Logic and Proof 2

Conjunctive NF

Push ∨ in, Simplify

Deducible

S ⊢ A, if there is a finite proof of A starting from elements of S

Entails

S ⊨ A: every interpretation satisfying S, satisfies A A ⊨ B if ⊨I A then ⊨I B for every I A ⊨ B iff ⊨ A → B

Modal Operators (W, R)

is a modal frame where W set of possible worlds & R: accessibility relation

⊨ A

means A is universally valid

⊨w,R A

means w ⊨ A for all w & all I

Resolution Heuristics

orderings to focus search, subsumption or deleting redundant clauses, indexing with elaborate data structures for speed, pre-processing to remove tautologies, symmetries, weighting to give priority to some clauses over those containing unwanted constants

If θ unifies t & u

then so does θ ◦ σ, t(θ ◦ σ) = tθσ = uθσ = u(θ ◦ σ)

w ⊨ A ∧ B <=>

w ⊨ A & w⊨ B,

w ⊨ □A <=>

v ⊨ A for all v s.t. R(w,v),

w ⊨◇A <=>

v ⊨ A for some v s.t. R(w,v)

Semantics of Propositional Modal Logic

w ⊨ A means A is true in word w, w ⊨ P <=> w ∈ I[P],

Free-variable Tableau Calculus

∀l inserts a new free variable, updates across entire proof tree, unification instantiates any free variable, don't use ∃l, skolemize instead. Algo: Negate, NNF, push ∃,∀ in, skolemise, apply, unify

□ A

A is necessarily true in all accessible worlds

◇A

A is possibly true in all accessible worlds

To convert BDDs Z = (P, X, Y) and Z' = (P', X', Y')

If P = P′ then convert (P, X ∧ X', Y ∧ Y') If P < P' then convert (P, X ∧ Z', Y ∧ Z') If P > P' then convert (P', Z ∧ X', Z ∧ Y')

Deduction Theorem

If S ∪ {A} ⊢ B then S ⊢ A → B

Soundness Theorem

If S ⊢ A then S ⊨ A.

Completeness Theorem

If S ⊨ A then S ⊢ A.

Negation NF

Remove →, Push ¬ in using DM Laws

Skolemization

Start with formula ∀x1...∀xk ∃y A. Choose a fresh k-place function symbol, say f. Delete ∃y and sub f for y: ∀x1...∀xk A[f(x1...xk)/y], repeat for all ∃ qualifiers

Herbrand's Theorem

unsatisfiable S <=> there is a finite unsatisfiable set S' of ground instances of clauses of S

K

pure modal logic

Quantifiers

universal ∀ and existential ∃

¬(∀x A)

≃ ∃x¬A

¬◇A

≃ □¬A ie. A cannot be true A must be false

Satisfies

⊨I A, with interpretation I, A evaluates to t Valid (tautology): ⊨ A, all interpretations satisfy A

T

□A → A (reflexive)

S4

□A→ □□A (transitive)

Sequent Calculus

A1,...,Am => B1,...,Bn means if A1 ∧ ... ∧ Am then B1 ∨ ... ∨ Bn, Ai are assumptions, Bi are goals

Modus Ponens

(A → B A) || B

Resolution Rule

({B,A₁,...,Am } {¬B,C₁,...,Cn}) || {A₁,...,Am,C₁,...,Cn }

Binary Resolution Rule

({B,A₁,...,Am } {¬D,C₁,...,Cn}) || {A₁,...,Am,C₁,...,Cn }σ provided Bσ=Dσ, σ is most general unfier of B and D

Paramodulation Rule

({B[t'],A₁,...,Am }{t=u,C₁,...,Cn}) || {B[u],A₁,...,Am,C₁,...,Cn }σ provided tσ = t'σ

Implication

A → B ≃ ¬A ∨ B A ↔ B ≃ (A → B) ∧ (B → A) ¬(A → B) ≃ A ∧ ¬B

S5

A → □◇A (symmetric)

Distributive Law

A ∨ (B ∧ C) ≃ (A ∨ B) ∧ (A ∨ C)

Equivalence

A ≃ B iff A ⊨ B and B ⊨ A A ≃ B iff ⊨ A ↔ B

Bound

All occurrences of x in ∀x A and ∃x A; Free if not bound Sub A[t/x]: sub t for x in A, t must be free in A

Prenex NF

Convert to NNF, Move quantifiers to front

Herbrand Universe

H₀ ≝ set of constants in S (non-empty) Hi+1 ≝ Hi ∪ {f(t1,..., tn) | t1,..., tn ∈ Hi, f is in S}. H ≝∪ Hi, i ≥ 0 H consists of terms in S that contain no variables (ground terms) Hi contains terms with ≤ i nested function apps.

Valuation

V variables → D, supplies values to free variables. IV[x] = V(x), IV[c] = I[c], IV[f(t1,...,tn)] = I[f](Iv[t1],...,IV[tn])

The Saturation Algorithm

all clauses start passive. Transfer a current clause to active, Form all resolvents between current & an active clause, Use new clauses to simplify both passive & active, Put new clauses into passive, Repeat.

Function Symbol

each symbol represents n-place function, constant is 0-place function symbol, variable ranges over all individuals, term is a variable, constant or function application f(t1,...,tn) where ti are terms

Relation Symbol

each symbol represents n-place relation, equality is 2 place relation =, atomic formula R(t1,...,tn) n-place relation with a formula built up from atomic formulae using ¬, ∧, ∨

Herbrand Model

every constant stands for itself, every function symbol stands for a term-forming operation, f denotes the function that puts 'f' in front of the given arguments.

Binary Decision Diagrams

canonical form for boolean expressions, decision trees with sharing, detects tautologies and exhibits models. Solid line if true, dashed if false, no duplicates & no redundant tests. To Build: Recursively convert operands to BDDs. Combine operand BDDs, respecting ordering & sharing. Delete redundant variable tests.

Davis-Putnam-Logeman-Loveland Method

decision procedure which finds a contradiction or model: Delete tautological clauses {P,¬P, . . .}. For each unit clause {L}: delete all clauses containing L & delete ¬L from all clauses. Delete all clauses containing pure literals. Perform a case split on some literal. STOP if a model is found

First-order Language Semantics

defined by interpretation I = (D,I), D is non-empty domain/universe, I maps symbols to elements, functions & relations. I[c] ∈ D, I[f] ∈ Dn → D, I[P] ∈ Dn → {t,f}

Herbrand Interpretation

defines n-place predicate P to denote a truth-valued function in Hn → {t,f} making P(t1,...,tn) true iff formula holds in our interpretation of clauses. Unsatisfiable S no Herbrand interpretation satisfies S

Simplification

delete disjunctions of P & ¬P, those that includes another (P ∨ Q) ∧ P ≃ P, (P ∨ A) ∧ (¬P ∨ A) ≃ A

Clause Form

disjunction of literals with empty clause {} equivalent to f meaning contradiction

θ is a unifier of terms t and u

if tθ = uθ.

θ is more general than φ

if φ = θ ◦ σ for some sub σ.

θ is most general

if more general than every unifier

Interpretation

maps variables to real objects, function from propositional letters to {t,f},

⊨w,R,I A

means w ⊨ A for all w in W

Unification Algo

represent terms by binary trees, each term is a variable/constant/pair. Constants do not unify with different constants or pairs, variable x & term t, unifier is [t/x] unless x occurs in t. θ ◦ θ' unifies (t,t') with (u,u') if θ unifies t with u and θ' unifies t'θ with u'θ

Consistent (satisfiable)

some interpretation satisfies all elements, else Inconsistent (Unsatisfiable)

Clause Form Proof

to prove A, translate ¬A into CNF as A1 ∧ • • • ∧ Am, transform the clause set, preserving consistency, deducing {} refutes ¬A, proving A, an empty clause set (all clauses deleted) means ¬A is satisfiable

Tableau Calculus

work in NNF, fewer connectives: ∧, ∨, ∀, ∃, (□,◇), sequents only need one side, eg left:

Factoring Rule

{B₁,...,Bk,A₁,...,Am }/{B₁,A₁,...,Am }σ provided B₁ σ=⋯=Bk σ

Conversion to Clauses

{P(x)} means ∀x P(x)

De Morgan's Laws

¬¬A ≃ A ¬(A ∧ B) ≃ ¬A ∨ ¬B ¬(A ∨ B) ≃ ¬A ∧ ¬B

◇A ≝

¬□¬A

(∀x A) ∧ (∀x B)

≃ ∀x(A ∧ B)

¬(∃x A)

≃ ∀x¬A