Logical and Rule-Based Reasoning Part II

Logical and Rule-Based Logical and Rule-Based Reasoning Reasoning

Part IIPart II

Logic for Cognitive ScienceLogic for Cognitive Science Good News: Good News:

Predicate Logic is sound and complete. A completely rigorous Predicate Logic is sound and complete. A completely rigorous and correct system for predicate logic can be computerized so and correct system for predicate logic can be computerized so that any correct pattern of reasoning in the language can be that any correct pattern of reasoning in the language can be discovered by a computer.discovered by a computer.

Turing's Thesis: any process that can be expressed with a finite Turing's Thesis: any process that can be expressed with a finite set of rules can processed by a digital computer that operates on set of rules can processed by a digital computer that operates on representations in the language of logic. representations in the language of logic.

Bad News: Bad News: A fully correct mechanism for logic problem solving may spend A fully correct mechanism for logic problem solving may spend

exponential time solving problems. So, systems have to sacrifice exponential time solving problems. So, systems have to sacrifice correctness to obtain efficiency. correctness to obtain efficiency.

Turing's Theorem: Predicate Logic has no decision procedure. Turing's Theorem: Predicate Logic has no decision procedure. Although good reasoning can always be discovered (eventually) Although good reasoning can always be discovered (eventually) by a logic problem solver, there is no guarantee that bad by a logic problem solver, there is no guarantee that bad reasoning can be identified as such in a finite amount of time.reasoning can be identified as such in a finite amount of time.

Logic for Cognitive ScienceLogic for Cognitive Science Good News: Good News:

Systems to handle belief, time and other so-called intensional concepts Systems to handle belief, time and other so-called intensional concepts have been developed, and are being adopted by AI researchers. have been developed, and are being adopted by AI researchers.

Bad News:Bad News: Predicate Logic doesn't let you take it back. Standard logic uses Predicate Logic doesn't let you take it back. Standard logic uses

monotonic reasoning, which means that the more information you have monotonic reasoning, which means that the more information you have the more you can prove from it. What if we find that something we knew the more you can prove from it. What if we find that something we knew is in fact false.is in fact false.

Good News:Good News: Non-monotonic logics have been developed that are modifications of Non-monotonic logics have been developed that are modifications of

predicate logic. In these systems you can say 'All birds fly', and then predicate logic. In these systems you can say 'All birds fly', and then assert 'Penguins don't fly' without causing contradictions. Logics with assert 'Penguins don't fly' without causing contradictions. Logics with exceptions.exceptions.

Bad News:Bad News: Predicate Logic doesn't (conveniently) let you handle matters of degree. Predicate Logic doesn't (conveniently) let you handle matters of degree.

If I write 'Tall(John)' then I have said that John really is tall. There is no If I write 'Tall(John)' then I have said that John really is tall. There is no way to say he is sort of tall, or somewhat tall. Similarly you can't (easily) way to say he is sort of tall, or somewhat tall. Similarly you can't (easily) say that the probability that John is tall is 90%. say that the probability that John is tall is 90%.

Good News: Good News: Many-Valued Logics, Logics of Probability, and Fuzzy Logics allow Many-Valued Logics, Logics of Probability, and Fuzzy Logics allow

expression of matters of degree. Fuzzy logics have been found to be expression of matters of degree. Fuzzy logics have been found to be quite useful in AI especially in controlling machines.quite useful in AI especially in controlling machines.

The Psychological Plausibility of LogicThe Psychological Plausibility of Logic

Primary complaint Primary complaint Logic is a poor model of human reasoning.Logic is a poor model of human reasoning. Logic a fine Logic a fine standardstandard for good reasoning but for good reasoning but

not necessarily how human beings actually not necessarily how human beings actually reason. reason.

There are two kinds of evidence for this There are two kinds of evidence for this claim:claim: Logic is too time intensive and full logical Logic is too time intensive and full logical

reasoning requires long derivations using reasoning requires long derivations using inference rulesinference rules

People do not live up to the rules of logic – People do not live up to the rules of logic – Recall the Wason's card selection task .Recall the Wason's card selection task .


The general finding for the Wason card selection task has been replicated again and again on a wide variety of problems

Tversky & Kahneman, 1974 People use heuristics for judgments under

uncertaintyRepresentativeness

Availability

Anchoring and adjustment

“People rely on a limited number of heuristic principles which reduce the complex tasks of assessing probabilities and predicting values to simpler judgmental operations. In general, these heuristics are quite useful, but sometimes they lead to severe and systematic errors”

RepresentativenessRepresentativenessLinda is 31 years old, single outspoken and very bright. She Linda is 31 years old, single outspoken and very bright. She majored in philosophy. As a student she was deeply concerned majored in philosophy. As a student she was deeply concerned with the issues of discrimination and social justice, and also with the issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.participated in anti-nuclear demonstrations.

Rank order the following statements by their probability, using Rank order the following statements by their probability, using 1 for the most probable and 8 for the least probable1 for the most probable and 8 for the least probable

a) Linda is a teacher in elementary school a) Linda is a teacher in elementary school b) Linda works in a bookstore and takes Yoga classesb) Linda works in a bookstore and takes Yoga classesc) Linda is active in the feminist movement c) Linda is active in the feminist movement d) Linda is a psychiatric social worker d) Linda is a psychiatric social worker e) Linda is a member of the League of Women voters e) Linda is a member of the League of Women voters f) Linda is a bank teller f) Linda is a bank teller g) Linda is an insurance salesperson g) Linda is an insurance salesperson h) Linda is a bank teller and is active in the feminist movementh) Linda is a bank teller and is active in the feminist movement

RepresentativenessRepresentativeness

Which sequence is more likely to be Which sequence is more likely to be produced by flipping a fair coin?produced by flipping a fair coin?

HHTHT HHTHT

HHHHHHHHHH

RepresentativenessRepresentativenessKahneman & Tversky: Kahneman & Tversky: people judge the probability of an outcome people judge the probability of an outcome

based on the extent to which it is representative based on the extent to which it is representative of the generating processof the generating process

The fact that The fact that HHTHTHHTHT looks representative of a fair looks representative of a fair coin and coin and HHHHHHHHHH does not reflects our implicit does not reflects our implicit theories of how the world works. theories of how the world works. Easy to imagine how a trick all-heads coin could work: Easy to imagine how a trick all-heads coin could work:

high prior probability.high prior probability. Hard to imagine how a trick “Hard to imagine how a trick “HHTHTHHTHT” coin could work: ” coin could work:

low prior probability.low prior probability.


Response to Time Complexity:Response to Time Complexity: Can be overcome using massively parallel Can be overcome using massively parallel

system like the brain. system like the brain. Poor reasoning of humans does not show that Poor reasoning of humans does not show that

people fail to have logical machinery in their people fail to have logical machinery in their brains. There are just limitations on how they brains. There are just limitations on how they use it: bad memory, poor attention, etc.use it: bad memory, poor attention, etc.

Response to “people aren’t logical”Response to “people aren’t logical” So what if humans don't use logic. TheySo what if humans don't use logic. They ought ought

to. Cognitive science is the study of to. Cognitive science is the study of intelligenceintelligence not stupidity. not stupidity.


Recall the modified Wason selection taskRecall the modified Wason selection task

So, people are in principle logical

In practice, however, they are not because their ideal logical abilities are confronted with sever limitations in working memory

Cannot generate indefinitely long chains of deductions

Cannot generate indefinitely many mental models May require a context to help in the selection of

correct logical rules

Another Approach: dual-process Another Approach: dual-process accounts of reasoningaccounts of reasoning

Divide reasoning into two modular componentsDivide reasoning into two modular components

System 1System 1 Early evolving set of many systemsEarly evolving set of many systems Common between man and animalsCommon between man and animals Innate processes and associative learningInnate processes and associative learning Rapid, parallel, automaticRapid, parallel, automatic Only their result is available to consciousnessOnly their result is available to consciousness

System 2System 2 Late evolving, unique to peopleLate evolving, unique to people Slow, serial processing involving working memorySlow, serial processing involving working memory Capable of abstract, hypothetical thoughtCapable of abstract, hypothetical thought

The Belief-Bias EffectThe Belief-Bias Effect

One of the key methods for demonstrating dual processes in reasoning tasks

Seeks to create a conflict between responses based upon a process of logical reasoning and those derived from prior belief about the truth of conclusions.

In belief-bias experiments, participants are instructed to treat the problem as a logical reasoning task and to endorse only conclusions that necessarily follow from the premises given. In spite of this, intelligent adult populations (undergraduate students) are consistently influenced by the prior believability of the conclusion given as well as by the validity of the arguments presented.

The Belief-Bias EffectThe Belief-Bias EffectTypically, syllogisms are presented for evaluation, which fall into one of the four following categories

The Belief-Bias EffectThe Belief-Bias EffectParticipants are substantially influenced by both the logic of the argument and believability of its conclusion, with more belief-bias on invalid argumentsDual-process accounts propose that although participants attempt to reason logically in accord with the instructions, the influence of prior beliefs is extremely difficult to suppress and effectively competes for control of the responses made.

Study of J. B. T. Evans, 1988Green = BelievableRed - Unbelievable

Rules as Mental RepresentationsRules as Mental RepresentationsRulesRules

People have mental rules. People have mental rules. People have procedures for using these rules to search a space People have procedures for using these rules to search a space

of possible solutions, and procedures for generating new rules. of possible solutions, and procedures for generating new rules.

Procedures for using and forming rules produce the Procedures for using and forming rules produce the behavior. behavior. Rule-based systems:Rule-based systems:

manipulation and transformation of symbolsmanipulation and transformation of symbols

Rule-based systems in AI and cognitive science:Rule-based systems in AI and cognitive science: Newell and Simon, GPS 1950s-60s Newell and Simon, GPS 1950s-60s Expert systems, 1970s-90s. MYCIN Expert systems, 1970s-90s. MYCIN ACT 1983. John Anderson. ACT 1983. John Anderson. SOAR, Newell and his students, 1980s, John E. Laird (Univ. SOAR, Newell and his students, 1980s, John E. Laird (Univ.

Michigan) Michigan) Prolog: logic programmingProlog: logic programming

Example of a rule-based systemExample of a rule-based system

How to get from Chicago to Madison?How to get from Chicago to Madison? IF you want to get to Madison, and you are in IF you want to get to Madison, and you are in

Chicago, and you have no car, THEN take the bus.Chicago, and you have no car, THEN take the bus. IF you want to take the bus from Chicago to Madison, IF you want to take the bus from Chicago to Madison,

THEN go to the bus depot and buy a ticket.THEN go to the bus depot and buy a ticket. IF you want to buy a ticket, THEN get some money.IF you want to buy a ticket, THEN get some money. IF you want to get some money, then go to the bank IF you want to get some money, then go to the bank

and withdraw it.and withdraw it. IF you want to get to Madison and you have a car, IF you want to get to Madison and you have a car,

THEN take highway 94.THEN take highway 94.

IF: conditions (antecedent)IF: conditions (antecedent)THEN: action (consequent)THEN: action (consequent)

Production RulesProduction RulesRules in a rule based systems are often called Rules in a rule based systems are often called production rules production rules Production rules are one of the most popular and widely Production rules are one of the most popular and widely used knowledge representation languages. used knowledge representation languages. Early expert systems used production rules as their main Early expert systems used production rules as their main knowledge representation language. knowledge representation language. Production rule system consists of three componentsProduction rule system consists of three componentsworking memoryworking memory - contains the information that the - contains the information that the system has gained about the problem thus farsystem has gained about the problem thus farrule baserule base - contains information that applies to all the - contains information that applies to all the problems that the system may be asked to solveproblems that the system may be asked to solveInterpreterInterpreter - solves the control problem, i.e., decide - solves the control problem, i.e., decide which rule to execute on each which rule to execute on each selection-execute cycleselection-execute cycle (also called (also called recognition-act cyclerecognition-act cycle).).

Example: Water Jug ProblemExample: Water Jug Problem

We have two jugs: one holds 4 gallons and the We have two jugs: one holds 4 gallons and the other 3 gallons of waterother 3 gallons of waterThere are no external measuring devicesThere are no external measuring devicesWe can fill-up a jug from a pump at any time We can fill-up a jug from a pump at any time We can pour water out of a jug or from one into We can pour water out of a jug or from one into the otherthe otherThe problem is to start from an initial state (each The problem is to start from an initial state (each state would be the status of the two jugs) and state would be the status of the two jugs) and get to a final state by a sequence of legal movesget to a final state by a sequence of legal moves e.g., from e.g., from [0,0][0,0] (both jugs are empty) to (both jugs are empty) to [2,0][2,0] (the 4- (the 4-

gallon jug has 2 gallons of water and the three gallon gallon jug has 2 gallons of water and the three gallon jug is empty). jug is empty).

Production Rules for the Water Jug Production Rules for the Water Jug ProblemProblem

1. Fill 4-gallon jug: ([x, y] | x < 4) [4, y]2. Fill 3-gallon jug: ([x, y] | y < 3) [x, 3]3. Empty 4-gallon jug: ([x, y] | x > 0) [0, y]4. Empty 3-gallon jug: ([x, y] | y > 0) [x, 0]

5. From 3-gallon jug to 4 gallon jug, until full:([x, y] | x+y 4 and y > 0) [4, y-(4-x)]

6. From 4-gallon jug to 3 gallon jug, until full:([x, y] | x+y 3 and x > 0) [x-(3-y)], 3]

7. All of 3-gallon jug into 4-gallon jug:([x, y] | x+y 4 and y > 0) [x+y, 0]

8. All of 4-gallon jug into 3-gallon jug:([x, y] | x+y 3 and x > 0) [0, x+y]

Suppose we start with two empty jugs, i.e., [0,0]. How do we get to the state [2,2]?

How different from logic?How different from logic?

Less informational power than logic: Less informational power than logic: rule-based system may not have full quantifiers and rules of rule-based system may not have full quantifiers and rules of

inference.inference.

But can be more computationally efficient, because it But can be more computationally efficient, because it focuses on the task to be accomplished.focuses on the task to be accomplished.Uses processes that are not inherently part of logic: e.g., Uses processes that are not inherently part of logic: e.g., subgoaling.subgoaling.Can be tied in with other processes, such as spreading Can be tied in with other processes, such as spreading activation to model human memory (ACT, PI). activation to model human memory (ACT, PI). Can be combined with other representations, such as Can be combined with other representations, such as concepts.concepts.

Advantages of Rule-Based SystemsAdvantages of Rule-Based Systems

Have been used in many commercial Have been used in many commercial systems (e.g., expert systems).systems (e.g., expert systems).Modular, easy to add to.Modular, easy to add to.Have modeled various kinds of Have modeled various kinds of psychological experiments.psychological experiments.Lots of human knowledge and ability Lots of human knowledge and ability naturally described in terms of rules.naturally described in terms of rules.Various techniques known for learning Various techniques known for learning rules automatically.rules automatically.

Expert SystemsExpert Systems

A production rule system combined with an A production rule system combined with an inference engine and a user interfaceinference engine and a user interface

It is often implemented as a “conversational” or It is often implemented as a “conversational” or interactive systeminteractive system Pre-defined questions are designed to reduce the Pre-defined questions are designed to reduce the

search space for rules whose antecedents are search space for rules whose antecedents are satisfied.satisfied.

WeaknessesWeaknessesInflexibilityInflexibility To change the behavior of the systems must change To change the behavior of the systems must change

the rulesthe rules

Over-generalityOver-generality Rules may not be able to deal with specific problems Rules may not be able to deal with specific problems

that are not part of the knowledge basethat are not part of the knowledge base

Control may be difficultControl may be difficult Actions are performed only if the rule conditions are Actions are performed only if the rule conditions are

satisfiedsatisfied

Knowledge acquisition is difficultKnowledge acquisition is difficult The most challenging part: knowledge engineeringThe most challenging part: knowledge engineering E.g., in expert systems, the knowledge of many E.g., in expert systems, the knowledge of many

experts and about many situations must be encoded experts and about many situations must be encoded as rules (often manually).as rules (often manually).

Rules and Cognitive ScienceRules and Cognitive ScienceJohn Anderson and ACT-RJohn Anderson and ACT-R Cognitive skills are realized by production rules. Cognitive skills are realized by production rules. Production rules are organized around a set of goals. Production rules are organized around a set of goals. Complex cognitive processes involve a sequence of Complex cognitive processes involve a sequence of

production rules. production rules. Productions are matched against working memory. Productions are matched against working memory. Rules are psychologically realistic, because they Rules are psychologically realistic, because they

describe many aspects of skilled behavior, and describe many aspects of skilled behavior, and predict the details of that behavior. predict the details of that behavior.

Rules and Problem SolvingRules and Problem SolvingForward chaining: Forward chaining:

do deductive reasoning forward by modus ponens: if p then q; p; do deductive reasoning forward by modus ponens: if p then q; p; so q. But use strategies to focus inference, e.g. use most so q. But use strategies to focus inference, e.g. use most specific rule.specific rule.

Good for planning. Start with initial state, work forward to goal.Good for planning. Start with initial state, work forward to goal. Reasoning is search through a space of states and operators.Reasoning is search through a space of states and operators.

Backward chaining: Backward chaining: if p then q; q; so check whether p can be accomplished. Good if p then q; q; so check whether p can be accomplished. Good

for diagnosis (explanation), planning. Start with goal, work for diagnosis (explanation), planning. Start with goal, work backward to current state.backward to current state.

Bidirectional search: forward and backward.Bidirectional search: forward and backward.Most of the successful commercial expert systems are Most of the successful commercial expert systems are rule-based.rule-based.Modularity of rules: just add more to the rule base.Modularity of rules: just add more to the rule base.

Learning RulesLearning RulesRules can be learnedRules can be learned

Generalization from examples: experience, imitationGeneralization from examples: experience, imitation Fa, Ga Fa, Ga all F are G. Induction. all F are G. Induction.

Rule compilation, chunking (same as transitivity)Rule compilation, chunking (same as transitivity) p p q, q q, q r then p r then p r. r. Important for skill acquisition: chunk together several rules into Important for skill acquisition: chunk together several rules into

one that can be quickly executed.one that can be quickly executed.

Learn by being told – Learn by being told – rote learningrote learning

AbductiveAbductive inference uses rules to form hypotheses inference uses rules to form hypotheses Babies with ear infections cry.Babies with ear infections cry. Adam is a baby and is crying.Adam is a baby and is crying. So maybe Adam has an ear infection.So maybe Adam has an ear infection.

Note that this is not a valid inference, but is a plausible use of Note that this is not a valid inference, but is a plausible use of backward chaining.backward chaining.

Psychological Plausibility of RulesPsychological Plausibility of RulesProduction systems have been used to model Production systems have been used to model performance on many tasks, e.g. chess, Tower of Hanoi.performance on many tasks, e.g. chess, Tower of Hanoi.Quantitative fit: power law of learning. Quantitative fit: power law of learning.

Rate of learning slows down.Rate of learning slows down.

Applies to many kinds of skill acquisition.Applies to many kinds of skill acquisition.Learning in rats: not just associations, but rules.Learning in rats: not just associations, but rules.

E.g. if tone then shock. Conditioning is learning rules.E.g. if tone then shock. Conditioning is learning rules.

Learning of social rules. Learning of social rules. If given something, say thank youIf given something, say thank you more interesting: in explaining other people's behavior more interesting: in explaining other people's behavior

(abduction).(abduction).

Knowledge of physical systems: mental models. Knowledge of physical systems: mental models. If you turn the key, the engine starts.If you turn the key, the engine starts.

Learning language: grammatical rulesLearning language: grammatical rules Past tense: learn Past tense: learn learned; turn learned; turn turned; shock turned; shock shocked shocked But, what about cry But, what about cry cryed? OR go cryed? OR go goed? goed?

Rules and LanguageRules and Language

Basic questions:Basic questions: What representations are required for our ability to understand What representations are required for our ability to understand

and produce language?and produce language? How is language learned?How is language learned?

Behaviorist answer:Behaviorist answer: Language is based on a set of associations, learned by trial and Language is based on a set of associations, learned by trial and

error.error.

Chomsky's revolutionChomsky's revolutionSyntactic Structures (1956) - Syntactic Structures (1956) - Rejected associationismRejected associationism

grammars are complex, rule-like structuresgrammars are complex, rule-like structures universal grammar is innateuniversal grammar is innate we are born with readiness to notice what kind of grammar our we are born with readiness to notice what kind of grammar our

native language has.native language has.

Rules and LanguageRules and LanguageEvidence supports Chomsky’s ViewEvidence supports Chomsky’s Viewproductivity of language: we can understand sentences productivity of language: we can understand sentences that have never been uttered before.that have never been uttered before.

"Colorless green ideas sleep furiously." Compare"Colorless green ideas sleep furiously." Compare

Ease of language learning: Ease of language learning: Almost all children acquire language with relatively little Almost all children acquire language with relatively little

feedback.feedback.

Example of Chomskyan grammar (earlier work):Example of Chomskyan grammar (earlier work): The girl kicked the ball The girl kicked the ball The ball was kicked by the girl. The ball was kicked by the girl.

A syntactic transformation produces a new structure.A syntactic transformation produces a new structure. This explains the productivity of language: transformations can This explains the productivity of language: transformations can

produce an unlimited number of structures.produce an unlimited number of structures.

Learning of grammars is relatively easyLearning of grammars is relatively easy We have innate expectations about structures and We have innate expectations about structures and

transformations.transformations.

Learning is a kind of abduction: children form Learning is a kind of abduction: children form hypotheses to explain the utterances they here.hypotheses to explain the utterances they here.

Rules and LanguageRules and LanguageChomsky's 1980's view: government and binding.Chomsky's 1980's view: government and binding.

Less emphasis on transformations.Less emphasis on transformations. More emphasis on constraints on what can count as grammatical.More emphasis on constraints on what can count as grammatical. Innate universal grammar: E.g. asymmetry of subject and object.Innate universal grammar: E.g. asymmetry of subject and object.

All languages have nouns, verbs, adjectives, and adpositions (pre- or All languages have nouns, verbs, adjectives, and adpositions (pre- or post-positions).post-positions).

XP = X - YP.XP = X - YP. For each X (verb, noun, adjective, preposition) there is a phrase YP For each X (verb, noun, adjective, preposition) there is a phrase YP

(noun phrase, etc.) that can follow it.(noun phrase, etc.) that can follow it.

Children merely need to learn parameters: set of switches to be set. Children merely need to learn parameters: set of switches to be set. But the basics of universal grammar are not learned, and could not But the basics of universal grammar are not learned, and could not be in the time available.be in the time available.

Concepts are also innate and preexisting: children just need to learn Concepts are also innate and preexisting: children just need to learn what labels to apply to them.what labels to apply to them.

Logical and Rule-Based Reasoning Part II

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