computational applied logic · propositional logic statements and their uses what things can one...

Computational Applied LogicCSC 503 Fall 2005

Jon Doyle

Department of Computer ScienceNorth Carolina State University

Propositional logic

NC State University 1 / 77CSC 503 Fall 2005

c© 2005 by Jon Doyle

Propositional logic Statements and their uses

What things can one express?

• Sounds/exclamations/marks• Words• Statements• Sets of statements = theories• Partial statements• Sets of partial statements• Sequences of statements or sets of statements




How to do things with sentences

• Declarative: facts and descriptions• Interrogative: questions• Imperative: commands and pleas




What can I express with statements?

• Knowledge/facts/opinions/conditions• Ignorance/uncertainty• Goals/desires/intentions• Procedures/methods• Propositional attitudes



Propositional logic Language

Statements• Complete statements = propositions

• “Snow is white”

• “Letters”• snow-is-white

• Four-score-and-seven-years-ago-our-fathers-brought-forth-on-this-continent-a-new-nation-conceived-in-Liberty-and-dedicated-to-the-proposition-that-all-men-are-created-equal

• Ignore spelling, just enumerate• A1, A2, . . .




Propositional connectives

∨ Disjunction “or”∧ Conjunction “and”¬ Negation “not”→ Conditional “implies”↔ Biconditional “iff = if and only if”+ Exclusive or “xor”| Sheffer stroke “nand”∨n

at least n

And more besides, when we visit description logicsNC State University 6 / 77

CSC 503 Fall 2005c© 2005 by Jon Doyle


Complex propositions

• All combinators use parentheses to provideunique parse tree.

• We omit parentheses when parse is clear.• p = ¬A ∨ B → C• p = (((¬A) ∨ B)→ C)

• Depth = depth of parse tree (root has depth 0)• Depth(p) = 3

• Support = set of letters appearing in tree• Support(p) = {A, B, C}



Propositional logic Meaning

Meaning of propositions

We only consider standard meanings at this time.• Standard meanings

• True/False (= T/F, 1/0, >/⊥)• Multivalued logics

• Elements of boolean lattices• Belnap 4-valued logic {TT , TF , FT , FF}

• Probabilistic logics• Probability values in [0, 1]

• Fuzzy logics• Possibility values in [0, 1]




Logicians are weird

• Logical meaning 6= English (etc.) meaning• “If 1=2, then I’m the Man in the Moon.”• “She is either a lawyer or a professor.”• “I’ve won every World Cup game I’ve played.”

• Logical meaning is atemporal• 10:00AM — Assert α• 10:01AM — Assert ¬α• Simple inconsistency, or change?




Truth functionality

• Basic connectives are truth functional• Truth of compound statement determined by

truth of the connected substatements• Truth of compound a function of truth of

constituents• Truth tables represent these functions




Connective truth tables

α β (α ∨ β)

T T TT F TF T TF F F

α β (α ∧ β)

T T TT F FF T FF F F

α (¬α)

T FF T





α β (α→ β)

T T TT F FF T TF F T

α β (α↔ β)

T T TT F FF T FF F T





α β (α + β)

T T FT F TF T TF F F

α β (α | β)

T T FT F TF T TF F T




Truth tables• Complete truth tables

• One column for each proposition in formation tree

• Abbreviated truth tables• Omit one or more intermediate propositions




Non-truth-functional connectives

Truth values of component propositions do notdetermine truth value of compound proposition.• α because β

• α causes β

• α necessarily implies β

• α preceded β

• α is a shorter statement than β

• α expresses more information than β

• α is more likely than β

• Alice believes α but Bob claims β




Truth valuations

• Truth assignment A : L→ {T , F}• Truth valuation V : L(L)→ {T , F}

• Required to respect truth tables in every connective

• Valuations must agree on a propositionwhenever they agree on the proposition’ssupport




Logical equivalence

α ≡ β

• Means α and β are logically equivalent• Logical equivalence = agreement w.r.t. every

valuation

• True just in case the truth table column forα↔ β contains only T ’s• Each row corresponds to a class of valuations• Truth table summarizes all valuations restricted to

support of a proposition




Lattice of meanings

Propositional equivalence classes• [α] = {β | α ≡ β}• 2n distinct truth tables over n letters• Thus 2n equivalence classes over n letters

• Form a Boolean lattice with respect to ∧, ∨, ¬• Define lattice order α ≤ β iff [α ∧ β] = [α]




Metalanguage vs. object language

≡ is part of the logical metalanguage• Part of the language we use to talk about logical

statements• Not part of the logical object language in which

propositions are expressed.

Other metalinguistic notions:• Entailment• Satisfiability• Provability




Adequacy

What can one say with a specific set of connectives?

• S is adequate iff every proposition is equivalentto some proposition constructed using onlyconnectives in S

• For every truth-functional α, there is some βover S such that α ≡ β

• Claim: {¬,∧,∨} is adequate. Why?• Claim: {¬,∨} is adequate. Why?




Normal forms

• Literal = letter or negation of a letter: A,¬A• Clause = disjunction of literals: A1 ∨ . . . ∨ An

• Conjunct = conjunction of literals: A1 ∧ . . . ∧ An

• CNF = Conjunctive normal form• Conjunction of clauses• (A1 ∨ . . . ∨ An) ∧ . . . ∧ (B1 ∨ . . . ∨ Bm)

• DNF = Disjunctive normal form• Disjunction of conjuncts• (A1 ∧ . . . ∧ An) ∨ . . . ∨ (B1 ∧ . . . ∧ Bm)




Linguistic expressiveness

Choose or change the basis connectives to improve• Consision of expression• Cardinality of expression• Complexity of expression• Clarity/comprehensibility/convenience of

expression




Validity

• α is valid iff every valuation makes it true• α is a tautology

• Taut = set of all tautologies

• α is nontrivial if neither α nor ¬α are valid




Satisfiability

• α is satisfiable iff some valuation makes it true• α is possibly true

• α is unsatisfiable iff no valuation makes it true• α is a contradiction• ¬α is a tautology




Consistency

• Σ is consistent just in case some valuationmakes every statement in Σ true

• For finite Σ, just in case∧

Σ is satisfiable

• Σ is inconsistent if no valuation makes allstatements true

• α and β are (in)consistent iff{α, β} is (in)consistent




Logical consequence

• α |= β means• V(β) = T whenever V(α) = T• α entails β

• Σ |= β means• V(β) = T whenever V(α) = T for each α ∈ Σ• Σ entails β

• Cn(Σ) is the set of consequences of Σ• Cn(Σ) = {α | Σ |= α}• Taut = Cn(∅)




Properties of consequences

• Monotonic: Σ ⊆ Σ′ implies Cn(Σ) ⊆ Cn(Σ′)• Supra-tautologous: Taut ⊆ Cn(Σ)

• Idempotent: Cn(Cn(Σ)) = Cn(Σ)

• Additive: Σ ⊆ Cn(Σ)




Completeness

A complete theory determines truth values for allpropositions• Σ is complete iff for each p either

• p ∈ Cn(Σ), or• ¬p ∈ Cn(Σ)

• Is {p,¬p} complete?




Models

Models = interpretations that make true• V a model of Σ iff V(α) = T for each α ∈ Σ

• M(Σ) = {V | ∀α ∈ Σ.V(α) = T} is the set of allmodels of Σ.

• Σ ⊆ Σ′ impliesM(Σ′) ⊆M(Σ)



Propositional logic Formalizing theories

So what good is logic?

• Precise concepts for expressing theories• Precise concepts for critiquing theories




The process of logical formalization

• Commence with initial formulation• Common sense• Expertise• Informed speculation• Wild guesses

• Critique the formulation with respect to thedesired qualities

• Correct the visible flaws as seems fit• Continue this process until convergence




The N“C” State Knowledge DiscoveryMethod

• Commence to continuously correct the contentvia the critique categories until convergence

Truth

Correctness**Consistency**CompletenessCategoricity**ContingencyChanceCoverageCourageousness

Goodness

Computability**Complexity**CardinalityCompromisesConvenienceCharityCompactness

Beauty

ClarityComprehensibilityCleavageCogencyCommonsensicalityContinuity

Perfection

ClosenessCumulativityConvergenceConstancy




Logical formalization as search

• Different critiques might suggest incompatiblecorrections; what to desire?

• Applied corrections might not work• Confusion or contradiction can suggest retreat

to prior formulation; divergence• View this process as a search for the right

formulation• Process state as position in a space of assessment

dimensions• Assessment criteria as elements of heuristic

evaluation function• Correction methods as possible actions



Propositional logic Inference

Tableau proofs

• Tableaux = tables• Labeled trees, built up from atomic tableaux• Various nice computational properties

We will consider other proof formalisms later




Atomic propositional tableaux

TA FAT (¬α)

Fα

F (¬α)

Tα





T (α ∧ β)

Tα

Tβ

F (α ∧ β)�

�

Tα

@@

Tβ

T (α ∨ β)�

�

Tα

@@

Tβ

F (α ∨ β)

Fα

Fβ





T (α→ β)�

�

Fα

@@

Tβ

F (α→ β)

Tα

Fβ

T (α↔ β)�

�

Tα

@@

Fα

Tβ Fβ

F (α↔ β)�

�

Tα

@@

Fα

Fβ Tβ




Tableaux construction rules

• Root is proposition under consideration• Apply atomic tableau to some proposition in tree

• Append atomic tableau at end of branch beneath• Head of appended tableau duplicates proposition

being reduced




Tableau properties

Tableau τ , path P on τ , and E an entry on P

• E is reduced iff all entries on the atomic tableauwith root E appear on P

• P is contradictory iff both Tα and Fα appear onP

• P is finished iff it is contradictory or every entryon P is reduced on P

• τ is finished iff every path is finished• τ is contradictory iff every path is contradictory




Tableau proof

Proof by refutation• Tableau proof of α = a contradictory tableau with

root Fα

• ` α means α is tableau provable

• Tableau refutation of α = a contradictory tableauwith root Tα

• α is tableau refutable iff it has a tableaurefutation




Complete systematic tableaux

• Construct increasing sequence of tableaux• Find highest level with unreduced noncontradictory

entry E• Find leftmost path containing such an entry• Adjoin atomic tableau with root E to each such path• Adjunction means τm ⊆ τm+1

• Limit (union) of this sequence is the CST




Properties of CST

• Every CST is finished

• If a CST is contradictory, it contains a finitecontradictory tableau τm

• Thus if a CST is a proof, it is a finite tableau.

• Every CST is finite




Soundness and completeness

What is the relationship between truth and proof?Between entailment and provability?

• Soundness means truth preserving• A logic is sound if ` p implies |= p

• Completeness means proof preserving• A logic is complete if |= p implies ` p

Logicians often will use “completeness proof” tomean a proof of both soundness and completeness




Soundness of tableau proof

• Each satisfying valuation of a formula mustagree with the labels on some path through thetableau

• No valuation can agree with a contradictory path• In a proof, all paths are contradictory




Completeness of tableau proof

• Each finished but noncontradictory pathprovides a counterexample

• Assign T to A if TA appears on the path• Assign F to A otherwise




Tableau proof from premises

• Allow a set Σ of premises for use in proofs• Add a new atomic tableau Tp for each premise p• Tp can be added to any path that does not

contradict it




Complete systematic tableaux frompremises

• Assume an enumeration of the premises• Add premises sequentially to each

noncontradictory finished path





• Sound: Σ ` p implies Σ |= p

• Complete: Σ |= p implies Σ ` p




Compactness

Propositional logic is a compact logic• Σ |= p iff Σ′ |= p for some finite subset Σ′ ⊆ Σ

• One only needs finitely many premises to getany particular consequence

• An infinite set Σ is satisfiable iff every finitesubset of Σ is satisfiable




Deductive closure

The deductive closure of a set of propositionscontains all the statements deducible from the set• Th(Σ) = {p | Σ ` p}

Soundness and completeness mean• Th(Σ) = Cn(Σ)




Deduction theorem

If Σ is finite and∧

Σ is the conjunction of thesestatements, the following conditions are equivalent:• Σ |= α

• |=∧

Σ→ α

• Σ ` α

• `∧

Σ→ α

This shows the desired matching of truth and proof




Alternative proof systems

• Proof by intimidation• “ ‘Shut up’, he explained.”• “Five-finger” argument

• Axiomatic proofs• Natural Deduction proofs




Axiomatic logics

• Axioms• Maybe lots• Axiom schemata

• (α→ (β → α))• ((α→ (β → γ))→ ((α→ β)→ (α→ γ)))• ((¬β → ¬α)→ ((¬β → α)→ β))

• Inference rules• Usually a small set• Modus ponens

• p, p → q ` q




Axiomatic proofs

• Proof = sequence of statements• Each statement either

• An axiom, or• A conclusion of an inference rule applied to preceding

statements

• Final statement is the “theorem”




Natural deduction proofs

• No axioms• Lots of inference rules

• Rules ` p correspond to axioms

• Introduction and discharge of assumptions• Dependency tracking




A sample proof

In the style of Kalish and Montague

Line Statement Justification Deps.1. A→ B Premise {1}2. B → C Premise {2}3. A Hypothesis {3}4. B MP 1,3 {1,3}5. C MP 2,4 {1,2,3}6. A→ C Discharge 3,5 {1,2}7. A→ B ∧ B → C ∧-introduction {1,2}8. (A→ B ∧ B → C)→ (A→ C) Discharge 7,6 {}



Propositional logic Resolution

Resolution

• Language• Inference method• Proof automation• Logic programming




Language

• CNF language:• Literals• Clauses• Formulas

• Set notation:• Clauses as finite sets of literals

• Empty clause � is always false• Formulae as finite sets of clauses

• Empty formula {} is always true

Sets mean syntactic irredundance




Linguistic models

• Partial truth assignment = consistent set ofliterals• Does not contain both A and ¬A• Literals in set = what is assigned T

• Complete truth assignment contains each letteror its negation

• A |= S• Means assignment A satisifies formula (set) S• For each C ∈ S, C ∩ A 6= ∅

• S (un)satisfiable iff there is an (no) assignmentthat satisfies S




Prolog

• Divide clauses into positive and negative literals• Interpret each clause as implication

• A1 ∨ . . . ∨ An ∨ ¬B1 ∨ . . . ∨ ¬Bm• B1 ∧ . . . ∧ Bm → A1 ∨ . . . ∨ An

• Horn clause: at most one positive literal• Program clause: exactly one positive literal• Prolog program = set of program clauses




Prolog notation

• Rule: some negative literals• Fact or unit clause: no negative literals• Goal clause: no positive literals

Rule A← B1, . . . , Bm A :− B1, . . . , BmFact A← A :−Goal ← B1, . . . , Bm :− B1, . . . , Bm

Nomenclature for clause parts:head :− bodygoal :− subgoals




Modus Ponens in Clausal Form

Modus Ponens:• From α and α→ β infer β

• From α and ¬α ∨ β infer β

Cut rule generalizes Modus Ponens:• From α ∨ γ and α→ β infer β ∨ γ

• From α ∨ γ and ¬α ∨ β infer β ∨ γ




Resolution rule

Resolving on literal A:• Clause {A} ∪ C1

• Clause {¬A} ∪ C2

• Infer C1 ∪ C2




Resolution deduction

A resolution deduction of C from S consists of• A finite sequence C1, . . . , Cn with Cn = C• Each Ci is either

• A clause in S or• The resolvent of two preceding clauses in the

sequence




Resolution refutations

A resolution refutation of S is a resolution proof of �from S• Resolution preserves satisfiability

• Clauses {A} ∪ C1, {¬A} ∪ C2• Resolvent C1 ∪ C2

• Hence refutation is sound




Resolution trees

A resolution tree deduction of C from S:• A labeled binary tree such that• The root is labeled with clause C• The leaves are labeled with the clauses of S• Each nonleaf node is labeled with resolvents of

its children




Resolution closure

The resolution closure R(S) of a set of clauses S isthe closure of S under the operation of takingresolutions• S ⊆ R(S)

• If C1, C2 ∈ R(S) and C is a resolvent of C1 andC2, then C ∈ R(S)

There is a resolution refutation of S iff � ∈ R(S)




Semantic analysis

Formula S, literal `

• Literal reductions:S(`) = {C ∈ R(S) | `,¬` /∈ C}• If S is unsatisfiable, then so is S(`)

• S` = {C − {`} | C ∈ S ∧ ` /∈ C}• Formula reduced by assuming ` is true• If S is unsatisfiable, both S` and S¬` must be

unsatisifable





• S is satisfiable iff either S` or S¬` is satisfiable• The unsatisifiable sentences U are generated

by• If � ∈ S, then S ∈ U• If S` ∈ U and S¬` ∈ U, then S ∈ U

• If S is unsatisfiable, then � ∈ R(S)




Computational complexity

SAT = set of all satisfiable formulae

Is S satisfiable?• Resolution answer

• 2-SAT is linear time• SAT is NP-complete• 3-SAT is NP-complete

This is good news too, not just bad;more on this later




Restricted resolution

• T -resolution: never resolve a tautology• Semantic resolution: one parent is falsified by

assignment A• Ordered resolution: order letters, always resolve

on highest-index letter possible• Support restriction: never resolve two clauses

outside support clauses

These are sound and complete, but others are not




Linear resolution

A linear resolution deduction of C from S is asequence of pairs 〈C0, B0〉, . . . , 〈Cn, Bn〉 such that• C = Cn

• C0 ∈ S• Each Bi is either in S or is some preceding Cj

• Each Ci+1 is a resolvent of Ci and Bi .

C is linearly deducible (refutable) from S if there is alinear deduction (refutation) of C from S




Nomenclature

• S = input clauses• C0 = starting clauses• Ci = center clauses• Bi = side clauses





• Linear resolution is sound (by restriction)• Linear resolution is complete




Linear input resolution

• Starts with goal clause• All side clauses are input clauses

• Incomplete in general• Consider all clauses of two literals

• Complete when all inputs are program clauses




Refinements• LD-resolution = linear definite resolution

• Ordered literals = definite clauses• Resolutions maintain ordering within insertions

• SLD-resolution = selected linear definiteresolution• Resolutions follow syntactic ordering of literals• Prolog: always resolve on first goal literal

Both are sound and complete




Search and backtracking

• Success and failure on resolution paths• Success = find � on path• Failure = end path with no �

• Search all paths until success or exhaustion

• Depth-first search, breadth-first search, etc.• Pure backtracking DFS can fail!• “Intelligent” backtracking schemes



computational applied logic · propositional logic statements and their uses what things can one...

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