FirstOrder Logic Chapter 8 Spring 2004 CS 471598

First-Order Logic Chapter 8 Spring 2004 CS 471/598 by H. Liu Copyright, 1996 © Dale Carnegie & Associates, Inc.

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. n n 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 n The D-Backs beat the Cardinals. (When? ) CS 471/598 by H. Liu 2

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. Liu 3

Syntax Symbols n n n (Fig 8. 4, Page 247) Constant symbols Predicate symbols - relations, tuples Functional symbols - relations Terms - objects, ground (constant symbols) & complex (functions) terms Atomic sentences n Brother(Richard, John), Married(Father(R), Mother(J)) Complex sentences formed by connectives n !Brother(Robin, John) CS 471/598 by H. Liu 4

Quantifiers Universal quantification ( ) - to avoid enumerating the objects by name combining with variables, we can do that: x Cat(x) Mammal(x) n x P(x) Q(x) makes a statement about everything, but not when P(x) is false n x P(x) ^ Q(x) leads to a too strong statement n w x King(x) ^Person(x) CS 471/598 by H. Liu 5

Existential quantification ( ) - make a statement about some object without naming it. n n n 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. Liu 6

Nested quantifiers Multiple quantifiers can be used. The order of quantification is important. n n x y Loves (y, x) y x Loves (y, x) When there is confusion, the variable belongs to the innermost quantifier that mentions it. n 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. Liu 7

Connections, Equality The two quantifiers are connected via negation. n De Morgan’s rules Do we really need both quantifiers? n Some examples Equality symbol: two terms refer the same object or not the same object n Some examples CS 471/598 by H. Liu 8

Using FOL Domain – some part of the world n The kinship domain Axioms - basic facts Definitions - concepts defined by axioms n x, y P(x, y) … Theorems - that are proved using axioms and definitions, or entailed by axioms n x, y Sibling(x, y) Sibling(y, x) Two important questions in building a KB n n Are axioms in the KB sufficient? Are all axioms in the KB necessary? CS 471/598 by H. Liu 9

What should be in a KB n n From a purely logical point of view From a practical point of view Adding sentences (assertions) to a KB n n Tell(KB, King(John)) Tell(KB, x King(x) Person(x)) Asking questions and getting answers n n Ask(KB, King(John)) - True Ask(KB, x Child(x, Spot)) – substitution x/Wonder CS 471/598 by H. Liu 10

The domain of sets and lists Epmty. Set – constant {} Member , Subset - predicates Intersection, Union, Adjoin - functions Eight axioms of sets (page 257): 1. the only sets are Empty. Set and those made by adjoining something to a set {x|s 2} where Set(s 2). The differences between lists and sets n Order and repetition of an element CS 471/598 by H. Liu 11

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. Liu 12

Constructing a logical agent Define the interface (percepts) between the environment and the agent n Including time using a time stamp w Percept([Stench, Breeze, Glitter, None], 5) Define actions n Actions: Turn(Right), Forward, Shoot, Grab, Release, Climb Provide an action: a Best. Action(a, 5) - a/Grab Modify the environment n n t, s, b, m, c Percept([s, b, Glitter, m, c], t) Glitter(t) t Glitter(t) Best. Action(Grab, t) w This implements a simple reflex behavior CS 471/598 by H. Liu 13

Deducing hidden properties Synchronic rules (vs. diachronic) n Causal rules specify the assumed direction of causality - model-based reasoning w Squares adjacent to pits are breezy w A pit causes … n Diagnostic rules infer hidden properties from the percept-derived information w If a location is smelly, the wumpus must either be in that location or in an adjacent location w If there is breeze, … CS 471/598 by H. Liu 14

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 n having gold and at home square Representing change is one of the most important tasks in knowledge representation n How to represent change? CS 471/598 by H. Liu 15

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 n n can’t reason about >1 situation simultaneously need to represent different situations/actions in one KB Situation calculus (Chapter 10. 3) n representing situations and actions as representing objects CS 471/598 by H. Liu 16

Situation calculus A particular way of describing change in FOL Each situation is a snapshot of the state n Situations are generated from previous situations by actions (Fig 10. 2, p 329) Give an extra situation argument for every relation/property that can change over time n it’s always the last one argument w At(Agent, [1, 1], S 0)^ At(Agent, [1, 2], S 1) n using Result(action, situation) w Result(Forward, S 2) = S 3 CS 471/598 by H. Liu 17

Special axioms Effect axioms - actions are described by stating their effects n n Holding-gold via Grab, !Holing-gold via Release Are the above enough? Frame axioms - describing how the world stays the same n n 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. Liu 18

Successor-state (SS) axioms - resulting from the combining of the E- and F- axioms n n n 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. Liu 19

Keep track of location What direction an agent is facing n Orientation(Agent, S 0) = 0 How locations are arranged (via a map) n n n x, y Location. Toward([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=Location. Toward(y, d) What’s known about the map n x, y Wall([x, y]) (x=0 or x=5 or y=0 or y=5) CS 471/598 by H. Liu 20

What actions change locations n Going forward changes location What actions change orientations n 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. Liu 21

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 n 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. Liu 22

A goal-based agent Certain actions lead to radical policy change: getting the gold -> returning n s Holding(Gold, s) Goal. Location([1, 1], s) Explicit goals allow many ways to work out a sequence of actions n n n Inference Search Planning CS 471/598 by H. Liu 23

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. . . CS 471/598 by H. Liu 24