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L ogics for D ata and K nowledge R epresentation

L ogics for D ata and K nowledge R epresentation. Propositional Logic. Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese. Outline. Introduction Syntax Semantics Truth valuation Satisfiability Validity Entailment

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L ogics for D ata and K nowledge R epresentation

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  1. Logics for Data and KnowledgeRepresentation Propositional Logic Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

  2. Outline • Introduction • Syntax • Semantics • Truth valuation • Satisfiability • Validity • Entailment • Logical implication • Reasoning Services • Normal Forms 2

  3. Introduction: propositions INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • All knowledge representation languages deal with sentences • Sentences denote propositions (by definition), namely they express something true or false (the mental image of what you mean when you write the sentence) • Logic languages deal with propositions • Propositional logic is the simplest logic that does this (no individuals, no quantifiers) “All men are mortals” “Obama is the president of the USA” ¬ Mortal ∨ Man President

  4. Introduction: example INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS Mental Model Language L World Monkey SEMANTIC GAP PROPOSITION SENTENCE 4

  5. Introduction: the famous triangle INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS CONCEPT (what we have in mind) refers to symbolizes Monkey REFERENT (the real object) SYMBOL (what we write) stands for 5

  6. Σ0 Descriptive Logical , , , … Variables they can be substituted by any proposition or formula P, Q, ψ … Constants one proposition only A, B, C … Language (Syntax) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Alphabet of symbols: The first step in setting up a formal language is to list the symbols of the alphabet • Propositional Logic (PL) is a symbolic language • Σ0: the alphabet of a PL language NOTE: not only characters but also words (composed by several characters) like “monkey” are descriptive symbols

  7. Additional Symbols INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Auxiliary symbols • Parentheses: ( ) • Defined symbols • Logical defined constants are, for all propositions P: ⊥ (falsehood symbol, false, bottom) ⊥=df P∧¬P T(truth symbol, true, top) T=df ¬⊥ • Defined symbols are not strictly necessary • However, they increase the usability by reducing the syntactic complexity of propositions

  8. Yes, ψ is correct! PARSER ψ, PL No Formation Rules (FR): well formed formulas INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Well formed formulas (wff) in PL can be described by the following BNF (*) grammar (codifying the rules): <Atomic Formula> ::= A | B | ... | P | Q | ... | ⊥ | ⊤ <wff> ::= <Atomic Formula> | ¬<wff> |<wff>∧ <wff> | <wff> ∨ <wff> • Atomic formulas are also called atomic propositions • Wff are propositional formulas (or just propositions) • A formula is correct if and only if it is a wff • Σ0 + FR define a propositional language (*) BNF = Backus–Naur form (formal grammar)

  9. Example: the Monkey-Banana problem INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Significant data: monkey, bananas, box • Significant knowledge: Low, High, Climb, GetBanana • All other data and knowledge are irrelevant! • Both data and knowledge are codified as (atomic or more complex) formulas in the language • Language (any language you want to define): MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana “There is a monkey in a laboratory with some bananas hanging out of reach from the ceiling. A box is available that will enable the monkey to reach the bananas if he climbs on it. The monkey and box have height Low, but if the monkey climbs onto the box he will have height High, the same as the bananas. At this point the monkey can get a banana [...]”

  10. Examples of formulas INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS NOTE: In PL we do not have individuals (it is not so expressive). Parentheses in inLab(Monkey) are not the auxiliary symbols like in e.g. (Monkey ∧ GetBanana) ∨ ¬ High(problem of ambiguity)

  11. Propositional Theory INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Propositional (or sentential) theory • A set of propositions • It is a (propositional) knowledge base (containing true facts) • It corresponds to a TBox (terminology) only, where no meaning is specified yet: it is a syntactic notion • Recall that: • knowledge is a set of facts, i.e. statements • we assumed statements are propositions

  12. Propositional Theories and Databases INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • A propositional theory is not a database! • Propositionsdon’t show data explicitly (no individuals) • In PL the form P(a) must be interpreted as P-a (so the use of form P(a) is ambiguous). • Example: In the sentence ‘inLab(Monkey)’, the string ‘Monkey’does not refer to an individual from some data domain

  13. Example: the Monkey-Banana problem INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS “There is a monkey in a laboratory with some bananas hanging out of reach from the ceiling. A box is available that will enable the monkey to reach the bananas if he climbs on it. The monkey and box have height Low, but if the monkey climbs onto the box he will have height High, the same as the bananas. At this point the monkey can get a banana [...]” L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana, , , } T = { (MonkeyLow  BananaHigh  MonkeyGetBanana)  (MonkeyLow  MonkeyClimbBox)  ( MonkeyLow  BananaHigh  MonkeyGetBanana)} NOTE: We still need to assign semantics to the symbols in the language.

  14. Semantics means providing an interpretation INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • So far the elements of our propositional language are simply strings of symbols without formal meaning • The meanings which are intended to be attached to the symbols and propositions form the intended interpretation ν (nu) of the language

  15. Semantics: formal model INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Intensional interpretation We must make sure to assign the formal meanings out of our intended interpretation to the (symbols of the) language, so that formulas (propositions) really express what we intended. • The mental model: What we have in mind? In our mind (mental model) we have a set of properties that we associate to propositions. We need to make explicit (as much as possible) what we mean. • The formal model This is done by defining a formal model M. Technically: we have to define a pair (M,⊨) for our propositional language • Truth-values In PL a sentence A is true (false) iff A denotes a formal object which satisfies (does not satisfy) the properties of the object in the real world.

  16. Example: Banana (I) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Bananas may differ in many ways, but when using the proposition Banana we have in mind some specific properties:e.g. a given shape, a given color ... • These specific properties determine the intension of the proposition Banana • The intentional interpretation of the proposition Banana determines its truth-value (true or false).

  17. Example: Banana (II) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS Mental Model Language L Language L’ World Banana Banana  (Yellow  Curve) PROPOSITION What we have in mind? 1) There is at least a banana 2) The banana is yellow 3) The banana has curve shape 4) No matter if opened SENTENCE SENTENCE OUR MENTAL MODEL MATCHES WITH THE REAL WORLD, SO THE SENTENCES ARE TRUE SEMANTIC GAP 17

  18. Example: Banana (III) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS Mental Model Language L Language L’ World Banana Banana  (Yellow  Curve) PROPOSITION What we have in mind? 1) There is at least a banana 2) The banana is yellow 3) The banana has curve shape 4) No matter if opened SENTENCE SENTENCE OUR MENTAL MODEL DOES NOT MATCH WITH THE REAL WORLD, SO THE SENTENCES ARE FALSE SEMANTIC GAP 18

  19. Example: Banana (IV) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS THE WORLD MENTAL MODEL MENTAL MODEL I mean banana as something yellow or red, which can be one or many together, no matter how big they are, and whether they are opened or not I mean banana as something yellow and curve, no matter if partially opened T F T F FORMAL MODEL M1 FORMAL MODEL M2 19

  20. Truth-values INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Definition: a truth valuation on a propositional language L is a mapping ν assigning to each formula A of L a truth value ν(A), namely in the domain D = {T, F} • ν(A) = T or F according to the modeler, with A atomic • ν(¬A) = T iff ν(A) = F • ν(A∧B) = T iff ν(A) = T and ν(B) = T • ν(A∨B) = T iff ν(A) = T or ν(B) = T • ν(⊥) = F (since ⊥=df P∧¬P) • ν(⊤) = T (since ⊤=df ¬⊥)

  21. Example of truth valuation ν INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana, , , } T = { (MonkeyLow  BananaHigh  MonkeyGetBanana)  (MonkeyLow  MonkeyClimbBox)  ( MonkeyLow  BananaHigh  MonkeyGetBanana)} MODEL Informal Semantics: “If the monkey is low and the banana is high in position, then the monkey cannot get the banana. “ Formal Semantics: ν(MonkeyLow) = T ν(BananaHigh) = T ν(MonkeyClimbBox) = F ν(MonkeyGetBanana) = F 21

  22. Truth Tables INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • To compute (in polynomial time) truth valuations, the method of truth tables was introduced (Wittgenstein, 1921). • Truth tables are well-known: Truth-table (TT) for the logical constants ¬ , ∧, ∨

  23. Truth Relation (Satisfaction Relation) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Let ν be a truth valuation on language L, we define the truth-relation (or satisfaction-relation) ⊨ and write ν⊨ A (read: ν satisfies A) iff ν(A) = True • Given a set of propositions Γ, we define ν ⊨ Γ iff if ν⊨ θ for all formulas θ ∈Γ

  24. Model and Satisfiability INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Let ν be a truth valuation on language L. ν is a model of a proposition P (set of propositions Γ) iff ν satisfies P (Γ). • P (Γ) is satisfiable if there is some (at least one) truth valuation ν such that ν⊨ P (ν⊨Γ).

  25. Truth and Validity INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Let ν be a truth valuation on language L. • P is true underν if ν⊨ P • P is valid if ν⊨ P for all ν (notation: ⊨ P) • P is called atautology

  26. Example INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana, , , } MODEL Informal Semantics: “If the monkey is low and the banana is high in position, then the monkey cannot get the banana. “ Formal Semantics: ν(MonkeyLow) = T ν(BananaHigh) = T ν(MonkeyClimbBox) = F ν(MonkeyGetBanana) = F • ν⊨ MonkeyLow and ν⊨ BananaHigh [SATISFIABLE] • ν⊨ MonkeyClimbBoxand ν⊨MonkeyGetBanana [SATISFIABLE] • MonkeyLow   MonkeyLow is a tautology [VALID] 26

  27. Entailment INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Propositional entailment  ⊨ψ where  = {θ1, ..., θn} is a finite set of propositions ν⊨θi for all θi in  implies ν⊨ ψ 27

  28. Example INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana, , , } MODEL Informal Semantics: “If the monkey is low and the banana is high in position, then the monkey cannot get the banana. “ Formal Semantics: ν(MonkeyLow) = T ν(BananaHigh) = T ν(MonkeyClimbBox) = F ν(MonkeyGetBanana) = F •  = {MonkeyLow, BananaHigh} ν ⊨ MonkeyLow and ν ⊨ BananaHigh implies ν⊨MonkeyGetBanana therefore  ⊨ MonkeyGetBanana [ENTAILMENT] 28

  29. Entailment INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Difference in notation ⊨ψiff  ⊨ψ Left side means we have already filtered out all the models that do not satisfy  Right side means that we start from all models. In any case the final result is the same. 29

  30. Implication: Premise (I) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Propositional entailment in the finite, i.e.  ⊨ψ where  = {θ1, ..., θn} is a finite set of propositions • How to embed such a notion directly in the language? As we will see, this would allow us to use truth tables and directly reformulate entailment in terms of the other reasoning services 30

  31. Implication: Premise (II) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Entailment can be viewed as thelogical implication (θ1 ∧ θ2 ∧ ... ∧ θn) → ψ to be read θ1 ∧ θ2 ∧ ... ∧ θnlogically impliesψ • → is a new symbol that we add to the language 31

  32. Implication and equivalence INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • We extend our alphabet of symbols with the following defined logical constants:→ (implication)↔ (double implication or equivalence) <Atomic Formula> ::= A | B | ... | P | Q | ... | ⊥ | ⊤ <wff> ::= <Atomic Formula> | ¬<wff> |<wff>∧ <wff> | <wff>∨ <wff> | <wff> →<wff> | <wff> ↔<wff> (new rules) • The operator precedence is (from higher to lower): ¬, ∧, ∨, →, ↔ • Let propositions ψ, θ, and finite set {θ1,...,θn} of propositions be given. We define: • ⊨θ → ψ iff θ⊨ψ • ⊨ (θ1∧...∧θn) → ψ iff {θ1,...,θn} ⊨ψ • ⊨θ ↔ψ iff θ → ψ and ψ → θ 32

  33. Truth Table of → INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • By defining the semantics of → in terms of ⊨we have the following truth-table for the logical implication P→Q: REMEMBER: Given ν, P⊨Q iff ν⊨P then ν⊨Q. 33

  34. Three Properties of Logical Implication INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Property 1: For all ψ and θ: ψ → θ iff ¬ψ ∨ θ • Property 2: For all ψ: ¬ψ iff ψ →⊥ • Property 3: For all ψ: ⊥ ⊨ψ(inconsistent theories imply any proposition) 34

  35. Examples (tautologies) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Laws for ∧, ¬ and →: (A ∧ ¬B) → ¬(A → B) ¬(A → B) → (A ∧ ¬B) • Pierce’s law: ((A → B) → A) → A • De Morgan’s laws:¬(A ∨ B) → (¬A ∧ ¬B) (¬A ∧ ¬B) → ¬(A ∨ B)¬(A ∧ B) → (¬A ∨ ¬B) (¬A ∨ ¬B) → ¬(A ∧ B) 35

  36. Exercise (tautologies) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Prove the following double implications (i.e., logical equivalences) by using truth tables. 36

  37. Exercise (tautologies) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Prove the following double implications (i.e., logical equivalences) by using truth tables. 37

  38. Reasoning Services INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • The basic reasoningtasks (or “services”) we can represent (and compute) using ⊨ are: • Model Checking (EVAL) • Satisfiability (SAT) • Unsatisfiability (UnSAT) • Validity (VAL) • Entailment (ENT)

  39. Reasoning Services: EVAL Yes EVAL P , ν No INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS Model Checking (EVAL) Is a proposition P true under a truth-valuation ν? Check ν⊨ P Does ν⊨MonkeyLow  MonkeyGetBanana ? 39

  40. Reasoning Services: SAT ν SAT P No Find a ν that ν⊨ MonkeyLow  GetBanana For instance, the truth-valuation νwhere both are true INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS Satisfiability (SAT) Is there a truth-valuation ν where P is true? find ν such that ν⊨ P 40

  41. Reasoning Services (unSAT) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • Unsatisfiability is the impossibility to find a truth-valuation ν • Monkey is satisfiable. • Monkey ∨¬ Monkey is satisfiable. • Monkey∧¬ Monkey is unsatisfiable (inconsistent). • In fact, for every truth-valuation ν, either • ν(A) = T, so ν(¬A) = F, hence ν(P) = F; or ν(A) = F, so ν(¬A) = T, hence ν(P) = T. 41

  42. Reasoning Services: VAL Yes VAL P No Is P = MonkeyLow ∨ MonkeyLow valid?Yes! INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS Validity (VAL) Is P true according to all possible truth-valuation ν? Check whether for all ν, ν⊨ P 42

  43. Reasoning Services (Entailment) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • A set Γ (eventually empty) of propositions θentailsa proposition ψ, written: Γ⊨ψ if ν⊨θ for all θ∈Γ, then ν⊨ψ (for a given ν or for all ν) • If Γ⊨ψ, then we say that ψ is a logical consequence of Γ, and that Γlogically impliesψ. NOTE: ⊨ P is the same as {} ⊨ P

  44. Reasoning Services: ENT Yes Γ, ψ, ν ENT No Does  MonkeyLow ⊨ MonkeyGetBanana ?No! Does MonkeyLow  BananaHigh⊨  MonkeyGetBanana? Yes! INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS Entailment (ENT) All θ ∈Γ true in ν (in all ν) implies ψ true in ν (in all ν) check Γ⊨ψ in ν (in all ν ) by checking that: given that ν⊨θ for all θ∈Γ implies ν⊨ψ 44

  45. Reasoning Services (Entailment) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS Exercises: • Demonstrate that {today} ⊨ today ∨ tomorrow • Demonstrate that {today} ⊭tomorrow • Suppose P = today ∨ tomorrow. Then: • Define Γ such that Γ⊨ P • Define Γ such that Γ⊭ P

  46. Reasoning Services: properties (I) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • EVAL is the easiest task. We just test one assignment. • SAT is NP complete. We need to test in the worst case all the assignments. We stop when we find one which is true. • UnSAT is CO-NP. We need to test in the worst case all the assignments. We stop when we find one which is true. • VAL is CO-NP. We need to test all the assignments and verify that they are all true. We stop when we find one which is false. • ENT is CO-NP. It can be computed using VAL (see next slide) 46

  47. Reasoning Services: properties (II) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • VAL(ψ) iff UnSAT( ψ) • SAT(ψ)   VAL(ψ) iff SAT( ψ)   VAL( ψ) •  ⊨ ψ, where = {θ1, …, θn} iff VAL(θ1 … θn ψ) ψ = (A   A) is valid, while ( A  A) is unsatisfiable ψ = A is satisfiable in all models where ν(A) = T. Informally, the left side says that there is at least an assignment which makes A true and there is at least an assignment which makes A false; the right side says that there is at least an assignment which makes  A true (in other words it makes A false) and there is at least an assignment which makes  A false (in other words it makes A true). This means that not all assignments make A true (false). Take A ⊨ A  B. the formula (A  A  B) is valid. 47

  48. Reasoning Services: properties (III) INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • EVAL is easy. • ENT can be computed using VAL • VAL can be computed using UnSAT • UnSAT is the opposite of SAT • All reasoning tasks can be reduced to SAT!!! SAT is the most important reasoning service. 48

  49. Normal Forms: Literals INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • A literal is either an atomic proposition or the negation of an atomic proposition, i.e., for all propositional variables P P and ¬P are literals. P is apositive literal ¬P is anegative literal Monkey HighBanana ¬GetBanana ¬ClimbBox 49

  50. Normal Forms: Clauses INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS • A (disjunctive) clause is a disjunction of literals α1… αn • A unit clause is a clause that contains only a single literal • An empty clause does not contain literals B ∨ ¬C ∨ ¬D Monkey ∨ ¬Monkey B and ¬C are unit clauses B ∨ ¬C is not 50

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