Assumptionbased Truth Maintenance Systems: Motivation

1
Assumption-based Truth Maintenance Systems
Motivation

• Problem solvers need to explore multiple contexts
  at the same time, instead of a single one (the
  JTMS case)
• Alternate diagnoses of a broken system
• Different design choices
• Competing theories to explain a set of data
• Problem solvers often need to compare contexts
  rapidly switching from one context to another. In
  JTMS, this can be done by enabling and retracting
  assumptions. In ATMS, re-labeling is avoided
  because alternative contexts are explicitly
  stored.
• ATMS contexts are monotonic.

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The idea behind ATMS

• The assumptions underlying conclusions are
  important in problem-solving
• Solutions can be concisely described as sets of
  assumptions
• States of the world can be represented by sets of
  assumptions
• Theories can be represented by sets of
  assumptions
• Identify sets of assumptions called here
  environments
• Organize problem solver around manipulating
  environments
• Facilitates reasoning with multiple hypotheses

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• ATMS keeps and manipulates sets of assumptions
  rather than sets of
• beliefs that directly support a given belief. It
  works with three types of
• nodes
• Premise nodes. These are always true, but they
  are of no special interest for ATMS.
• Assumption nodes. These extend the incomplete
  description of the domain in different possible
  ways. Once made, assumptions are never retracted.
• Contradictions. These are defined by means of
  assumptions that originate them. Such sets of
  assumptions are called nogoods.
• ATMS justifications are Horn formulas of the
  form
• Jk I1, I2, , In ? Ck, where I1, I2, , In are
  the antecedents, and Ck is the consequent of
  justification Jk.

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• Basic ATMS terminology
• Similar to JTMS, the primary task of the ATMS is
  to answer queries
• about whether a node holds in a given set of
  beliefs. The following
• definitions are setting up the ATMS-related
  terminology to address this task.
• Definition. A set of assumptions upon which a
  given node depends is called
• an environment. Example A,B,C
• Definition. A label is a set of environments.
  Example A,B,C, ,D,F
• That is, the label is the assumptions upon which
  the node ultimately
• depends major difference from JTMS label, where
  labels are simple, IN
• or OUT.
• Definition. An ATMS-node, Nk is a triplet
• ltdatum, label(status),
  justificationsgt

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• Basic ATMS terminology (cont.)
• Definition. A node, Nk, holds in a given
  environment, E, iff it is propositionally
• derivable from E given the set of justifications,
  J.
• Definition. Let E be a (consistent) environment,
  and N be a set of nodes
• propositionally derivable from E. Then, E ? N is
  called the context of E.
• Definition. A characterizing environment for the
  context is a set of
• assumptions where every node belonging to that
  context holds.
• Each context is completely specified by its
  characterizing environment. This is
• why, the question about whether a node holds in a
  given set of beliefs (or
• context) is reduced to defining whether the node
  holds in a given environment.

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• Relations between environments
• Because environments are monotonic, set inclusion
  between
• environments implies logical subsumption of
  consequences.
• Example
• E1 C
• E2 C, D
• E3 D, E
• E1 subsumes E2
• E2 is subsumed by E1
• E1 neither subsumes or is subsumed by E3

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• How ATMS answers queries about whether a node
• holds in a given environment?
• The easiest way associate with each node all of
  the environments where this nodes holds. However,
  if a node holds universally (that is, in all
  possible environments, the number of which can be
  as much as 2n, where n is the number of
  assumptions) this may result in a huge data
  structure associated with a node.
• The better way given the fact that ATMS is a
  monotonic system, we can record only those
  environments which satisfy the following four
  properties
• Soundness, i.e. a node holds in any of the
  environments associated with it.
• Consistency, i.e. no environment is a nogood.
• Completeness, i.e. every consistent environment
  where a node holds is either associated with it,
  or is a superset of some environment associated
  with it.
• Minimality, i.e. no environment is a proper
  subset of any other.

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• ATMS labels more complex than JTMS labels
• Examples Consider the following dependency
  network
• 
  
• Is H believed?
• Yes, because its label is non-empty.
• Is H believed under B, C, D, Z, X?
• Yes, because B, C, D ? B, C, D, Z,
  X
• Is H believed under C, D?
• No.

A
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• Contradictions
• Like JTMS, in ATMS certain nodes can be declared
  as contradictions, which
• here suggests that every environment which would
  allow a contradiction to be
• believed is inconsistent. Inconsistent
  environments are called nogoods.
• Example

A,B
F
A,B,C
G
B,C
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• There are two special labels in ATMS
• Case 1 Label (empty label)
• This means that there is no known consistent
  environment in which
• the node is believed, i.e. either there is no
  path from assumptions to
• it, or all environments for it are
  inconsistent.
• Case 2 Label (empty environment)
• This means that the node is believed in every
  consistent environment,
• i.e. the node is either a premise or can be
  derived strictly from premises.

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Label propagation
L
H
K
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Label propagation enable A
A
A
A
L
H
K
13
Label propagation enable B
14
Label propagation enable C
A, B
A
A, B
A,C, B,C
C
B
L
H
C
C
K
C
15
Label propagation enable D
A, B
A
A, B
A,C, B,C
C
B
L
H
C
C,D
K
C
D
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ATMS algorithms

• Logical specification of the ATMS
• ATMS does propositional reasoning over nodes.
• ATMS justifications are Horn clauses.
• Contradictions are characterized by nogoods.
• Every ATMS operation which changes a node label
  can be viewed as
• adding a justification, i.e. this is the only
  operation we have to be
• concerned here is label update as a result of
  adding a justification.
• This operation is carried out in two step
• Step 1 Compute a tentative new (locally
  correct) label for the affected node as follows
• Lnew ?k x x ?i xi , where xi ?
  Lik
• Step 2 All nogoods and subsumed
  environments are removed from Lnew to achieve
  global correctness.

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Propagating label changes

• To update node Ni compute its new label as
  described.
• If the label has not changed DONE.
• If Ni is a contradiction node do
• Mark all environments in its label as nogoods.
• For every node in the network, check its label
  for environments newly marked as nogoods and
  remove them from every node label.
• If Ni is not a contradiction node, then
  recursively update all of its consequences.
• This algorithm is guaranteed to terminate with
  correct labels for all of the
• nodes in the network. However, it is inefficient,
  because it constantly
• re-computes nodes labels.

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Constructing solutions

• ATMS can answer a variety of questions
• 1. Does a node hold in a given context?
• 2. Is a context originated by a given set of
  assumptions consistent?
• 3. Why does a node hold in a given context?
• 4. Which assumptions underlie a given node?
• What exactly is expected by the ATMS (the
  solution) depends on the task
• the system is intended to solve. This defines a
  major difference between
• JTMS and ATMS, namely
• In single-context JTMS, a solution is a
  contradiction-free state of the TMS which
  contains beliefs of particular types. Search
  procedures push the TMS into appropriate states,
  and the answers are read out of the database.
• In multiple-context ATMS, Solution Environment
  that supports beliefs of particular types. We
  must ensure that such environments are created
  somehow.

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Constructing solutions by using goal nodes

• If a problem-solving task can be solved by using
  choice sets (i.e. the
• solution is generated by picking up one choice
  from each choice set), then
• we can read off the solution from the label of
  one goal node. The algorithm
• for generating the solution becomes
• Construct a node (or nodes) whose label
  constitutes valid solutions.
• Solve a problem by
• Building the appropriate dependency structure
  (including nogoods)
• Enabling assumptions
• Reading off the label of the goal node.
• Optimizations
• Interleave assumption-making with building the
  dependency network
• Use multiple goal nodes to decompose search space
  
• See textbook, example p.437

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Applications of ATMS

• In class, we shall discuss the following two
  applications of the ATMS
• Model-based diagnosis. Diagnosis in general is
  intended to identify the cause (or causes) of a
  system failure, which is signaled by one or more
  failure symptoms. The traditional approach to
  diagnosis is the rule-based approach, where rules
  describe the relationship between symptoms and
  the underlying causes. The major problem with
  this easy-to-implement approach is that the
  actual cause for the failure may not be
  explicitly represented into the KB, in which case
  the system will either fail to reach a conclusion
  or will reach a wrong conclusion instead.
  Model-based diagnosis provides an alternative,
  much more comprehensive approach to building
  diagnostic systems. (Notes to be distributed in
  class.)
• Constraint satisfaction. Given a finite sets of
  variables, possible values of those variables and
  a set of constraints on those variables, the
  constraint satisfaction problem (CSP) consists in
  defining substitutions for variables from
  corresponding sets of values, so that all of the
  constraints are satisfied. The traditional
  solution to this problem relies on
  generate-and-test search and chronological
  backtracking. Because of the complexity of this
  type of search, this solution is good only for
  small problems. The ATMS approach to the CSP
  allow the PS to keep track of already evaluated
  constraints, thus improving the efficiency of the
  search. (Notes to be distributed in class.)