective
  semantics.)

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Background Descriptive Complexity

• The set of order-invariant formulas of LFP (in
  a vocabulary with lt) fails to be a logic for
  PTIME on unordered structures, since this set of
  formulas is not recursive.
• No one knows how to define a (polynomial time)
  logic that allows access to an arbitrary
  ordering, yet guarantees that all definable
  queries are order-invariant.

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Choiceless Polynomial Time (Blass, Gurevich,
Shelah, 1999)

• Intention
• Prohibit introducing an ordering.
• Equivalently, prohibit arbitrary choices.
• Allow everything else parallelism, fancy data
  structures.

• Implementation
• Work with input structure plus hereditary finite
  sets over it.
• Compute using abstract state machines (ASMs).
• Polynomially bound the number of computation
  steps and the number of active elements.

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Choiceless Polynomial Time Overview

• The state of an abstract state machine is a
  first-order structure.
• The machines program tells how to update certain
  dynamic function symbols.
• The program is executed repeatedly until the
  computation is complete.
• Active elements are (1) the arguments and values
  involved in updates and (2) all members of sets
  that are active elements.

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Choiceless Polynomial Time States

• The vocabulary of the structure HF(A) of
  hereditarily finite sets over an input structure
  A has symbols for the relations of A and symbols
  for basic set-theoretic notions ?, ?, U , ,,
  the set of atoms (i.e. A), and the unique
  element of. (Note Ux z (?y ? x) z ?
  y.)
• HF(A) is the smallest set containing A and all
  finite subsets of itself. (Ur)elements of A are
  called atoms. Atoms have rank 0. The rank of a
  set x ? HF(A) is 1 the maximal rank of an
  element of x.

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Choiceless Polynomial Time Rules

• Dynamic function symbols have value ? initially
  and are modified during the computation. They
  include Halt and Output. (The computation ends
  when Halt True.)
• Terms are built as in first-order logic with the
  additional constructor
• t(x) x ? r ?(x).
• Rules are built from updates of dynamic function
  symbols
• f(t1,,tn) t0,
• by conditional branching of the form
• if ? then R1 else R2,
• and by parallel combination of the form
• do for all x ? r, R(x) enddo.

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Choiceless Polynomial Time Programs

• A program is a closed rule R together with
  polynomials p and q bounding (1) the length of
  the computation and (2) the number of active
  elements.
• Terms can compute (mostly) everything youd
  expect, e.g.,
• union x ? y Ux,y
• bounded quantification
• (?x ? r) ?(x) ?? x x ? r ?(x) r
• ?? ? x ? r ??(x) ?
• When a set is produced, all its members have been
  looked at but not ordered.

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Choiceless Polynomial Time Expressive Power

• Power
• More expressive than least-fixed-point logic and
  the relational machines of Abiteboul and Vianu.
• Can compute any PTIME query that depends only on
  a tiny part of the input structure log n/log
  log n in an input structure of size n.
• Limitations
• Cant count, not even modulo 2.
• Even worse Has a zero-one law! (Shelah, 2000)

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Counting

• (BGS, 2002) Give Choiceless PTIME the ability to
  count. For each term t, let Card(t) be a new
  term which evaluates to the cardinality of t (as
  a finite von Neumann ordinal) if t is a set, or ?
  if t is an atom.
• The new logic is called Choiceless PTIME with
  Counting (or CPTCard).

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Counting

• Blass, Gurevich, Shelah (2002) give a number of
  innovative CPTCard algorithms for PTIME
  problems, e.g.. perfect matching in bipartite
  graphs. They also propose a few candidate
  problems for the separation CPTCard lt? PTIME.
  Among these is the Cai-Furer-Immerman problem,
  originally used to show LFPCard lt PTIME.

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LFPCard ? C???

• LFP ? L???, so Ehrenfeucht-Fraisse k-pebble
  games for Lk?? can be used to show
  inexpressibility in LFP.
• Similarly, LFPCard ? C???, so k-bijection
  games for Ck?? can be used to show
  inexpressibility in LFPCard.

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Bijection game for Ck?? on structures A and B

• Spoiler chooses a structure X ? A, B and an
  index i ? 1,k.
• Duplicator chooses a bijection f X ? Y where Y
  A, B X.
• Spoiler chooses x ? X.
• The ith pebbles are placed on x and f(x).
• Spoiler wins if the pebble-map is not a partial
  isomorphism.

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Cai-Furer-Immerman Graphs
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Cai-Furer-Immerman Graphs
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Cai-Furer-Immerman Graphs
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Cai-Furer-Immerman Graphs
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Cai-Furer-Immerman Graphs
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Cai-Furer-Immerman Graphs
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Cai-Furer-Immerman Graphs
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Cai-Furer-Immerman Graphs
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Results

• Theorem 1
• The (pre-ordered) Cai-Furer-Immerman problem is
  computable in Choiceless PTIME.
• We give an explicit Choiceless PTIME program
  that solves the CFI problem. The algorithm does
  not use counting, but requires activating sets of
  unbounded rank.

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Results

• Theorem 2
• The (pre-ordered) Cai-Furer-Immerman problem is
  not computable by any CPTCard program that
  involves only sets of bounded rank.
• The proof generalizes the Support Theorem and
  form-and-matter construction from (Blass,
  Gurevich, Shelah, 1999).

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Open Problems

• Is the non-pre-ordered Cai-Furer-Immerman problem
  in CPT or even CPTCard? (The answer is yes
  for certain restricted classes of graphs, e.g.
  complete graphs, hypercube graphs.)
• Does CPTCard PTIME?

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Choiceless Linear Algebra

• Consider a function m I ? I ? F where F is a
  finite field and I is a finite (unordered) set.
  m can be seen as an (unordered) square matrix.
  Notice that det m can be perfectly well defined.
• Gaussian elimination cannot be implemented in
  Choiceless PTIME. However, we can compute det m
  using a variant of Csankys algorithm.