Non-Resolution Theorem Proving Various different techniques have - PowerPoint PPT Presentation

9ai Non-Resolution Theorem Proving Various different techniques have been considered as suitable alternatives to resolution and clausal form for automated reasoning. These include: • Relax adherence to clausal form: Non-clausal resolution (Murray, Manna and Waldinger) Semantic tableau /Natural deduction but with unification ( Hahlne, AUTOMATED REASONING Manna and Waldinger, Reeves, Broda, Dawson, Letz, Baumgartner, Hahlne, Beckert and many others) SLIDES 9 • Relax restriction to first order classical logic: Modal logics, resource logics (Constable, Bundy,Wallen , D'Agostino, McRobbie,) Add sorts (Walther, Cohn, Schmidt Schauss) SEMANTIC TABLEAUX Higher order logics (Miller, Paulson) Standard Tableaux Temporal logic with time parameters (Reichgeldt, Hahlne, Gore) Labelled deduction (Gabbay, D'Agostino, Russo, Broda} Free Variable Tableaux • Heuristics and Metalevel reasoning: Soundness and Completeness Use metalevel rules to guide theorem provers - includes rewriting, Different strategies paramodulation (Bundy, Dershowitz, Hsiang, Rusinowitz, Bachmair) Abstractions (Plaistead) Lean Tap Procedural rules for natural deduction (Gabbay) Unification for assoc.+commut. operators (Stickel) KB - AR - 09 Use models / analogy (Gerlenter, Bundy) Inductive proofs, proof plans (Boyer and Moore, Bundy et al) Tacticals / interactive proof / proof checkers, Isabelle (Paulson) Considered in Part II are Tableaux methods and rewriting for equality. For your interest, some non-examinable notes on using analogy are given in Slides Extra. Theorem Proving with Semantic Tableaux 9bi 9bii a ∧ w → p Example : Proof Method : Refutation (but no translation into clausal form) i ∨ a a ∧ w → p Here ¬e is goal: Show by construction that no model can exist for given sentences ¬ w → m i ∨ a Givens negated = ¬¬e i.e. that all potential models are contradictory. Do this by following the ¬ w → m ¬ p Givens consequences of the data - aim is to show that they all lead to a contradiction. e → ¬ i ∧ ¬ m ¬ p Development Rules: e → ¬ i ∧ ¬ m ¬¬e 1. Conclusion is negated and added to givens (if conclusion is distinguished). ¬¬e e 2. A tree is developed s.t. sentences in each branch give a partial model. e 4. Non-splitting and Splitting branch development rules (see below): ¬ i ∧ ¬ m ¬ (a ∧ w ) p 5. Rules can be applied in any order and branches may be developed in any ¬ i order s.t. all sentences in a branch B are eventually developed. ¬ e ¬ m ¬ a ¬ w 6. In the first order case develop branches to a given maximum depth to avoid possibly infinite tableaux. i Non - splitting ( α rules): a m i a ¬ ¬ w A ∧ B ¬(A ∨ B ) Splitting (or β rules): ¬¬ w A ¬A ¬ e A ∨ B ¬ ( A ∧ B) A → B A ↔ B ¬ i ∧ ¬ m ¬ i ∧ ¬ m m ¬ e ¬ (a ∧ w ) B ¬B p ¬ i ¬ i ¬(A → B ) A B ¬A ¬B ¬ A B A ¬A ¬ m ¬ m ¬¬A B ¬B ¬ a ¬w A (¬(A ↔ B) ≡ ¬A ↔ B ) A ¬B

Semantic Tableaux 9biv What if the Givens do not imply the conclusion? 9biii Semantic tableaux were introduced in 1954 by Beth. The standard method was automated in 1985, when the free variable method was also automated. The Model Elimination approach was Example : Givens do not |= ¬e a ∧ w → p introduced by both Kowalski and Loveland in 1970, but as a resolution refinement, not as a There is a model of Givens and e i ∨ a tableau method. This was later related and extended to tableau methods in 1990 onwards. The Givens ¬ w → m TABLEAUX Workshops (now part of IJCAR) are devoted to tableaux and related theorem The branch ending in w does not close. proving methods. e → ¬ i ∧ ¬ m All data has been processed in the branch. The initial branch of a semantic tableau contains the given sentences that are to be refuted. (It is called saturated.) ¬¬e Reasoning progresses by making assertions about the satisfiability of sub-formulas based on the e satisfiability of the larger formulas of which they are a part. The α -rules (slide 9bi) can be read as Since the assumption that a model of Givens ¬ i ∧ ¬ m "if α is in a branch and there is a model of the sentences in the branch, then there is a model of the exists is not contradicted the branch is ¬ i sentences in the branch extended by α1 and α2 " and the β -rules as "if β is in a branch and there is a consistent. ¬ e ¬ m model of the sentences in the branch, then there is a model of either the sentences in the branch A model can be read from the literals in the extended by β 1, or the sentences in the branch extended by β 2", where α 1, α 2/ β 1, β 2 are the two branch. Each atom that occurs positively will i sub-formulae of the rules. If X and ¬X are in the branch then the sentences in the branch are clearly a be true in the model and each atom that inconsistent and the branch is closed . If all branches in a tableau are closed (it is also said to be occurs negatively will be false. Other atoms closed ), then there are no possible consistent derivations of sub-formulas from the initially given ¬¬ w (that don't occur in the open branch) can be sentences, and these sentences are unsatisfiable. true or false but are usually made false. m ¬ (a ∧ w ) Two examples of closed propositional tableaux are on slide 9bii, illustrating that there can be p differently sized tableaux for the same set of initial sentences. For propositional sentences the Here: a, e, p, w =True and i, m=False. development of a tableau will always terminate as there is no need to develop a sentence in a branch more than once – to do so would duplicate one or more sub-formulas or atoms in that Note that no atom can occur both positively ¬ a ¬w w branch, which adds no information to the branch. However, every sentence in a branch must be and negatively in an open branch. Why? developed in it. A fully developed (or completed) branch is called open if it is not closed, and similarly the tableau . A fully developed open branch will yield a model of the initial data. 9bv The invariant property SATISFY: (1) div(x,x), (2) less(1,n), (3) div(u,w) ∧ div(w,z) → div(u,z) 9bvi • Each tableau extension rule maintains satisfiability: (4) ¬(div(g(x),x) ∧ less(1,g(x)) ∧ less(g(x),x) ) → pr(x) if the sentences in a branch are satisfiable and a rule is applied, the new (5) less(1,x) ∧ less(x,n) → div(f(x),x) ∧ pr(f(x)) Show ∃ y (pr(y) ∧ div(y,n)) sentences in at least one descendant branch are satisfiable. e.g. If M is a model of a branch including i ∨ a, then M must assign true to at ¬ ∃ y (pr(y) ∧ div(y,n)) (**) (Negated conclusion) least one of i or a. Hence at least one extended branch is satisfied by M. ¬(pr(n) ∧ div(n,n)) (guess n is a good value for y in (**) ) • A branch that contains both X and ¬ X is unsatisfiable and can be closed by the closure rule . (4) ¬¬(div(g(n)n) ∧ less(1,g(n)) ∧ less(g(n),n)) Quantifier rules ( ∀ ∀) ∀ ∀ ) ) ( γ rules) for Standard version (not often used now ) pr(n) div(g(n)n), less(1,g(n)), less(g(n),n) except in proofs about tableau properties): ¬div(n,n) ∀ x P[x] ¬ ∃ x P[x] (5) e.g . ∀ x P(x,f(x) ) ¬pr(n) (1) | | ¬(less(1,g(n) div(f(g(n)), g(n)) ⇒ P(s,f(s) ) ∧ less(g(n),n)) P[t1] ¬ P[t1] div(n,n) pr(f(g(n)) ¬ (pr(f(g(n)) ∧ div(f(g(n)),n)) (**) where t1 is a ground term from the language. A closed tableau using the Standard γ γ rules : γ γ ¬ div(f(g(n)),n) In the standard version, each ∀ sentence in B should eventually be used for ¬less(1,g(n) ¬ pr(f(g(n)) 1, n are constants; (Use (3) next ) every name that "occurs" in a sentence in B (unless the branch is closed). This all variables in data are includes all names constructed from constants and functors in the branch ¬less(g(n),n) (**) Guess f(g(n)) universally quantified. is a good value for y