The Complexity of Theorem Proving in Autoepistemic Logic Olaf - PowerPoint PPT Presentation

The Complexity of Theorem Proving in Autoepistemic Logic Olaf Beyersdorff School of Computing University of Leeds, UK 1

What is autoepistemic logic? Autoepistemic Logic ◮ a non-monotone logic, introduced 1985 by Moore ◮ models common-sense reasoning ◮ The language of classical propositional logic is augmented augmented by an unary modal operator L . ◮ Intuitively, for a formula ϕ , the formula L ϕ means that ϕ is believed by a rational agent. 2

Semantics of autoepistemic logic The language L ae consists of the language L of classical propositional logic augmented by an unary modal operator L . Entailment ◮ An assignment is a mapping from all propositional variables and formulas L ϕ to { 0 , 1 } . ◮ For Φ ⊆ L ae and ϕ ∈ L ae , Φ | = ϕ iff ϕ is true under every assignment which satisfies all formulas from Φ. ◮ deductive closure Th (Φ) = { ϕ ∈ L ae | Φ | = ϕ } . 3

Semantics of autoepistemic logic Stable Expansions [Moore 85] ◮ Informally: a stable expansion corresponds to a possible view of an agent, allowing him to derive all statements of his view from the given premises Σ together with his believes and disbelieves. ◮ Formally: a stable expansion of Σ ⊆ L ae is a set ∆ ⊆ L ae satisfying the fixed-point equation ∆ = Th (Σ ∪ { L ϕ | ϕ ∈ ∆ } ∪ {¬ L ϕ | ϕ �∈ ∆ } ) . 4

Examples Exactly one expansion If Σ only consist of objective formulas (no L operators), then the only expansion is the deductive closure of Σ (together with closure under L ). Several expansions { p ↔ Lp } has two stable expansions: ◮ one containing p and Lp ◮ the other containing both ¬ p and ¬ Lp No expansions { Lp } has no stable expansion. 5

Two important problems Credulous Reasoning Problem Instance: a formula ϕ ∈ L ae and a set Σ ⊆ L ae Question: Is there a stable expansion of Σ that includes ϕ ? Sceptical Reasoning Problem Instance: a formula ϕ ∈ L ae and a set Σ ⊆ L ae Question: Does every stable expansion of Σ include ϕ ? 6

Previous results ◮ Semantics and complexity of autoepistemic logic have been intensively studied. ◮ Credulous Reasoning is Σ p 2 -complete. [Gottlob 92] ◮ Sceptical Reasoning is Π p 2 -complete. [Gottlob 92] ◮ Bonatti and Olivetti (ACM ToCL’02) introduced the first purely axiomatic formalism using sequent calculi. 7

Our results ◮ We give the first proof-theoretic analysis of the sequent calculi of Bonatti and Olivetti. ◮ The calculus for credulous autoepistemic reasoning obeys almost the same bounds on the proof size as Gentzen’s system LK , i. e., proof lengths are polynomially related. ◮ For the calculus for sceptical autoepistemic reasoning we show an exponential lower bound to the proof size (even to the number of steps). 8

The proof systems Bonatti and Olivetti’s sequent calculi for autoepistemic logic consist of three main ingredients: ◮ classical sequents and rules from LK , ◮ antisequents to refute non-tautologies, ◮ autoepistemic rules for the L operator. 9

Gentzen’s LK Initial sequents: A ⊢ A , 0 ⊢ , ⊢ 1 Γ ⊢ ∆ Γ ⊢ ∆ (weakening) A , Γ ⊢ ∆ Γ ⊢ ∆ , A Γ 1 , A , B , Γ 2 ⊢ ∆ Γ ⊢ ∆ 1 , A , B , ∆ 2 (exchange) Γ 1 , B , A , Γ 2 ⊢ ∆ Γ ⊢ ∆ 1 , B , A , ∆ 2 Γ 1 , A , A , Γ 2 ⊢ ∆ Γ ⊢ ∆ 1 , A , A , ∆ 2 (contradiction) Γ 1 , A , Γ 2 ⊢ ∆ Γ ⊢ ∆ 1 , A , ∆ 2 Γ ⊢ ∆ , A A , Γ ⊢ ∆ ( ¬ introduction) ¬ A , Γ ⊢ ∆ Γ ⊢ ∆ , ¬ A A , Γ ⊢ ∆ A , Γ ⊢ ∆ Γ ⊢ ∆ , A Γ ⊢ ∆ , B ( ∧ rules) A ∧ B , Γ ⊢ ∆ B ∧ A , Γ ⊢ ∆ Γ ⊢ ∆ , A ∧ B A , Γ ⊢ ∆ B , Γ ⊢ ∆ Γ ⊢ ∆ , A Γ ⊢ ∆ , A Γ ⊢ ∆ , B ∨ A ( ∨ rules) A ∨ B , Γ ⊢ ∆ Γ ⊢ ∆ , A ∨ B Γ ⊢ ∆ , A A , Γ ⊢ ∆ (cut rule) Γ ⊢ ∆ 10

The Antisequent Calculus Axioms: Γ � ∆ where Γ and ∆ are disjoint sets of variables. Γ � Σ , α Γ , α � Σ ( ¬ � ) ( � ¬ ) Γ , ¬ α � Σ Γ � Σ , ¬ α Γ , α, β � Σ Γ � Σ , α Γ � Σ , β ( ∧ � ) ( � •∧ ) ( � ∧• ) Γ , α ∧ β � Σ Γ � Σ , α ∧ β Γ � Σ , α ∧ β Γ � Σ , α, β Γ , α � Σ Γ , β � Σ ( � ∨ ) ( •∨ � ) ( ∨• � ) Γ � Σ , α ∨ β Γ , α ∨ β � Σ Γ , α ∨ β � Σ Theorem (Bonatti 93) The antisequent calculus is sound and complete, i. e., Γ � Σ is derivable iff there is an assignment satisfying Γ , but falsifying Σ . Observation The antisequent calculus is polynomially bounded. 11

The credulous autoepistemic calculus Definition ◮ A provability constraint is of the form L α or ¬ L α with a formula α . ◮ A set E of formulas satisfies a constraint L α if α ∈ Th ( E ). ◮ Similarly, E satisfies ¬ L α if α �∈ Th ( E ). Definition ◮ A credulous autoepistemic sequent Σ; Γ | ∼ ∆ consists of a set Σ of provability constraints, and Γ , ∆ ⊆ L ae . ◮ Semantically, Σ; Γ | ∼ ∆ is true, if there exists a stable expansion of Γ which satisfies all constraints in Σ and contains � ∆. 12

The credulous autoepistemic calculus CAEL Uses rules from LK , the anti-sequent calculus and Γ ⊢ ∆ ( cA1 ) (Γ ∪ ∆ ⊆ L ) ; Γ | ∼ ∆ Γ ⊢ α Σ; Γ | ∼ ∆ ( cA2 ) ( α ∈ L ) L α, Σ; Γ | ∼ ∆ Γ �⊢ α Σ; Γ | ∼ ∆ ( cA3 ) (Γ ∪ { α } ⊆ L ) ¬ L α, Σ; Γ | ∼ ∆ ¬ L α, Σ; Γ[ L α/ ⊥ ] | ∼ ∆[ L α/ ⊥ ] ( cA4 ) Σ; Γ | ∼ ∆ L α, Σ; Γ[ L α/ ⊤ ] | ∼ ∆[ L α/ ⊤ ] ( cA5 ) Σ; Γ | ∼ ∆ In ( cA4 ) and ( cA5 ) L α is a subformula of Γ ∪ ∆ and α ∈ L . 13

The credulous autoepistemic calculus Theorem (Bonatti, Olivetti 02) The calculus is sound and complete, i.e., a credulous autoepistemic sequent is true if and only if it is derivable in CAEL. 14

The credulous autoepistemic calculus Theorem (Bonatti, Olivetti 02) The calculus is sound and complete, i.e., a credulous autoepistemic sequent is true if and only if it is derivable in CAEL. Theorem The length of proofs in CAEL and in LK are polynomially related. The same holds for the number of steps. More precisely s LK ( n ) ≤ s CAEL ( n ) ≤ n ( s LK ( n ) + n 2 + n ) and t LK ( n ) ≤ t CAEL ( n ) ≤ n ( t LK ( n ) + n + 1). where for a proof system P s ∗ P ( x ) = min {| w | : P ( w ) = x } and s P ( n ) = max { s ∗ P ( x ) : | x | ≤ n } 14

A typical derivation Proofs in CAEL are very structured LK ( cA1 ) Γ ′| ∼ ∆ ′ LK / AC ( cA2 ) or ( cA3 ) σ ; Γ ′| ∼ ∆ ′ ( cA2 ) or ( cA3 ) . . . Σ ′′ ; Γ ′| ∼ ∆ ′ LK / AC ( cA2 ) or ( cA3 ) Σ ′ ; Γ ′| ∼ ∆ ′ ( cA4 ) or ( cA5 ) . . . Σ; Γ | ∼ ∆ 15

Sceptical reasoning Simpler sequents ◮ Sequents now only consist of two components Γ , ∆ ⊆ L ae . ◮ An SAEL sequent is such a pair � Γ , ∆ � , denoted by Γ | ∼ ∆. ∼ ∆ is true, if � ∆ holds in ◮ Semantically, the SAEL sequent Γ | all expansions of Γ. 16

The sceptical autoepistemic calculus SAEL The sceptical autoepistemic calculus uses rules from LK , the anti-sequent calculus, and Rules for autoepistemic formulas ¬ L α, Γ | ∼ α L α, Γ �⊢ α Γ ⊢ ∆ ( sA1) ( sA2) ¬ L α, Γ | ( sA3) L α, Γ | Γ | ∼ ∆ ∼ ∆ ∼ ∆ where Γ ∪ { L α } is complete wrt. ELS (Γ ∪ { α } ) in rule ( sA3 ) L α, Γ | ∼ ∆ ¬ L α, Γ | ∼ ∆ ( sA4) ( L α ∈ LS (Γ ∪ ∆)) Γ | ∼ ∆ Theorem (Bonatti, Olivetti 02) The calculus SAEL is sound and complete, i.e., an SAEL sequent Γ | ∼ ∆ is derivable in SAEL if and only if it is true. 17

An exponential lower bound Theorem There exist sequents Γ n | ∼ ∆ n of size O ( n ) such that ∼ ∆ n has 2 Ω( n ) steps. every SAEL-proof of Γ n | Therefore, s SAEL ( n ) ∈ 2 Ω( n ) . Sketch of proof ◮ Let Γ n = ( p i ↔ Lp i , p i ↔ q i ) i =1 ,..., n n � ∆ n = ( Lp i ↔ Lq i ) i =1 ∼ ∆ n contains 2 n ◮ We will prove that each SAEL -proof of Γ n | applications of rule ( sA4 ). ◮ The antecendent Γ n has exactly 2 n stable expansions. ◮ But Γ n ⊢ ∆ n is not classically valid, i.e., not provable in LK . 18

The wider picture Other non-monotonic logics ◮ default logic [Reiter 80] ◮ sequent calculi [Bonatti & Olivetti 02] ◮ proof-theoretic analysis ◮ first-order [Egly & Tompits 01] ◮ propositional [B., Meier, M¨ uller, Thomas & Vollmer 11] Propositional default logic: credulous reasoning ◮ decision complexity: Σ p 2 -complete [Gottlob 92] ◮ proof complexity: close link to LK [BMMTV 11] Propositional default logic: sceptical reasoning ◮ decision complexity: Π p 2 -complete [Gottlob 92] ◮ proof complexity: exponential lower bound [BMMTV 11] 19

Attempting an explanation Proposition Let L be a language in Σ p 2 and let f be any monotone function. Then ◮ for each propositional proof system P with s P ( n ) ≤ f ( n ) ◮ there exists a proof system P ′ for L with s P ′ ( n ) ≤ p ( n ) f ( p ( n )) for some polynomial p . Our situation ◮ LK corresponds to Bonatti and Olivetti’s CAEL for autoepistemic logic. ◮ same correspondence in default logic Consequently The sequent calculi of Bonatti and Olivetti for credulous reasoning are as good as one can hope for from a proof complexity perspective. 20