First-Order Logic Copyright, 1996 © Dale Carnegie & Associates, Inc. Chapter 8 Spring 2004

CS 471/598 by H. Liu2 Why and what FOL makes a stronger set of ontological commitments (more than facts) shown in Fig 8.1 The world consists of objects and relations. Objects - things with individual identities Properties - sth distinguishing them from others Relations - sth between objects Functions - special relations with one value Facts refer to objects, properties or relations The D-Backs beat the Cardinals. (When?)

CS 471/598 by H. Liu3 FOL FOL is universal - it can express anything that can be programmed - what else do we want? FOL is the most studied and best understood scheme yet devised. Its syntax and semantics

CS 471/598 by H. Liu4 Syntax Symbols (Fig 8.4, Page 247) Constant symbols Predicate symbols - relations, tuples Functional symbols - relations Terms - objects, ground (constant symbols) & complex (functions) terms Atomic sentences Brother(Richard, John), Married(Father(R),Mother(J)) Complex sentences formed by connectives !Brother(Robin,John)

CS 471/598 by H. Liu5 Quantifiers Universal quantification (  ) - to avoid enumerating the objects by name combining with variables, we can do that:  x Cat(x)  Mammal(x)  x P(x)  Q(x) makes a statement about everything, but not when P(x) is false  x P(x) ^ Q(x) leads to a too strong statement   x King(x) ^Person(x)

CS 471/598 by H. Liu6 Existential quantification (  ) - make a statement about some object without naming it.  x P(x) ^ Q(x) - at least one x such that P(x) and Q(x) is true  x P(x)  Q(x) leads to a too weak statement No uniqueness is claimed  is used with , ^ with 

CS 471/598 by H. Liu7 Nested quantifiers Multiple quantifiers can be used. The order of quantification is important.  x  y Loves (y,x)  y  x Loves (y,x) When there is confusion, the variable belongs to the innermost quantifier that mentions it.  x [Cat(x) v (  x Brother(Richard,x))] Well-formed formula (wff) - sentences that have all their variables properly introduced.

CS 471/598 by H. Liu8 Connections, Equality The two quantifiers are connected via negation. De Morgan’s rules Do we really need both quantifiers? Some examples Equality symbol: two terms refer the same object or not the same object Some examples

CS 471/598 by H. Liu9 Using FOL Domain – some part of the world The kinship domain Axioms - basic facts Definitions - concepts defined by axioms  x,y P(x,y)  … Theorems - that are proved using axioms and definitions, or entailed by axioms  x,y Sibling(x,y)  Sibling(y,x) Two important questions in building a KB Are axioms in the KB sufficient? Are all axioms in the KB necessary?

CS 471/598 by H. Liu10 What should be in a KB From a purely logical point of view From a practical point of view Adding sentences (assertions) to a KB Tell(KB, King(John)) Tell(KB,  x King(x)  Person(x)) Asking questions and getting answers Ask(KB, King(John)) - True Ask(KB,  x Child(x,Spot)) – substitution x/Wonder

CS 471/598 by H. Liu11 The domain of sets and lists EpmtySet – constant {} Member, Subset - predicates Intersection, Union, Adjoin - functions Eight axioms of sets (page 257): 1. the only sets are EmptySet and those made by adjoining something to a set {x|s2} where Set(s2). The differences between lists and sets Order and repetition of an element

CS 471/598 by H. Liu12 Logical agents for Wumpus Reflex agents classify percepts and act Model-based agents have an internal representation Goal-based agents form goals and achieve them The first-order axioms are much more concise than propositional logic axioms

CS 471/598 by H. Liu13 Constructing a logical agent Define the interface (percepts) between the environment and the agent Including time using a time stamp  Percept ([Stench, Breeze,Glitter, None,None], 5) Define actions Actions: Turn(Right), Forward, Shoot, Grab, Release, Climb Provide an action:  a BestAction(a, 5) - a/Grab Modify the environment  t,s,b,m,c Percept([s,b,Glitter,m,c],t)  Glitter(t)  t Glitter(t)  BestAction(Grab,t)  This implements a simple reflex behavior

CS 471/598 by H. Liu14 Deducing hidden properties Synchronic rules (vs. diachronic) Causal rules specify the assumed direction of causality - model-based reasoning  Squares adjacent to pits are breezy  A pit causes … Diagnostic rules infer hidden properties from the percept-derived information  If a location is smelly, the wumpus must either be in that location or in an adjacent location  If there is breeze, …

CS 471/598 by H. Liu15 Representing change Storing a complete percept sequence is tedious and inefficient to search for actions An internal model allows an agent to know its current status having gold and at home square Representing change is one of the most important tasks in knowledge representation How to represent change?

CS 471/598 by H. Liu16 Ways of representing change The latest case only, forget about the past = having a shallow memory and no history = repeating errors Each state represented by a KB can’t reason about >1 situation simultaneously need to represent different situations/actions in one KB Situation calculus (Chapter 10.3) representing situations and actions as representing objects

CS 471/598 by H. Liu17 Situation calculus A particular way of describing change in FOL Each situation is a snapshot of the state Situations are generated from previous situations by actions (Fig 10.2, p329) Give an extra situation argument for every relation/property that can change over time it’s always the last one argument  At(Agent,[1,1],S 0 )^ At(Agent,[1,2],S 1 ) using Result(action, situation)  Result(Forward, S 2 ) = S 3

CS 471/598 by H. Liu18 Special axioms Effect axioms - actions are described by stating their effects Holding-gold via Grab, !Holing-gold via Release Are the above enough? Frame axioms - describing how the world stays the same Holding-sth not releasing it, then holding it next state !Holding-sth not (grab or present or portable) The two types of axioms together describe the world in change.

CS 471/598 by H. Liu19 Successor-state (SS) axioms - resulting from the combining of the E- and F- axioms true afterwards  [an action made it true v true already and no action made it false] One SS axiom is needed for each predicate changing with time A SS axiom must list all the ways in which the predicate can become true or false

CS 471/598 by H. Liu20 Keep track of location What direction an agent is facing Orientation(Agent,S 0 ) = 0 How locations are arranged (via a map)  x,y LocationToward([x,y],90)=[x,y+1] Location l ahead of agent p:  p,l,s At(p,l,s)   x,y Adjacent(x,y)   d x=LocationToward(y,d) What’s known about the map  x,y Wall([x,y])  (x=0 or x=5 or y=0 or y=5)

CS 471/598 by H. Liu21 What actions change locations Going forward changes location What actions change orientations Turning changes orientation There are still many research issues: frame problems - the property remains unchanged qualification problem - an action guaranteed to work ramification problem - implicit consequences of an action

CS 471/598 by H. Liu22 Which action Different actions can achieve the same goal depending on constraints Separating facts about actions from facts about goals as goals describe the desirability of outcome states desirability scale: great, good, medium,risky, deadly Defining the desirability of actions, leaving the inference to choose an action that has the highest desirability

CS 471/598 by H. Liu23 A goal-based agent Certain actions lead to radical policy change: getting the gold -> returning  s Holding(Gold,s)  GoalLocation([1,1],s) Explicit goals allow many ways to work out a sequence of actions Inference Search Planning

CS 471/598 by H. Liu24 Summary FOL is a general-purpose representation language based on objects and relations BNF of FOL A logical agent using FOL Situation calculus to handle changes Causal rules are often more flexible and entail a wider range of consequences We’re ready to infer in FOL...