Symbolic Representation and Reasoning an Overview Stuart C. Shapiro Department of Computer Science and Engineering, Center for Multisource Information Fusion, and Center for Cognitive Science University at Buffalo, The State University of New York 201 Bell Hall, Buffalo, NY

September, 2004S. C. Shapiro2 Introduction Knowledge Representation Reasoning Symbols Logics

September, 2004S. C. Shapiro3 Knowledge Representation A subarea of Artificial Intelligence Concerned with understanding, designing, and implementing ways of representing information in computers So that programs can use this information to derive information that is implied by it, to converse with people in natural languages, to plan future activities, to solve problems in areas that normally require human expertise.

September, 2004S. C. Shapiro4 Reasoning Deriving information that is implied by the information already present is a form of reasoning. Knowledge representation schemes are useless without the ability to reason with them. So, Knowledge Representation and Reasoning

September, 2004S. C. Shapiro5 Knowledge vs. Belief Knowledge: Justified True Belief KR systems operate the same whether or not the information stored is justified or true. So, Belief Representation and Reasoning would be better. But we’ll stick with KR.

September, 2004S. C. Shapiro6 What Is a Symbol? “A symbol token is a pattern that can be compared to some other symbol token and judged equal with it or different from it… Symbols may be formed into symbol structures by means of a set of relations… The `objects’ that symbols designate may include … objects in an external environment of sensible (readable) stimuli.” [Newell & Simon, Concise Encyclopedia of CS, 2004]

September, 2004S. C. Shapiro7 What Is Logic? The study of correct reasoning. Not a particular KR language. There are many systems of logic. With slight abuse, we call a system of logic a logic. KR research may be seen as the search for the correct logic(s) to use in intelligent systems.

September, 2004S. C. Shapiro8 Parts of Specifying a Logic Syntax Semantics Proof Theory

September, 2004S. C. Shapiro9 Syntax The specification of a set of atomic symbols, and the grammatical rules for combining them into well-formed expressions (symbol-structures).

September, 2004S. C. Shapiro10 Syntactic Expressions Atomic symbols Individual constants: Tom, Betty, white Variables: x, y, z Function symbols: motherOf Predicate symbols: Person, Elephant, Color Propositions: P, Q, BdT Terms Individual constants: Tom, Betty, white Variables : x, y, z Functional terms: motherOf(Fred) Well-formed formulas (wffs) Propositions (Proposition symbols) : P, Q, BdT Atomic formulas: Color(x, white), Duck(motherOf(Fred)) Non-atomic formulas: TdB  Td  Bp

September, 2004S. C. Shapiro11 Semantics The specification of the meaning (designation) of the atomic symbols, and the rules for determining the meanings of the well-formed expressions from the meanings of their parts.

September, 2004S. C. Shapiro12 Semantic Values Terms could denote Objects Categories of objects Properties… Wffs could denote Propositions Truth values

September, 2004S. C. Shapiro13 Truth Values Could be 2, 3, 4, …, ∞ different truth values. Some truth values are “distinguished” Needn’t have anything to do with truth in the real world. By default, we’ll assume 2 truth values. Call distinguished one True (T) Call other False (F)

September, 2004S. C. Shapiro14 Proof Theory The specification of a set of rules, which, given an initial collection of well-formed expressions, specify what other well-formed expressions can be added to the collection.

September, 2004S. C. Shapiro15 Proof / Knowledge Base The collection could be A proof A knowledge base The initial collection could be Axioms Hypotheses Assumptions Domain facts & rules The added expressions could be Theorems Derived facts & rules

September, 2004S. C. Shapiro16 Example Logic: Standard Propositional Logic Domain: CarPool World Atomic Proposition Symbols: BdT, TdB, Bd, Td, Bp, Tp Unary wff-forming connective:  Binary wff-forming connectives: , , , 

September, 2004S. C. Shapiro17 Intended Interpretation (Intensional Semantics) BdT: Betty drives Tom TdB: Tom drives Betty Bd: Betty is the driver Td: Tom is the driver Bp: Betty is the passenger Tp: Tom is the passenger

September, 2004S. C. Shapiro18 Extensional (Denotational) Semantics BdTTTTFF TdBTTFTF BdTTTFF TdTFFTF BpTFFTF TpTFTFF Bd  Tp TFTFF Td   Td TTTTT Td   Td FFFFF 5 of 2 6 = 64 possible situations

September, 2004S. C. Shapiro19 Properties of Wffs Satisfiable T in some situation BdTTTTFF TdBTTFTF BdTTTFF TdTFFTF BpTFFTF TpTFTFF Bd  Tp TFTFF Td   Td TTTTT Td   Td FFFFF

September, 2004S. C. Shapiro20 Properties of Wffs Contingent T in some, F in some BdTTTTFF TdBTTFTF BdTTTFF TdTFFTF BpTFFTF TpTFTFF Bd  Tp TFTFF Td   Td TTTTT Td   Td FFFFF

September, 2004S. C. Shapiro21 Properties of Wffs Valid T in all situations BdTTTTFF TdBTTFTF BdTTTFF TdTFFTF BpTFFTF TpTFTFF Bd  Tp TFTFF Td   Td TTTTT Td   Td FFFFF

September, 2004S. C. Shapiro22 Properties of Wffs Contradictory T in no situation BdTTTTFF TdBTTFTF BdTTTFF TdTFFTF BpTFFTF TpTFTFF Bd  Tp TFTFF Td   Td TTTTT Td   Td FFFFF

September, 2004S. C. Shapiro23 Logical Implication P 1, …, P n logically imply Q P 1, …, P n |= Q In every situation that P 1, …, P n are True, so is Q.

September, 2004S. C. Shapiro24 Example: CarPool World KB Let KB CPW = Bd   Bp Td   Tp BdT  Bd  Tp TdB  Td  Bp TdB  BdT

September, 2004S. C. Shapiro25 Extensional (Denotational) Semantics BdTTF TdBFT BdTF TdFT BpFT TpTF Only 2 of the 64 situations where KB CPW are T So, e.g., KB CPW, BdT |= Bd   Bp This is how a KB constrains a model to the domain we want.

September, 2004S. C. Shapiro26 Proof Theory Some Rules of Inference P Q P  Q P Q P  Q P Q P P  Q Modus Ponens or  Elimination  Elimination  Elimination  Introduction

September, 2004S. C. Shapiro27 Derivation from Assumptions Q is derivable from P 1, …, P n P 1, …, P n |- Q Starting from the collection P 1, …, P n, one can repeatedly apply rules of inference, and eventually get Q.

September, 2004S. C. Shapiro28 Example: CarPool World Proof BdT  Bd  Tp BdT Bd  Tp Bd Bd   Bp  Bp So, KB CPW, BdT |- Bd   Bp Bd   Bp

September, 2004S. C. Shapiro29 Theoremhood If Q is derivable from no assumptions, |- Q We say that Q is provable, and that Q is a theorem.

September, 2004S. C. Shapiro30 Deduction Theorem P 1, …, P n |= Q iff |= (P 1  · · ·  P n )  Q P 1, …, P n |- Q iff |- (P 1  · · ·  P n )  Q So theorem-proving is relevant to reasoning.

September, 2004S. C. Shapiro31 Properties of Logics Soundness If |- P then |= P (If P is a provable, then P is valid.) Completeness If |= P then |- P (If P is valid, then P is a provable.)

September, 2004S. C. Shapiro32 Soundness vs. Completeness Soundness is the essence of correct reasoning Completeness is less important because it doesn’t indicate how long it might take.

September, 2004S. C. Shapiro33 Commutativity Diagram for Sound and Complete Logics P1, …, Pn |= Q |= (P1  · · ·  Pn )  Q P1, …, Pn |- Q |- (P1  · · ·  Pn )  Q completeness soundness So, whenever you want one, you can do another.

September, 2004S. C. Shapiro34 Use of Commutativity Diagram Refutation proof techniques, such as resolution refutation or semantic tableaux, prove that there can be no situation in which P 1, …, and P n are True and Q is False. These are semantic proof techniques.

September, 2004S. C. Shapiro35 Decision Procedure A procedure that is guaranteed to terminate and tell whether or not P is provable.

September, 2004S. C. Shapiro36 Semidecision Procedure A procedure that, if P is a theorem is guaranteed to terminate and say so. Otherwise, it may not terminate.

September, 2004S. C. Shapiro37 A Tour of Some Classes of Logics Propositional Logics Elementary Predicate Logics Full First-Order Logics

September, 2004S. C. Shapiro38 Propositional Logics Smallest Unit: Proposition/Sentence  propositional logics that are Sound Complete Have decision procedures

September, 2004S. C. Shapiro39 What You Can Do with Propositional Logic BettyDrivesTom  TomDrivesBetty BettyDrivesTom  NearTomBetty TomDrivesBetty  NearTomBetty  NearTomBetty Can derive conclusions even though the “facts” aren’t entirely known.

September, 2004S. C. Shapiro40 Elementary Predicate Logics Propositions plus Predicate (Relation) symbols, Individual terms, variables, quantifiers  elementary predicate logics that are Sound Complete Have decision procedures

September, 2004S. C. Shapiro41 What You Can Say with Elementary Predicate Logic  x[Elephant(x)  HasA(x, trunk)] Can state generalities before all individuals are known.  x[Elephant(x)  Color(x, white)] Can describe individuals Even when they are not specifically known.

September, 2004S. C. Shapiro42 Full First-Order Logics Elementary predicate logic plus Function symbols/ functional terms  full first-order logics that are Sound None are Complete Have decision procedures

September, 2004S. C. Shapiro43 What You Can Say with Full First-Order Logic  p[HasProp(0, p)  x[HasProp(x, p)  HasProp(x+1, p)]   x HasProp(x, p)] Principle of induction.

September, 2004S. C. Shapiro44 Example of Undecidability Large KB about ducks, etc.  x[  y (Duck(y)  WalksLike(x,y))   y (Duck(y)  TalksLike(x,y))  Duck(x)]  x Duck(motherOf(x))  Duck(x) Duck(Fred)? If Fred is not a duck, possible ∞ loop.

September, 2004S. C. Shapiro45 Unsound Reasoning Induction From Raven(a)  Black(a) Raven(b)  Black(b)  Raven(c)  Black(c)  Raven(d)   Black(d) … Raven(n)  Black(n) To  x[Raven(x)  Black(x)]

September, 2004S. C. Shapiro46 Unsound Reasoning Abduction From  x[Person(x)  Injured(x)  Bandaged(x)] Person(Tom) Bandaged(Tom) To Injured(Tom)

September, 2004S. C. Shapiro47 What’s “First-Order” about First-Order Logics Can’t quantify over Function symbols Predicate symbols Propositions

September, 2004S. C. Shapiro48 Examples of SNePS Reasoning Using a Logic Designed for KRR

September, 2004S. C. Shapiro49 SNePS, A “Higher-Order” Logic : all(R)(Transitive(R) => (all(x,y,z)(R(x,y) and R(y,z) => R(x,z)))). : Bigger(elephants, lions). : Bigger(lions, mice). : Transitive(Bigger). : Bigger(elephants, mice)? Bigger(elephants,mice) Really a higher-order language for a first-order logic

September, 2004S. C. Shapiro50 “Higher-Order” Example 2 : all(source)(Trusted(source) => all(p)(Says(source, p) => p)). : Trusted(Agent007). : Says(Agent007, Dangerous(Dr_No)). : Dangerous(Dr_No)? Dangerous(Dr_No)

September, 2004S. C. Shapiro51 Designing New Connectives : andor(1,1){OnFloor(G2), OnFloor(G1), OnFloor(1), OnFloor(2)}. : OnFloor(G1). : OnFloor(?where)? ~OnFloor(G2) ~OnFloor(1) ~OnFloor(2) OnFloor(G1)

September, 2004S. C. Shapiro52 Belief Change : andor(1,1){OnFloor(G2), OnFloor(G1), OnFloor(1), OnFloor(2)}. : {OnFloor(G2), OnFloor(G1)} => {Location(belowGround)}. : {OnFloor(1), OnFloor(2)} => {Location(aboveGround)}. : perform believe(OnFloor(G2)) : Location(?where)? Location(belowGround) : perform believe(OnFloor(2)) : Location(?where)? Location(aboveGround)

September, 2004S. C. Shapiro53 Summary 1 Symbolic KRR uses logic. There are many logics. The question is which to use.

September, 2004S. C. Shapiro54 Summary 2 A logic has a Syntax Semantics Proof Theory Logics may Be sound Be complete Have a decision procedure

September, 2004S. C. Shapiro55 Summary 3 Logics provide non-atomic wffs That can describe situations Without knowing all specifics

September, 2004S. C. Shapiro56 Summary 4 One can design and build Useful new logics And reasoning systems using them.