1. First-Order Logic Copyright, 1996 © Dale Carnegie & Associates, Inc. Chapter 8 Spring 2007

2. CSE 471/598, CBS 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 D-Backs beat Padres (when?). Squares neighboring the wumpus are smelly

3. CSE 471/598, CBS 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

4. CSE 471/598, CBS 598 by H. Liu4 Domain and domain elements

5. CSE 471/598, CBS 598 by H. Liu5 Syntax Symbols (Fig 8.3, BNF, 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(Richard),Mother(John)) Complex sentences formed by connectives !Brother(Robin,John)

6. CSE 471/598, CBS 598 by H. Liu6 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 if P(x) is true, but not when P(x) is false  x P(x) ^ Q(x) leads to a too strong statement   x King(x) ^ Person(x)

7. CSE 471/598, CBS 598 by H. Liu7 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 , and ^ with 

8. CSE 471/598, CBS 598 by H. Liu8 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.

9. CSE 471/598, CBS 598 by H. Liu9 Connections, Equality The two quantifiers are connected via negation. De Morgan’s rules Do we really need both quantifiers? Some examples  x Likes(x, Icecream) ≡ !  x !Likes(x, Icecream) !  x P ≡  x !P Equality symbol: two terms refer the same object or not the same object Some examples (x = y, !(x = y))

10. CSE 471/598, CBS 598 by H. Liu10 Using FOL Domain – some part of the world The kinship domain (P 254)  Parent, Sibling, Brother, Sister, Child, Daughter, Son, … Axioms - basic facts One’s mother is one’s female parent One’s husband is one’s male spouse 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), as it can be derived from the definition of Sibling(x,y) Two important questions in building a KB Are axioms in the KB sufficient? Are all axioms in the KB necessary?

11. CSE 471/598, CBS 598 by H. Liu11 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 (queries) and getting answers Ask(KB, King(John)) - True Ask(KB,  x Child(x,Spot)) – substitution x/Wonder

12. CSE 471/598, CBS 598 by H. Liu12 The domain of sets and lists in FOL 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

13. CSE 471/598, CBS 598 by H. Liu13 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

14. CSE 471/598, CBS 598 by H. Liu14 Constructing a W-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 Define adjacency of any two squares (x, y) and (a, b)  x,y,a,b Adjacent([x,y],[a,b])  …

15. CSE 471/598, CBS 598 by H. Liu15 Deducing hidden properties Two major types of rules: causal and diagnostic 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, …

16. CSE 471/598, CBS 598 by H. Liu16 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?

17. CSE 471/598, CBS 598 by H. Liu17 Ways of representing change The latest case only, forget about the past = having a shallow memory and no history = repeating errors Situation calculus (Chapter 10.3) A particular way of describing change in FOL representing situations and actions as representing objects  Fluents are functions and predicates that vary from one situation to the next E.g., location, aliveness

18. CSE 471/598, CBS 598 by H. Liu18 Situation calculus 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

19. CSE 471/598, CBS 598 by H. Liu19 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.

20. CSE 471/598, CBS 598 by H. Liu20 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

21. CSE 471/598, CBS 598 by H. Liu21 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)

22. CSE 471/598, CBS 598 by H. Liu22 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 (representational and inferential) qualification problem - an action guaranteed to work, the impossibility to list all the preconditions ramification problem - implicit consequences of an action

23. CSE 471/598, CBS 598 by H. Liu23 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

24. CSE 471/598, CBS 598 by H. Liu24 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

25. CSE 471/598, CBS 598 by H. Liu25 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 now ready to infer in FOL...