Dichotomies in CSP

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Dichotomies in CSP

• Karl Lieberherr

inspired by the paper Dichotomies and Duality
in First-order Model Checking Problems by Barnaby
Martin The 11th Mons Days of Theoretical Computer
Science 2006 Irisa - Rennes
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Generalized model checking

• Find an interpretation that satisfies a fraction
  t (between 0 and 1) of the constraints.
• The generalized model checking problem over the
  logic called positive existential conjunctive
  fragment of FOL, and, exists-FOL, takes as
  input a structure A and a sentence f in and,
  exists-FOL and asks whether there exists an
  interpretation for f satisfying at least the
  fraction t of the weighted conjuncts.
• This problem is equivalent to the maximum
  constraint satisfaction problem (MaxCSP).

FOL first-order logic
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Dichotomy for and, exists-FOL

• The class MaxCSP(A) exhibits a dichotomy For all
  structures A (set of binary relations) there
  exists an algebraic constant tA between 0 and 1
  such that the set of A-formulas f in and,
  exists-FOL satisfying
• Fraction(f, tA) are in P and (1)
• Fraction(f, tAe) is NP-complete for any e gt 0.
• Fraction(f, t) there exists an interpretation
  for f satisfying at least fraction t of the
  weighted constraints.
• There is a universal polynomial algorithm
  parameterized by A for case (1). This is called a
  P-optimal algorithm.
• We use the terminology MaxCSP(A) and MinCSP(A)
  for the maximization and minimization version. In
  the minimization version we replace at least by
  at most.

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Example

• A OneInThree where OneInThree(x1, x2, x3)
  x1x2x3.
• t OneInThree 4/9.
• See Lieberherr/Specker JACM 1981 and Lieberherr
  Journal of Algorithms 1982.
• http//www.ccs.neu.edu/home/lieber/p-optimal/READM
  E.html

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More examples

• R AllRenaming(Orn), tR 1-1/(2n)
• R Or1 union AllRenaming(Orgt2), tR
  (sqrt(5)-1)/2 0.618
• R AllRenaming(Orltn), tR ½ for all n gt 1.

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Minimization

• We use the terminology MaxCSP(A) and MinCSP(A)
  for the maximization and minimization version. In
  the minimization version we replace at least by
  at most.
• We reinterpret tA as tMax,A and we introduce by
  analogy tMin,A.
• Find an A so that tMax,A is different from
  tMin,A.

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A more general context

• First order predicate logic
• A conjunctive formula must be true, i.e., all
  conjuncts must be true. Drop weights.
• A model checking problem over a logic L takes as
  input a structure A and a sentence f of L and
  asks A f (before we had A, t f), where
  0lttlt1.
• Parameterize over A or f.

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L FOL (first-order logic)

• Alphabet G1 union G2, where G1 not, and, or,
  exists, for all, , G0 (,),R,v,0,1
• R(v1,v2, ,vn) is a formula with free variables
  v1,v2, ,vn.
• vivj is a formula with free variables vi, vj
• if f1 and f2 are formulas, then f1and f2, f1
  or f2 and not f1 are also formulae (having as
  free variables those free in the constituent
  formulae)

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FOL (continued)

• if f contains the free variable v, then exists v
  f and for all v f are formulae whose free
  variables are exactly those of f less v.
• A sentence is a formula with no free variables.
• We currently study and, exists-FOL but similar
  questions can be asked for other subsets of FOL.

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Standard Definition of Model Checking

• Model checking definition Efficiently deciding
  whether a temporal logic formula is satisfied in
  a finite state machine model.

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Model checking

• The model is usually given as a source code
  description in an industrial hardware description
  language or a special-purpose language. Such a
  program corresponds to a finite state machine,
  i.e., a directed graph consisting of nodes (or
  vertices) and edges. A set of atomic propositions
  is associated with each node, typically stating
  which memory elements are one. The nodes
  represent states of a system, the edges represent
  possible transitions which may alter the state,
  while the atomic propositions represent the basic
  properties that hold at a point of execution.
• Formally, the problem can be stated as follows
  given a desired property, expressed as a temporal
  logic formula p, and a model M with initial state
  s, decide if M,s p . If M is finite, as it is in
  hardware, model checking reduces to a graph
  search.

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Going full circle

• Symbolic algorithms avoid ever building the graph
  for the FSM instead, they represent the graph
  implicitly using a formula in propositional logic
  (BDDs). More recently, SAT solvers (see Boolean
  satisfiability problem) are used to perform the
  graph search.

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Model checking andtraversal specifications

• M,s p
• M an object graph OG, s a node in OG
• p a formula expressing a desired node, e.g.,
• bypassing X,Y via Z bypassing R to T
• a strategy graph with source and target
• Meta level M,s p must hold, otherwise
  compile-time error message.

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Modular ImplementationKiczales / Mezini

• it is textually local
• there is a well-defined interface that describes
  how it interacts with the rest of the system
• the interface is an abstraction of the
  implementation, in that it is possible to make
  material changes to the implementation without
  violating the interface
• an automatic mechanism enforces that every module
  satisfies its own interface and respects the
  interface of all other modules
• the module can be automatically composed e.g.,
  by a compiler with other modules to produce a
  complete system

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• A, t f