Tableaux Implemented

1. Tableaux Implemented • A Tableaux Prover • Representation of tableaux • Translate tableaux rules to Prolog • Bring together this week‘s results in a combined demo system to see whether our efforts have been worthwhile

2. Representing Signed Formulae in Prolog true(walk(john) v talk(john)) false(walk(john)v talk(john))

3. Preprocessing • Use logical equivalences: • A  B  (A  B) • A  B  (A  B) • true(walk(john) v talk(john)) => true(~(~walk(john)&~talk(john))) • A simple prolog predicate does this for us (naonly/2).

4. The Driver Predicate theorem(Formula) :- naonly(Formula,ConvFormula), \+ tabl(false(ConvFormula),[],_). There is no way of falsifying the converted input Formula. If tabl/3 succeeds: Formula not valid. If tabl/3 fails: Formula valid. How is the tableaux represented?

5. We collect literals. One list per branch: [L1,L2,L3] [L1,L2,L4,L5] [L1,L2,L4,L6] …no more representation of the whole tableau. Representation of Tableaux L1 L2 L3 L4 L5 L6

6. Tableaux Rules in Prolog Scheme: tabl(InFormula,InBranch,OutBranch) :- … For each connective and sign: tabl(false(~A),InBranch,OutBranch) :- … tabl(true(A&B),InBranch,OutBranch) :- … The tabl/3 rules are straightforward translations of the tableaux rules. But some details are tricky. Input 1: The formula to be analyzed Input 2: A list of literals (branch above) Output: Literals from above plus literals from analyses of InFormula

7. Conjunctive Expansion: Negation tabl(true(~A),InBranch,OutBranch) :- !,tabl(false(A),InBranch,OutBranch). tabl(false(~A),InBranch,OutBranch) :- !,tabl(true(A),InBranch,OutBranch). Negation of input Formula is eliminated by a recursive tabl/3 call with reversed sign.

8. Conjunctive Expansion: Conjunction tabl(true(A&B),InBranch,OutBranch) :- !,tabl(true(A),InBranch,K), tabl(true(B),K,OutBranch). true(sleep(john)&~snort(john)) A=sleep(john) InBranch = [true(sleep(mary))] B=~snort(john) K = [true(sleep(mary)),true(sleep(john))] OutBranch = [true(sleep(mary)),true(sleep(john)), false(snort(john))] Two recursive calls of tabl/3. One for each conjunct. Intermediate branch is „handed over“ in variable K.

9. General Idea: Test branches one after another. Advantage: Two alternatives. If we come to the right branch, we can forget about the left one. Disjunctive Expansion I F2 F3 F4 … F5 … …

10. Disjunctive Expansion II tabl(false(A & _),InBranch,OutBranch) :- tabl(false(A),InBranch, OutBranch). tabl(false(_ & B),InBranch,OutBranch) :- !,tabl(false(B),InBranch, OutBranch). • If we succeed on the left branch to „prove“ false(A) without a clash, we stop (a model for the negation of our suspected theorem has been found). • Else, we have to try the right branch. • Use Prolog‘s backtracking mechanism to test one branch and then the other. • Two clauses for tabl(false(A&B),…)

11. Base Case: Atomic Formula tabl(AF,InBranch,OutBranch) :- OutBranch = [AF|InBranch], \+ clash(OutBranch). clash(Branch) :- member(true(A),Branch), member(false(A),Branch). • If we reach a literal on some branch and there is no clash, we stop: a counter model has been found. • Else, we fail. This triggers backtracking for further „waiting“ disjunctive expansions.

12. Main Ideas of Our Implementation • No explicit representation of the whole tableaux: We always work on one branch at a time. • Branches are represented as lists of literals (nothing else matters). • Use a sequence of Prolog calls to do tableaux construction: • Conjunctive expansion: one clause per conjunctive/sign with recursive call(s). • Disjunctive expansion: two clauses with a recursive call, backtracking if fail in first clause (i.e. no model found on left branch).

13. tabl(true(~A),InBranch,OutBranch) :- !,tabl(false(A),InBranch,OutBranch). tabl(false(~A),InBranch,OutBranch) :- !,tabl(true(A),InBranch,OutBranch). tabl(true(A & B),InBranch,OutBranch) :- !,tabl(true(A),InBranch,K), tabl(true(B),K,OutBranch). tabl(false(A & _),InBranch,OutBranch) :- tabl(false(A),InBranch,OutBranch). tabl(false(_ & B),InBranch, OutBranch) :- !,tabl(false(B),InBranch,OutBranch). tabl(AF,InBranch,OutBranch) :- OutBranch = [AF|InBranch], \+ clash(OutBranch).

14. Online Tableaux Interface • See example: Interface.htm • Course Page • Link to tableaux interfaced • All Prolog code for download • The reader • This week‘s slides • Corrected page of reader • Do not hesitate to email us Try it out!

15. The Proof of the Pudding is in the Eating Towards the end of this course… … we show a system that combines semantics construction with our tableaux prover. Given a model, we check whether a sentence is • Informative (is the information not already contained in the model) • Contradictory • Consistent

16. Conversational Maxims • Speaker and hearer obey some rules • H.P. Grice: • Maxim of Quality: Tell the truth! • Maxim of Manner: Be relevant! • … • Maximes (and violation hereof) is used in conversations (studied by Pragmatics) A:„Did you tell her you love her?“ B:„Oh, nice weather today“

17. Our System • Example model M: • John is a man. • John does not smoke. • Every man is human. • Every human works. • As large conjunctive Formula: man(john)&forall(x,human(x)>work(x))&forall(y,man(y)>human(y))&~smoke(john)

18. check :- lambda(F), M = man(john)&forall(x,human(x)>work(x))&forall(y,man checkFormula(F,M). checkFormula(F,M) :- theorem(M>F), nl, write('I know, I know...'). checkFormula(F,M) :- theorem(~(M&F)), nl, write('Impossible!'). checkFormula(_,_) :- nl, write('If you say so...').

19. Checking Consistency theorem(~(M&F)) Why does this ckeck whether F is inconsistent with M? • Mcontains no contradictions. • If  (M  F) is a theorem, then M  F is a contradiction. This must be due to F because of (1). System Demo

20. Extending our System • Argumentation, Syllogisms • Model extension • Extend model with NL input • Question answering • Question semantics, extend DCG… • Check it out! • Other Systems using NLP and inference • Curt (Blackburn/Bos) • Doris (Bos)

21. Thanks for your Interest... …enjoy Computational Semantics!