Pattern-directed inference systems

1
Pattern-directed inference systems

• We can describe any problem domain in terms of 2
  types of knowledge
• Declarative knowledge facts about the domain,
  which can be expressed as assertions (statements
  in some language). Examples
• (Today is a beautiful day) Assertion
  in natural language
• ( (Today) (beautiful-day) Assertion
  in FOL language
• (Today beautiful-day) Assertion
  in OPS language
• Procedural knowledge represents dependencies
  among facts, and can be expressed as if then
  rules. Example
• (gt temperature 650F) and (no-rain) ?
  beautiful-day
• In an AI program, the declarative component is
  called a "knowledge
• base", and the procedural component is called a
  "rule base".

2
Pattern-directed inference systems basic
architecture

• Adding new assertions
• 
  
  
• 
  
  Inference
• 
  
  Engine
• Adding new rules

Fact 1 Fact 2 .... Fact n
represented as sentences in some KR language
Assertion/fact base
Rule 1 Rule 2 ..... Rule k
define what follows from the facts in the KB
Rule base
Data base
3
Knowledge representation expressing knowledge in
a form understandable by a computer.

• Choosing an appropriate language to represent
  knowledge is the first and the
• most important step in building an intelligent
  system. Each language has 2 sides
• Syntax defines how to build sentences
  (formulas).
• Semantics defines the meaning and the truth
  value of sentences by connecting them to the
  facts in the outside world.
• If the syntax and the semantics of a language are
  precisely defined, we call that
• language a logic.
• Logic Syntax Semantics

4
Connection between sentences in a KR language and
facts in the outside world

• Internal
  Entails
• Representation Sentences
  Sentences
• Outside world Facts
  Facts
• 
  Follows
• There must exist an exact correspondence between
  the sentences entailed by
• the logic and the facts that follow from other
  facts in the outside world. If this
• requirement does not hold, the logic will be
  unpredictable and irrational.
  

5
Entailment and inference

• Entailment defines if sentence A is true with
  respect to a given KB (we denote it as
• KB A). Inference defines if sentence A can
  be derived from the KB (we denote it
• as KB -- A).
• Let the KB contains only true sentences
  representing explicit knowledge about some
• domain. To find all consequences (or derived
  knowledge) that follow from that KB,
• the system must run an inference procedure. If
  this inference procedure generates
• only entailed sentences, then it is called sound,
  and if the inference procedure
• generates all entailed sentences, then it is
  called complete. Ideally, we want an
• inference procedure to be both, sound and
  complete. In many cases, however, this
• may not be possible (for example, if the KB is
  infinite) and we are willing to drop the
• requirement for completeness.
• If A is derivable from a KB by a sound inference
  procedure, i, i.e. KB --i A, then the
• derivation process is called a proof of A.

6
Knowledge Representation Languages

• To formally express knowledge we need a language
  which is expressive
• and concise, unambiguous and context independent,
  and computationally
• efficient. Among the languages that fulfill at
  least partially these requirements are
• Propositional Logic (PL). It can represent only
  facts, which are true or false.
• First-Order Logic (FOL). It can represent
  objects, facts and relations between objects and
  facts, which are true or false.
• Temporal Logic. This is an extension of FOL which
  takes the time into account.
• Probabilistic Logic. Limits the representation to
  facts only, but can these facts can be uncertain,
  true or false. To express uncertainty, it
  attaches a degree of belief (0..1) to each fact.
• Truth Maintenance Logic. Represents facts only,
  but these can be unknown and uncertain as well as
  true and false.
• Fuzzy Logic. Represents facts which degree of
  truth can be explicitly defined.

7
Interpretation and model of a representation

• Interpretation establishes a connection between
  sentences of a selected KR
• language and facts from the outside world.
• Example Assume that A, B and C are
  sentences of our logic. If we refer to the Moon
  world, A may have the following interpretation
  The moon is green, B -- There are people on
  the moon, and C -- It is sunny and nice on the
  moon, and people there eat a lot of green
  cheese".
• Given an interpretation, a sentence can be
  assigned a truth value. In PL, for
• example, it can be true or false, where true
  sentences represent facts that hold
• in the outside world, and false sentences
  represent facts that do not hold.
• Any world in which a sentence is true under a
  particular interpretation is called a
• model of that sentence under that interpretation.

8

• Sentences may have different interpretations
  depending on the meaning given to them.
• Example Consider English language. The word
  Pope is to be understood as a microfilm, and
  the word Denver is to be understood as
  pumpkin on the left side of the porch. In this
  interpretation, sentence The Pope is in Denver
  means the microfilm is in the pumpkin.
• Assume that we can enumerate all possible
  interpretations in all possible worlds
• that can be given to the sentences from our
  representation. Then
• A sentence is called valid (or tautology) if it
  is true in all these interpretations. Example (A
  v not A) is always true even if we refer to the
  Moon world (There are people on the moon or
  there are no people on the moon).
• A sentence is called satisfiable if it is true in
  some interpretation. Example The snow is red
  and the day is hot is satisfiable if this is the
  case on Mars.
• A sentence is called unsatisfiable if it is not
  true in any interpretation.

9
Propositional logic

• To define any logic, we must address the
  following three questions
• 1. How to make sentences (i.e. define the
  syntax).
• 2. How to relate sentences to facts (i.e.
  define the semantics).
• 3. How to generate implicit consequences
  (i.e. define the proof theory).
• From the syntactic point of view, sentences are
  finite sequences of primitive
• symbols. Therefore, we must first define the
  alphabet of PL. It consists of the
• following classes of symbols
• propositional variables A, B, C ...
• logical constants true and false
• parentheses (, )
• logical connectives , v, ltgt, gt, not

10
Well-formed formulas (wff)

• Given the alphabet of PL, a wff (or sentence, or
  proposition) is inductively
• defined as
• a propositional variable
• A v B, where A, B are sentences
• A B, where A, B are sentences
• A gt B, where A, B are sentences
• A ltgt B, where A, B are sentences
• not A, where A is a sentence
• true is a sentence
• false is a sentence.
• The following hierarchy is imposed on logical
  operators not, , v, gt, ltgt.
• Composite statements are evaluated with respect
  to this hierarchy, unless
• parentheses are used to alter it.
• Example ((A B) gt C) is equivalent to A B gt
  C
• (A (B gt C)) is a different
  sentence.

11
The semantics of PL is defined by specifying the
interpretation of wwf and the meaning of logical
connectives.

• If a sentence is composed by only one
  propositional symbol, then it may have any
  possible interpretation. Depending on the
  interpretation, the sentence can be either true
  or false (i.e. satisfiable).
• If a sentence is composed by a logical constant
  (true or false), then its interpretation is
  fixed
• true has as its interpretation a true fact
• false has as its interpretation a false fact.
• If a sentence is composite (complex), then its
  meaning is derived from the meaning of its parts
  as follows (such semantics is called
  compositional, and this is known as a truth table
  method)
• P Q not P P Q P
  v Q P gt Q P ltgt Q
• F F T F
  F T T
• F T T F
  T T F
• T F F F
  T F F
• T T F T
  T T T

12
Example using a truth table, define the validity
of P (Q R) ltgt (P Q) R

• P Q R Q R P (Q R) (P Q)
  R P (Q R)ltgt(P Q) R
• F F F F F
  F T
• T F F F F
  F T
• F T F F F
  F T
• T T F F F
  F T
• F F T F F
  F T
• T F T F F
  F T
• F T T T F
  F T
• T T T T T
  T T
• This formula is valid, because it is true in all
  possible interpretations of its
• propositional variables. It is known as the
  associativity of conjunction law.

13
Inference this is a process of building a proof
of a sentence.

• Inference is carried out by inference rules,
  which allow one formula to be
• inferred from a set of other formulas. For
  example, A -- B meaning that
• B can be derived from A.
• An inference procedure is sound if and only
  if (iff) its inference rules are sound. In turn,
  an inference rule is sound iff its conclusion is
  true whenever the rule premises are true.
• PL inference rules are sound (this can be proven
  by means of truth tables)
• therefore they can be used for building proofs of
  other formulas.

14
PL inference rules

• Modus ponens if sentence A and implication A gt
  B hold, then B also holds, i.e. (A, A gt B)
  -- B.
• Example Let A means lights are off,
• A gt B means if
  lights are off, then there is no one
• 
  in the office
• B means there is
  no one in the office
• AND-elimination if conjunction A1 A2 ...
  An holds, then any of its conjuncts also holds,
  i.e. A1 A2 ... An -- Ai.
• AND-introduction if a list of sentences holds,
  then their conjunction also holds, i.e. A1,
  A2,...,An -- (A1 A2 ... An).
• OR-introduction If Ai holds, then any
  disjunction containing Ai also holds, i.e. Ai
  -- (A1 v ... v Ai v ... v An).
• Double-negation elimination states that a
  formula can be either true or false, i.e. (not
  (not A)) -- A

15
PL inference rules (cont.)

• Unit resolution (A v B), not B -- A. Note
  that (A v B) is equivalent to (not B gt A), i.e.
  unit resolution is a modification of modus
  ponens.
• Resolution (A v B), (not B v C) -- (A v C).
  Note that
• (A v B) is equivalent to
  (not A gt B),
• (not B v C) is equivalent to
  (B gt C)
• By eliminating the intermediate
• conclusion, we get
  (not A gt C).
• The soundness of each one of these rules can be
  checked by means of the
• truth table method. Once the soundness of a rule
  has been established, it
• can be used for building proofs. Proofs are
  sequences of applications of
• inference rules, starting with sentences
  initially contained in the KB

16
Complexity of propositional inference

• 1. Propositional inference is complete, i.e. any
  valid formula can be proved by means of the truth
  tables method.
• 2. However, the truth table may have as much as
  2N rows, where N is the number of propositional
  variables in the KB. To build such a table, takes
  time proportional to N, i.e. the problem of
  proving the validity of a PL formula is
  NP-complete.
• 3. Inference rules in PL are monotonic. That is,
• if KB1 -- A, then KB1 U KB2 -- A
• Here, KB2 can be contradictory to KB1, which
  makes monotonicity of PL rule a major
  representation problem.
• 4. Inference rules in PL are local, i.e. they
  depend only on their premises (this is a
  consequent of the monotonicity property). This,
  in turn, makes the inference procedure much
  better than exponential, because only a small
  number of propositions are involved in each
  inference.

17
Horn formulas

• Although inference in PL in the worst case takes
  an exponential time, for one
• special class of PL formulas, there exists a
  polynomial time inference
• procedure. These formulas are called Horn
  formulas, and they have the
• following form
• A1 A2 ... An ? B,
• where A1, A2, ..., An, B are
  positive literals.
• A literal is a formula or its
  negation.
• If the KB can be represented as a collection of
  Horn formulas, then by just
• applying modus ponens, we can infer all
  conclusions.

18
Pattern-matching and unification

• Assume that the KB consists of Horn formulas,
  whose literals are not simple
• propositional variables or constants, but
  assertions such as
• (Robot Robbie)
• (Robot ?x) ? (Can_reason ?x), where ?x is
  called a pattern variable.
• We can apply MP in this case, if our inference
  procedure is augmented with a
• pattern-matching facility. It allows pattern
  (Robot ?x) to be matched to
• data (Robot Robbie). Note that the match must be
  propagated to the right hand
• side of the implication making (Can_reason
  Robbie) the conclusion of this
• inference step.
• In some cases, there is a need to match two
  patterns (rather than a pattern and a
• data). This requires a special procedure, called
  unification, which finds all values
• of pattern variables that make the two patterns
  identical.