On Axiomatic Rejection for the Description Logic ALC

On Axiomatic Rejection for the Description Logic ALC Hans Tompits Vienna University of Technology Institute of Information Systems Knowledge-Based Systems Group Joint work with Gerald Berger

Context The traditional view about proof calculi is that they are assertional their aim is to axiomatise the valid propositions of a logic. 1

Context The traditional view about proof calculi is that they are assertional their aim is to axiomatise the valid propositions of a logic. But we can also have a complementary view: Instead of axiomatising the valid sentences we may axiomatise the invalid ones. In such a system, false propositions are deduced from other (elementary) false ones. 1

Context (ctd.) Ich bin der Geist der stets verneint! Und das mit Recht; denn alles, was entsteht, Ist wert, dass es zugrunde geht. ( I am the spirit, ever, that denies! And rightly so; since everything created, In turn deserves to be annihilated. ) J.W. von Goethe, Faust I 2

Main Contributions We introduce a Gentzen-type rejection systems for description logic ALC. 3

Main Contributions We introduce a Gentzen-type rejection systems for description logic ALC. Gentzen-type systems are well-known calculi optimised for proof search. 3

Main Contributions We introduce a Gentzen-type rejection systems for description logic ALC. Gentzen-type systems are well-known calculi optimised for proof search. Description logics are important knowledge-representation languages for modelling ontologies provide the formal underpinning for semantic-web reasoning. Our calculus axiomatises concept non-subsumption. That is, a sequent C D is provable in our calculus iff C D does not hold. We also analyse the relationship between our calculus and a well-known tableau procedure for ALC. 3

Historical Remarks Investigation of invalid arguments traces back to Aristotle in his analysis of logical fallacies in On Sophistical Refutations of the Organon. Raphael s Scuola di Atene 4

Historical Remarks (ctd.) The first system for axiomatic rejection was introduced by Jan Lukasiewicz in 1957 in his book Aristotle s syllogistic from the standpoint of modern formal logic 5

Historical Remarks (ctd.) The first system for axiomatic rejection was introduced by Jan Lukasiewicz in 1957 in his book Aristotle s syllogistic from the standpoint of modern formal logic There, he axiomatised invalid syllogisms of Aristotle by means of a Hilbert-type system using the detachment rule if ϕ ψ is asserted and ψ is rejected, then ϕ is rejected too. 5

Relevance of Axiomatic Rejection Complementary calculi are relevant for axiomatising nonmonotonic logics. 6

Relevance of Axiomatic Rejection Complementary calculi are relevant for axiomatising nonmonotonic logics. E.g., default rule if A and no evidence for B then C amounts to inference A B C 6

Relevance of Axiomatic Rejection Complementary calculi are relevant for axiomatising nonmonotonic logics. E.g., default rule if A and no evidence for B then C amounts to inference A B C Indeed, Gentzen-type axiomatisations of central nonmonotonic logics (like default logic, circumscription, autoepistemic logic) rely on rejection calculi (Bonatti & Olivetti, 2002). Rejection calculi become relevant in studying proof systems of nonmonotonic extensions of description logics. 6

Description Logic ALC Syntax The vocabulary of ALC includes the following elements: concept names A, B,..., role names p, q, r,..., individual names a, b, c,..., 7

Description Logic ALC Syntax The vocabulary of ALC includes the following elements: concept names A, B,..., role names p, q, r,..., individual names a, b, c,..., concept intersection, concept union, concept negation, value restriction, existential restriction. 7

Description Logic ALC Semantics An interpretation is a pair I = I, I, where I is a non-empty set, called domain, I is a mapping ensuring that every concept name A is mapped to some subset A I I, every role name r is mapped to a binary relation r I I I. I satisfies the usual truth conditions concerning the concept constructors; the semantics of the quantifiers is given by ( r.c) I = {x y : (x, y) r I y C I }, ( r.c) I = {x y : (x, y) r I and y C I }. That is: ( r.c) corresponds to y(r(x, y) C(y)); ( r.c) corresponds to y(r(x, y) C(y)). 8

Description Logic ALC Semantics (ctd.) A general concept inclusion (GCI) is an expression of the form C D, where C and D are concepts. 9

Description Logic ALC Semantics (ctd.) A general concept inclusion (GCI) is an expression of the form C D, where C and D are concepts. An interpretation I satisfies a GCI C D iff C I D I. 9

Description Logic ALC Semantics (ctd.) A general concept inclusion (GCI) is an expression of the form C D, where C and D are concepts. An interpretation I satisfies a GCI C D iff C I D I. A concept D subsumes a concept C if every interpretation satisfies the GCI C D. 9

Rejection Calculus SC c ALC An anti-sequent is a pair Γ, where Γ and are finite multi-sets of concepts. 10

Rejection Calculus SC c ALC An anti-sequent is a pair Γ, where Γ and are finite multi-sets of concepts. An interpretation refutes an anti-sequent Γ if it does not satisfy the GCI γ δ; γ Γ δ (the empty concept intersection is ; the empty concept union is ). 10

Rejection Calculus SC c ALC (ctd.) Axioms of SC c ALC : any anti-sequent Γ 0 0 s.t. Γ 0 0 =, where Γ 0 and 0 consist of concept names only; any anti-sequent of form r 1.Γ 1,..., r n.γ n r 1. 1,..., r n. n. Structural Rules: Γ, C (w 1, l) Γ Γ, C (c 1, l) Γ, C, C Γ, C (w 1, r) Γ Γ C, (c 1, r) Γ C, C, 11

Rejection Calculus SC c ALC (ctd.) Propositional Rules: Γ, C, D (, l) Γ, C D Γ, C (, l)1 Γ, C D Γ, D (, l)2 Γ, C D Γ C, (, l) Γ, C Γ Γ, ( ) Γ C, D, (, r) Γ C D, Γ C, (, r)1 Γ C D, Γ D, (, r)2 Γ C D, Γ, C (, r) Γ C, Γ Γ, ( ) 12

Rejection Calculus SC c ALC (ctd.) Quantifier Rules: Γ 0 0 Γ r1,..., Γ rn r1,..., rn (Mix), Γ 0, Γ r1,..., Γ rn 0, r1,..., rn where Γ 0 0 is a propositional axiom. Γ r k r k, Ck Γr l r l, Cl Γ r1,..., Γ rn r1,..., rn (Mix, ) Γ r1,..., Γ rn r1,..., rn, r k.c k,..., r l.c l Γ r k, Ck r k Γr l, Cl r l Γ r1,..., Γ rn r1,..., rn (Mix, ) Γ r1,..., Γ rn, r k.c k,..., r l.c l r1,..., rn where 1 k l n. Notation: Γ r denotes any finite multi-set of concepts containing only concepts of form r.c or r.c. Γ := {C r.c Γ} and Γ := {C r.c Γ}. 13

Properties of SC c ALC SC c ALC is an analytic calculus, i.e., it enjoys the subformula property. 14

Properties of SC c ALC SC c ALC is an analytic calculus, i.e., it enjoys the subformula property. SC c ALC is sound and complete, i.e., an anti-sequent Γ is refutable iff it is provable in SC c ALC. Countermodels (in the form of tree models) can be extracted from a proof in SC c ALC : Assigning, from bottom to top, each anti-sequent in the proof a node of the tree. The end-sequent is the root of the tree. New nodes are created for each application of (Mix, ) and (Mix, ). For an axiom Γ 0 0 with assigned node v, we ensure that v C I for each C Γ 0 and v D I for each D 0. 14

Example: Extracting a Counter Model We refute r.c r.d r.(c D). 15

Example: Extracting a Counter Model We refute r.c r.d r.(c D). D C D C D C D (, r) 2 C C D r.(c D) (, r) 1 (Mix, ) r.c r.(c D) (Mix, ) r.c, r.d r.(c D) (, l) r.c r.d r.(c D) 15

Example: Extracting a Counter Model We refute r.c r.d r.(c D). D C D C D C D (, r) 2 C C D r.(c D) (, r) 1 (Mix, ) r.c r.(c D) (Mix, ) r.c, r.d r.(c D) (, l) r.c r.d r.(c D) I = {v 0, v 1, v 2 }, I, r I = {(v 0, v 1 ), (v 0, v 2 )}, C I = {v 1 }, D I = {v 2 }. v 0 r r C v 1 v 2 D 15

Relation to Tableau Algorithms Tableau algorithms are the most common reasoning procedures for description logics. 16

Relation to Tableau Algorithms Tableau algorithms are the most common reasoning procedures for description logics. They rely on the construction of a canonical model which witnesses the satisfiability of a concept or a knowledge base. Well known algorithm for ALC (Baader & Sattler, 2001) is based on the notion of a completion graph. A model can be extracted from a complete completion graph. A complete completion graph can be obtained from a proof in SC c ALC. 16

A Rejection Calculus for Multi-Modal Logic K ALC can be translated into a multi-modal version of the modal logic K. 17

A Rejection Calculus for Multi-Modal Logic K ALC can be translated into a multi-modal version of the modal logic K. Based on this translation, we obtain a rejection calculus for multi-modal K, generalising a rejection calculus for standard K by Goranko (1994). In multi-modal K, we have different modal operators [α], where α is a modality. Semantically, multi-modal K is based on Kripke models M = W, {R α } α τ, V, where W is a non-empty set of worlds, R α W W defines an accessibility relation for each modality α, and V defines which propositional variables are true at which worlds. 17

A Rejection Calculus for Multi-Modal Logic K (ctd.) The translation of ALC into multi-modal K is simply by viewing concepts of form r.c as modal formulae of form [α]c, C is the corresponding translation of the concept C, each role name corresponds to one and only one modality. Based on this translation, we can directly translate our calculus into corresponding modal rules. For instance, the rule (Mix) becomes Γ 0 0 [α 1 ]Γ 1,..., [α n ]Γ n [α 1 ] 1,..., [α n ] n (Mix) Γ 0, [α 1 ]Γ 1,..., [α n ]Γ n 0, [α 1 ] 1,..., [α n ] n where Γ 0, 0 are disjoint sets of propositional variables and [α]γ := {[α]ϕ ϕ Γ}. 18

Conclusion We presented a sound and complete rejection system axiomatising non-subsumption in ALC. Rejection proofs are witnesses for tree-like counterexamples. 19

Conclusion We presented a sound and complete rejection system axiomatising non-subsumption in ALC. Rejection proofs are witnesses for tree-like counterexamples. Future work: Provide a calculus for the general case of dealing with reasoning from knowledge bases, i.e., taking TBox reasoning into account. Study calculi for nonmonotonic extensions of description logics. Can a combination of traditional proof methods and rejection calculi yield potentially more efficient reasoning systems? 19