Knowledge bases

They are set of sentences in a formal language

It is a declarative approach to building an agent (or other system)

Tell it what it needs to know
Then it can Ask itself what to do—answers should follow from the KB

Agents can be viewed at the knowledge level

i.e., what they know, regardless of how implemented

at the implementation level i.e., data structures in KB and algorithms that manipulate them

A simple knowledge-based agent

The agent must be able to:

Represent states, actions, etc.
Incorporate new percepts
Update internal representations of the world
Deduce hidden properties ofthe world
Deduce appropriate actions

pseudocode

function KB-Agent(percept) returns an action
  static: KB, a knowledge base
          t, a counter, initially 0, indicating time
  Tel l(KB,Make-Percept-Sentence(percept,t)) 
  action ← Ask(KB,Make-Action-Query(t)) 
  Tell(KB, Make-Action-Sentence(action, t))
  t ← t + 1
  return action

Example: Wumpus World

PEAS description

Performance measure:

gold +1000, death -1000
-1 per step, -10 for using the arrow

Environment:

Squares adjacent to wumpus are smelly
Squares adjacent to pit are breezy
Glitter iff gold is in the same square
Shooting kills wumpus if you are facing it
Shooting uses up the only arrow
Grabbing picks up gold if in same square
Releasing drops the gold in same square

Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot

Sensors: Breeze, Glitter, Smell

Wumpus world characterization

Observable: No—only local perception Deterministic: Yes—outcomes exactly specified Episodic: No—sequential at the level of actions Static: Yes—Wumpus and Pits donotmove Discrete: Yes Single-agent: Yes—Wumpus is essentially a natural feature

Logic in general

Logics are formal languages for representing information such that conclusions can be drawn
Syntax definesthe sentencesin the language
Semantics define the “meaning” of sentences;
- i.e., define truth of a sentence in a world

Entailment

Entailment means that one thing follows from another:

\[KB |= α\]

Knowledge base $KB$ entails sentence $α$, if and only if $α$ is true in all worlds where $KB$ is true

Example:

$x + y= 4$ entails $4= x + y$
The KB containing “the Giants won” and “the Reds won” entails “Either the Giants won or the Reds won”

Entailment is a relationship between sentences (i.e., syntax) that is based on semantics

Models

They are formally structured worlds with respect to which truth can be evaluated

We say $m$ is a model of a sentence $α$ if $α$ is true in $m$

$M (α)$ is the set of all models of $α$

Then $KB|=α$ if and only if $M(KB) ⊆ M (α)$ Example: $KB$ = Giants won and Reds won $α$ = Giants won

Inference

$KB f–i α = \text{ sentence } α$ can be derived from $KB$ by procedure $i$

Consequences of $KB$ area haystack; $α$ is a needle.
Entailment = needle in haystack; inference = finding it

Soundness: i is sound if whenever KB f–iα, it is also true that KB |= α

Completeness: i is complete if whenever $KB |= α$, it is also true that $KB f–iα$

We will define a logic (first-order logic) which is expressive enough to say almost anything of interest, and for which there exists a sound and complete inference procedure
The procedure will answer any question whose answer follows from what is known by the KB.

Propositional logic

Syntax

Propositional logic is the simplest logic—illustrates basic ideas

The proposition symbols P1, P2 etc are sentences

If $S$ is a sentence, $¬S$ is a sentence (negation)
If $S_1$ and $S_2$ are sentences, $S_1 ∧ S_2$ is a sentence (conjunction)
If $S_1$ and $S_2$ are sentences, $S_1 ∨ S_2$ is a sentence (disjunction)
If $S_1$ and $S_2$ are sentences, $S_1 ⇒ S_2$ is a sentence (implication)
If $S_1$ and $S_2$ are sentences, $S_1 ⇔ S_2$ is a sentence (biconditional)

Semantics

Each model specifies true/false for each proposition symbol

Rules for evaluating truth with respect to a model m:

$¬S$ is true if $S$ is false
$S_1 ∧ S_2$ is true if $S_1$ is true and $S_2$ is true
$S_1 ∨ S_2$ is true if $S_1$ is true or $S_2$ is true
$S_1 ⇒ S_2$ is true if $S_1$ is false or $S_2$ is true
$S_1 ⇒ S_2$ is false if $S_1$ is true and $S_2$ is false
$S_1 ⇔ S_2$ is true if $S_1 ⇒ S_2$ is true and $S_2 ⇒ S_1$ is true

Simple recursive process evaluates an arbitrary sentence Example: $¬P_{1,2} ∧ (P_{2,2} ∨ P_{3,1}) = true ∧ (false ∨ true) = true ∧ true = true$

Wumpus world sentences

Let P_{i,j} be true if there is a pit in $[i, j]$. Let B_{i,j} be true if there is a breeze in $[i, j]$.

Pits cause breezes in adjacent squares $B_{1,1} ⇔ (P_{1,2} ∨ P_{2,1})$ $B_{2,1} ⇔ (P_{1,1} ∨ P_{2,2} ∨ P_{3,1})$

A square is breezy if and only if there is an adjacent pit

Inference by enumeration

Depth-first enumeration of all models is sound and complete

function TT-Entails?(KB,α) returns true or false
  inputs: KB, the knowledge base, a sentence in propositional logic
    α, the query, a sentence in propositional logic
  symbols← a list of the proposition symbols in KB and α
  return TT-Check-A ll(KB,α,symbols,[])
function TT-Check-All(KB, α, symbols,model) returns true or false
  if Empty?(symbols) then
    if PL-True?(KB, model) then return PL-True?(α, model)
    else return true
  else do
    P ← First(symbols); rest← Rest(symbols)
    return TT-Check-All(KB, α, rest,Extend(P ,true,model)) and
      TT-Check-All(KB, α, rest,Extend(P ,false,model))

$O(2^n)$ for $n$ symbols; problem is co-NP-complete

Logical equivalence

Two sentences are logically equivalent if true in same models: $α ≡ β$ if and only if $α |= β$ and $β |= α$

$(α ∧ β) ≡ (β ∧ α)$ commutativity of $∧$
$(α ∨ β) ≡ (β ∨ α)$ commutativity of $∨$
$((α ∧ β) ∧ γ) ≡ (α ∧ (β ∧ γ))$ associativity of $∧$
$((α ∨ β) ∨ γ) ≡ (α ∨ (β ∨ γ))$ associativity of $∨$
$¬(¬α) ≡ α$ double-negation elimination
$(α ⇒ β) ≡ (¬β ⇒ ¬α)$ contraposition
$(α ⇒ β) ≡ (¬α ∨ β)$ implication elimination
$(α ⇔ β) ≡ ((α ⇒ β) ∧(β ⇒ α))$ biconditional elimination
$¬(α ∧ β) ≡ (¬α ∨ ¬β)$ De Morgan
$¬(α ∨ β) ≡ (¬α ∧ ¬β)$ De Morgan
$(α ∧ (β ∨ γ)) ≡ ((α ∧ β) ∨(α ∧ γ))$ distributivity of $∧$ over $∨$
$(α ∨ (β ∧ γ)) ≡ ((α ∨ β) ∧(α ∨ γ))$ distributivity of $∨$ over $∧$

Validity and satisfiability

A sentence is valid if it is true in all models,

Example: $True, A ∨ ¬A, A ⇒ A, (A ∧ (A ⇒ B)) ⇒ B$

Validity is connected to inference via the Deduction Theorem: KB |= α if and only if (KB ⇒ α) is valid

A sentence is satisfiable if it is true in some model

Example: $A ∨ B, C$

A sentence is unsatisfiable if it is true in no models

Example: $A ∧ ¬A$

Satisfiability is connected to inference via the following: $KB |= α$ if and only if $(KB ∧ ¬α)$ is unsatisfiable i.e., prove $α$ by reductio adabsurdum

Proof methods

Proof methods divide into (roughly) two kinds:

Application of inference rules
- Legitimate (sound) generation of new sentences from old
- Proof = a sequence of inference rule applications Can use inference rules as operators in a standard search alg.
- Typically require translation of sentences into a normal form
Model checking
- truth table enumeration (always exponential in n)
- improved backtracking, e.g., Davis–Putnam–Logemann–Loveland
- heuristic search in model space (sound but incomplete) e.g., min-conflicts-like hill-climbing algorithms

Resolution

CNF—universal:

Conjunctive Normal Form
conjunction of disjunctions of literals

Resolution inference rule (for CNF): complete for propositional logic

\[\frac{J_1 ∨ ... ∨ J_k \ \ m_1 ∨ ... ∨ m_n}{J_1 ∨ ... ∨ J_{i-1} ∨ J_{i+1} ∨ ... ∨ J_n ∨ m_1 ∨ ... ∨ m_{j-1} ∨ m_{j+1} ∨ ... ∨ m_n}\]

where $J_i$ and $m_j$ are complementary literals.

Resolution is sound and complete for propositional logic

Conversion to CNF

$B_{1,1} ⇔(P_{1,2} ∨ P_{2,1})

Eliminate ⇔, replacing $α ⇔ β$ with $(α ⇒ β) ∧ (β ⇒ α)$.
$(B_{1,1} ⇒ (P_{1,2} ∨ P_{2,1})) ∧ ((P_{1,2} ∨ P_{2,1}) ⇒ B_{1,1})$
Eliminate ⇒, replacing α ⇒ β with ¬α ∨β.
$(¬B_{1,1} ∨ P_{1,2} ∨ P_{2,1}) ∧ (¬(P_{1,2} ∨ P_{2,1}) ∨ B_{1,1})$
Move $¬$ inwards using de Morgan’s rules and double-negation:
$(¬B_{1,1} ∨ P_{1,2} ∨ P_{2,1}) ∧ ((¬P_{1,2} ∧ ¬P_{2,1}) ∨ B_{1,1})$
Apply distributivity law (∨ over ∧) and flatten:
$(¬B_{1,1} ∨ P_{1,2} ∨ P_{2,1}) ∧ (¬P_{1,2} ∨ B_{1,1}) ∧ (¬P_{2,1} ∨ B_{1,1})$

Resolution algorithm

function PL-Resolution(KB, α) returns true or false
  inputs: KB, the knowledge base, a sentence in propositional logic
        α, the query, a sentence in propositional logic
  clauses ← the set of clauses in the CNF representation of KB ∧ ¬α
  new ← { }
  loop do
    for each Ci, Cj in clauses do
      resolvents ← PL-Resol ve(Ci,Cj)
      if resolvents contains the empty clause then return true 
      new ← new∪ resolvents
    if new ⊆ clauses then return false 
    clauses ← clauses ∪ new

Forward and backward chaining

These algorithms are very natural and run in linear time

Forward chaining

Idea: fire any rule whose premises are satisfied in the KB, add its conclusion to the KB, until query is found

Forward chaining algorithm

function PL-FC-Entails?(KB, q) returns true or false
  inputs: KB, the knowledge base, a set of propositional Horn clauses
    q, the query, a proposition symbol
  local variables: count, a table, indexed by clause, initially the number of premises
    inferred, a table, indexed by symbol, each entry initially false
    agenda, a list of symbols, initially the symbols known in KB

while agenda is not empty do 
  p← Pop(agenda) 
  unless inferred[p] do
    Inferred[p]← true
    for each Horn clause c in whose premise p appears do
      decrement count[c]
      if count[c] = 0 then do
        if Head[c] = q then return true
        Push(Head[c],agenda)
  return false

Forward Chaining

Proof of completeness FC derives every atomic sentence that is entailed by KB

FC reaches a fixed point where no new atomic sentences are derived
Consider the final state as a model m, assigning true/false to symbols
Every clause in the original KB is true in m Proof: Suppose a clause $a_1 ∧ … ∧ a_k ⇒ b$ is false in $m$ Then $a_1 ∧ … ∧ a_k$ is true in $m$ and $b$ is false in $m$ Therefore the algorithm has not reached a fixed point!
Hence $m$ is a model of $KB$
If $KB = q, q$ is true in every model of $KB$, including $m$

General idea: construct any model of $KB$ by sound inference, check $α$

Backward chaining

Idea: work backwards from the query q: to prove q by BC, check if q is known already, or prove by BC all premises of some rule concluding q

Avoid loops: check if new subgoal is already on the goal stack

Avoid repeated work: check if new subgoal 1) has already been proved true, or 2) has already failed

Backward Chaining

Forward vs. Backward chaining

Forward Chaining is data-driven, cf. automatic, unconscious processing,

example: object recognition, routine decisions

May dolots of work that is irrelevant to the goal

BC is goal-driven, appropriate for problem-solving,

example: Where are my keys? How do I get into a PhD program?

Complexity of BC can be much less than linear in size of $KB$

Effective Propositional Model Checking

Davis–Putnam algorithm with three improvements over TT-ENTAILS • Early termination: detect T/F • Pure symbol heuristic: same sign in all clauses • Unit Clause heuristic: clause with one literal

Local search algorithms such as hill-climbing & simulated annealing.

In recent years, there has been a great deal of experimentation to find a good balance between greediness and randomness.

WALKSAT: On every iteration, the algorithm picks an unsatisfied clause and picks a symbol in the clause to flip.

Summary

Logical agents apply inference to a knowledge base to derive new information and make decisions

Basic concepts of logic: – syntax: formal structure of sentences – semantics: truth of sentences wrt models – entailment: necessary truth of one sentence given another – inference: deriving sentences from other sentences – soundess: derivations produce only entailed sentences – completeness: derivations can produce all entailed sentences

Wumpus world requires the ability to represent partial and negated information, reason by cases, etc

Forward, backward chaining are linear-time, complete for Horn clauses

Resolution is complete for propositional logic.

Propositional logic lacks expressive power

Local search methods (WALKSAT) find solutions (sound but not complete).

Constraint Satisfaction Problems

Susquehanna International Group: Elevating Trading Acumen Through Poker Training