On Axiomatic Rejection for the Description Logic ALC
|
|
- Arline Jefferson
- 5 years ago
- Views:
Transcription
1 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
2 Context The traditional view about proof calculi is that they are assertional their aim is to axiomatise the valid propositions of a logic. 1
3 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. 1
4 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
5 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. Calculi axiomatising the invalid sentences of a logic are called rejection systems or complementary calculi. 1
6 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
7 Main Contributions We introduce a Gentzen-type rejection systems for description logic ALC. 3
8 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
9 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 3
10 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. 3
11 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. 3
12 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
13 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. Finally, we also obtain a calculus for the multi-modal version of modal logic K by the relation of ALC with this logic 3
14 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. Finally, we also obtain a calculus for the multi-modal version of modal logic K by the relation of ALC with this logic generalises a rejection calculus for standard K by Goranko (1994). 3
15 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
16 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
17 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
18 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. Subsequently, other rejection systems were introduced for intuitionistic logic, different modal logics, and many-valued logics. 5
19 Relevance of Axiomatic Rejection Complementary calculi are relevant for axiomatising nonmonotonic logics. 6
20 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
21 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). 6
22 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
23 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. Nonmonotonic DLs are the topic of recent investigations, e.g., by Casini et al. (DL 2013) and Giordano et al. (AIJ, 2013). 6
24 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
25 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
26 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. Syntax of ALC-concepts: C ::= A C C C C C r.c r.c A denotes a concept name, while r denotes a role name, 7
27 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. 8
28 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; 8
29 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
30 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)). A concept C is satisfiable iff there exists a finite tree-shaped interpretation T such that v 0 C T, where v 0 is the root of T. 8
31 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
32 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
33 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
34 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. In what follows, we introduce the sequential rejection system SC c ALC which axiomatises non-subsumption. 9
35 Rejection Calculus SC c ALC An anti-sequent is a pair Γ, where Γ and are finite multi-sets of concepts. 10
36 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
37 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. 11
38 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
39 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
40 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
41 Properties of SC c ALC SC c ALC is an analytic calculus, i.e., it enjoys the subformula property. 14
42 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. 14
43 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
44 Example: Extracting a Counter Model We refute r.c r.d r.(c D). 15
45 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
46 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
47 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 = ( r.c r.d) I = {v 0 } but ( r.(c D)) I =. 15
48 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 = ( r.c r.d) I = {v 0 } but ( r.(c D)) I =. 15
49 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 = ( r.c r.d) I = {v 0 } but ( r.(c D)) I =. 15
50 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 = ( r.c r.d) I = {v 0 } but ( r.(c D)) I =. 15
51 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 = ( r.c r.d) I = {v 0 } but ( r.(c D)) I =. 15
52 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 = ( r.c r.d) I = {v 0 } but ( r.(c D)) I =. 15
53 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 = ( r.c r.d) I = {v 0 } but ( r.(c D)) I =. 15
54 Relation to Tableau Algorithms Tableau algorithms are the most common reasoning procedures for description logics. 16
55 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. 16
56 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. 16
57 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. 16
58 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
59 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. Although tableaux are usually syntactic variants of standard Gentzen systems, in the case of description logics, tableaux axiomatise satisfiability 16
60 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. Although tableaux are usually syntactic variants of standard Gentzen systems, in the case of description logics, tableaux axiomatise satisfiability they therefore correspond to a rejection calculus. 16
61 A Rejection Calculus for Multi-Modal Logic K ALC can be translated into a multi-modal version of the modal logic K. 17
62 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). 17
63 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. 17
64 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
65 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. 18
66 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. 18
67 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
68 Conclusion We presented a sound and complete rejection system axiomatising non-subsumption in ALC. Rejection proofs are witnesses for tree-like counterexamples. 19
69 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. 19
70 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. 19
71 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
72 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? E.g., by effects of simulating versions of the cut rule. 19
Gentzen-type Refutation Systems for Three-Valued Logics
Gentzen-type Refutation Systems for Three-Valued Logics Johannes Oetsch and Hans Tompits Technische Universität Wien, Institut für Informationssysteme 184/3, Favoritenstraße 9-11, A-1040 Vienna, Austria
More informationStructured Descriptions & Tradeoff Between Expressiveness and Tractability
5. Structured Descriptions & Tradeoff Between Expressiveness and Tractability Outline Review Expressiveness & Tractability Tradeoff Modern Description Logics Object Oriented Representations Key Representation
More informationFuzzy Description Logics
Fuzzy Description Logics 1. Introduction to Description Logics Rafael Peñaloza Rende, January 2016 Literature Description Logics Baader, Calvanese, McGuinness, Nardi, Patel-Schneider (eds.) The Description
More informationChapter 2 Background. 2.1 A Basic Description Logic
Chapter 2 Background Abstract Description Logics is a family of knowledge representation formalisms used to represent knowledge of a domain, usually called world. For that, it first defines the relevant
More informationPhase 1. Phase 2. Phase 3. History. implementation of systems based on incomplete structural subsumption algorithms
History Phase 1 implementation of systems based on incomplete structural subsumption algorithms Phase 2 tableau-based algorithms and complexity results first tableau-based systems (Kris, Crack) first formal
More informationChapter 11: Automated Proof Systems (1)
Chapter 11: Automated Proof Systems (1) SYSTEM RS OVERVIEW Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. Automated systems
More informationTableau vs. Sequent Calculi for Minimal Entailment
Electronic Colloquium on Computational Complexity, Report No. 32 (2014) Tableau vs. Sequent Calculi for Minimal Entailment Olaf Beyersdorff and Leroy Chew School of Computing, University of Leeds, UK Abstract.
More informationCHAPTER 11. Introduction to Intuitionistic Logic
CHAPTER 11 Introduction to Intuitionistic Logic Intuitionistic logic has developed as a result of certain philosophical views on the foundation of mathematics, known as intuitionism. Intuitionism was originated
More informationComputational Logic. Davide Martinenghi. Spring Free University of Bozen-Bolzano. Computational Logic Davide Martinenghi (1/30)
Computational Logic Davide Martinenghi Free University of Bozen-Bolzano Spring 2010 Computational Logic Davide Martinenghi (1/30) Propositional Logic - sequent calculus To overcome the problems of natural
More informationPropositional Logic: Deductive Proof & Natural Deduction Part 1
Propositional Logic: Deductive Proof & Natural Deduction Part 1 CS402, Spring 2016 Shin Yoo Deductive Proof In propositional logic, a valid formula is a tautology. So far, we could show the validity of
More informationDESCRIPTION LOGICS. Paula Severi. October 12, University of Leicester
DESCRIPTION LOGICS Paula Severi University of Leicester October 12, 2009 Description Logics Outline Introduction: main principle, why the name description logic, application to semantic web. Syntax and
More informationNonmonotonic Reasoning in Description Logic by Tableaux Algorithm with Blocking
Nonmonotonic Reasoning in Description Logic by Tableaux Algorithm with Blocking Jaromír Malenko and Petr Štěpánek Charles University, Malostranske namesti 25, 11800 Prague, Czech Republic, Jaromir.Malenko@mff.cuni.cz,
More information185.A09 Advanced Mathematical Logic
185.A09 Advanced Mathematical Logic www.volny.cz/behounek/logic/teaching/mathlog13 Libor Běhounek, behounek@cs.cas.cz Lecture #1, October 15, 2013 Organizational matters Study materials will be posted
More informationQuasi-Classical Semantics for Expressive Description Logics
Quasi-Classical Semantics for Expressive Description Logics Xiaowang Zhang 1,4, Guilin Qi 2, Yue Ma 3 and Zuoquan Lin 1 1 School of Mathematical Sciences, Peking University, Beijing 100871, China 2 Institute
More informationClassical Propositional Logic
The Language of A Henkin-style Proof for Natural Deduction January 16, 2013 The Language of A Henkin-style Proof for Natural Deduction Logic Logic is the science of inference. Given a body of information,
More informationcse371/mat371 LOGIC Professor Anita Wasilewska Fall 2018
cse371/mat371 LOGIC Professor Anita Wasilewska Fall 2018 Chapter 7 Introduction to Intuitionistic and Modal Logics CHAPTER 7 SLIDES Slides Set 1 Chapter 7 Introduction to Intuitionistic and Modal Logics
More informationModal logics: an introduction
Modal logics: an introduction Valentin Goranko DTU Informatics October 2010 Outline Non-classical logics in AI. Variety of modal logics. Brief historical remarks. Basic generic modal logic: syntax and
More informationHypersequent Calculi for some Intermediate Logics with Bounded Kripke Models
Hypersequent Calculi for some Intermediate Logics with Bounded Kripke Models Agata Ciabattoni Mauro Ferrari Abstract In this paper we define cut-free hypersequent calculi for some intermediate logics semantically
More informationAn Introduction to Modal Logic III
An Introduction to Modal Logic III Soundness of Normal Modal Logics Marco Cerami Palacký University in Olomouc Department of Computer Science Olomouc, Czech Republic Olomouc, October 24 th 2013 Marco Cerami
More informationAn Introduction to Description Logics: Techniques, Properties, and Applications. NASSLLI, Day 2, Part 2. Reasoning via Tableau Algorithms.
An Introduction to Description Logics: Techniques, Properties, and Applications NASSLLI, Day 2, Part 2 Reasoning via Tableau Algorithms Uli Sattler 1 Today relationship between standard DL reasoning problems
More informationPropositional and Predicate Logic - V
Propositional and Predicate Logic - V Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - V WS 2016/2017 1 / 21 Formal proof systems Hilbert s calculus
More informationNonclassical logics (Nichtklassische Logiken)
Nonclassical logics (Nichtklassische Logiken) VU 185.249 (lecture + exercises) http://www.logic.at/lvas/ncl/ Chris Fermüller Technische Universität Wien www.logic.at/people/chrisf/ chrisf@logic.at Winter
More informationDescription Logics. an introduction into its basic ideas
Description Logics an introduction into its basic ideas A. Heußner WS 2003/2004 Preview: Basic Idea: from Network Based Structures to DL AL : Syntax / Semantics Enhancements of AL Terminologies (TBox)
More informationModal Logic XX. Yanjing Wang
Modal Logic XX Yanjing Wang Department of Philosophy, Peking University May 6th, 2016 Advanced Modal Logic (2016 Spring) 1 Completeness A traditional view of Logic A logic Λ is a collection of formulas
More informationDeductive Systems. Lecture - 3
Deductive Systems Lecture - 3 Axiomatic System Axiomatic System (AS) for PL AS is based on the set of only three axioms and one rule of deduction. It is minimal in structure but as powerful as the truth
More informationDeveloping Modal Tableaux and Resolution Methods via First-Order Resolution
Developing Modal Tableaux and Resolution Methods via First-Order Resolution Renate Schmidt University of Manchester Reference: Advances in Modal Logic, Vol. 6 (2006) Modal logic: Background Established
More informationAn Introduction to Description Logics
An Introduction to Description Logics Marco Cerami Palacký University in Olomouc Department of Computer Science Olomouc, Czech Republic Olomouc, 21.11.2013 Marco Cerami (UPOL) Description Logics 21.11.2013
More informationSeminar Topic Assignment
Seminars @ Knowledge-based Systems Group 184/3 WS 2017/18 Seminar Topic Assignment Institut für Informationssysteme Arbeitsbereich Wissensbasierte Systeme www.kr.tuwien.ac.at Seminar Topics Seminars: Seminar
More informationDescription Logics: an Introductory Course on a Nice Family of Logics. Day 2: Tableau Algorithms. Uli Sattler
Description Logics: an Introductory Course on a Nice Family of Logics Day 2: Tableau Algorithms Uli Sattler 1 Warm up Which of the following subsumptions hold? r some (A and B) is subsumed by r some A
More informationA Sequent Calculus for Skeptical Reasoning in Autoepistemic Logic
A Sequent Calculus for Skeptical Reasoning in Autoepistemic Logic Robert Saxon Milnikel Kenyon College, Gambier OH 43022 USA milnikelr@kenyon.edu Abstract A sequent calculus for skeptical consequence in
More informationKripke Semantics and Tableau Procedures for Constructive Description Logics
UNIVERSITÀ DEGLI STUDI DELL INSUBRIA DIPARTIMENTO DI INFORMATICA E COMUNICAZIONE Dottorato di Ricerca in Informatica XXIII Ciclo 2007-2010 Ph.D. Thesis Kripke Semantics and Tableau Procedures for Constructive
More informationChapter 11: Automated Proof Systems
Chapter 11: Automated Proof Systems SYSTEM RS OVERVIEW Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. Automated systems are
More informationOverview of Logic and Computation: Notes
Overview of Logic and Computation: Notes John Slaney March 14, 2007 1 To begin at the beginning We study formal logic as a mathematical tool for reasoning and as a medium for knowledge representation The
More informationTABLEAUX VARIANTS OF SOME MODAL AND RELEVANT SYSTEMS
Bulletin of the Section of Logic Volume 17:3/4 (1988), pp. 92 98 reedition 2005 [original edition, pp. 92 103] P. Bystrov TBLEUX VRINTS OF SOME MODL ND RELEVNT SYSTEMS The tableaux-constructions have a
More informationA Refined Tableau Calculus with Controlled Blocking for the Description Logic SHOI
A Refined Tableau Calculus with Controlled Blocking for the Description Logic Mohammad Khodadadi, Renate A. Schmidt, and Dmitry Tishkovsky School of Computer Science, The University of Manchester, UK Abstract
More informationRelational Reasoning in Natural Language
1/67 Relational Reasoning in Natural Language Larry Moss ESSLLI 10 Course on Logics for Natural Language Inference August, 2010 Adding transitive verbs the work on R, R, and other systems is joint with
More informationTR : Tableaux for the Logic of Proofs
City University of New York (CUNY) CUNY Academic Works Computer Science Technical Reports Graduate Center 2004 TR-2004001: Tableaux for the Logic of Proofs Bryan Renne Follow this and additional works
More information3 Propositional Logic
3 Propositional Logic 3.1 Syntax 3.2 Semantics 3.3 Equivalence and Normal Forms 3.4 Proof Procedures 3.5 Properties Propositional Logic (25th October 2007) 1 3.1 Syntax Definition 3.0 An alphabet Σ consists
More informationBasic Algebraic Logic
ELTE 2013. September Today Past 1 Universal Algebra 1 Algebra 2 Transforming Algebras... Past 1 Homomorphism 2 Subalgebras 3 Direct products 3 Varieties 1 Algebraic Model Theory 1 Term Algebras 2 Meanings
More informationRole-depth Bounded Least Common Subsumers by Completion for EL- and Prob-EL-TBoxes
Role-depth Bounded Least Common Subsumers by Completion for EL- and Prob-EL-TBoxes Rafael Peñaloza and Anni-Yasmin Turhan TU Dresden, Institute for Theoretical Computer Science Abstract. The least common
More informationPropositional Logic Language
Propositional Logic Language A logic consists of: an alphabet A, a language L, i.e., a set of formulas, and a binary relation = between a set of formulas and a formula. An alphabet A consists of a finite
More informationON THE ATOMIC FORMULA PROPERTY OF HÄRTIG S REFUTATION CALCULUS
Takao Inoué ON THE ATOMIC FORMULA PROPERTY OF HÄRTIG S REFUTATION CALCULUS 1. Introduction It is well-known that Gentzen s sequent calculus LK enjoys the so-called subformula property: that is, a proof
More informationReasoning About Typicality in ALC and EL
Reasoning About Typicality in ALC and EL Laura Giordano 1, Valentina Gliozzi 2, Nicola Olivetti 3, and Gian Luca Pozzato 2 1 Dipartimento di Informatica - Università del Piemonte Orientale A. Avogadro
More informationLogics for Data and Knowledge Representation
Logics for Data and Knowledge Representation 4. Introduction to Description Logics - ALC Luciano Serafini FBK-irst, Trento, Italy October 9, 2012 Origins of Description Logics Description Logics stem from
More informationALC + T: a Preferential Extension of Description Logics
Fundamenta Informaticae 96 (2009) 1 32 1 DOI 10.3233/FI-2009-185 IOS Press ALC + T: a Preferential Extension of Description Logics Laura Giordano Dipartimento di Informatica Università del Piemonte Orientale
More informationTR : Binding Modalities
City University of New York (CUNY) CUNY Academic Works Computer Science Technical Reports Graduate Center 2012 TR-2012011: Binding Modalities Sergei N. Artemov Tatiana Yavorskaya (Sidon) Follow this and
More informationFirst-Order Logic. Chapter Overview Syntax
Chapter 10 First-Order Logic 10.1 Overview First-Order Logic is the calculus one usually has in mind when using the word logic. It is expressive enough for all of mathematics, except for those concepts
More informationDecidability of SHI with transitive closure of roles
1/20 Decidability of SHI with transitive closure of roles Chan LE DUC INRIA Grenoble Rhône-Alpes - LIG 2/20 Example : Transitive Closure in Concept Axioms Devices have as their direct part a battery :
More informationInducing syntactic cut-elimination for indexed nested sequents
Inducing syntactic cut-elimination for indexed nested sequents Revantha Ramanayake Technische Universität Wien (Austria) IJCAR 2016 June 28, 2016 Revantha Ramanayake (TU Wien) Inducing syntactic cut-elimination
More informationApplied Logic. Lecture 1 - Propositional logic. Marcin Szczuka. Institute of Informatics, The University of Warsaw
Applied Logic Lecture 1 - Propositional logic Marcin Szczuka Institute of Informatics, The University of Warsaw Monographic lecture, Spring semester 2017/2018 Marcin Szczuka (MIMUW) Applied Logic 2018
More informationTeil III: Wissensrepräsentation und Inferenz. Kap.11: Beschreibungslogiken
Vorlesung Künstliche Intelligenz Wintersemester 2006/07 Teil III: Wissensrepräsentation und Inferenz Kap.11: Beschreibungslogiken Mit Material von Carsten Lutz, Uli Sattler: http://www.computationallogic.org/content/events/iccl-ss-
More informationEXERCISE 10 SOLUTIONS
CSE541 EXERCISE 10 SOLUTIONS Covers Chapters 10, 11, 12 Read and learn all examples and exercises in the chapters as well! QUESTION 1 Let GL be the Gentzen style proof system for classical logic defined
More informationChapter 3: Propositional Calculus: Deductive Systems. September 19, 2008
Chapter 3: Propositional Calculus: Deductive Systems September 19, 2008 Outline 1 3.1 Deductive (Proof) System 2 3.2 Gentzen System G 3 3.3 Hilbert System H 4 3.4 Soundness and Completeness; Consistency
More informationRelations to first order logic
An Introduction to Description Logic IV Relations to first order logic Marco Cerami Palacký University in Olomouc Department of Computer Science Olomouc, Czech Republic Olomouc, November 6 th 2014 Marco
More informationCompleting Description Logic Knowledge Bases using Formal Concept Analysis
Completing Description Logic Knowledge Bases using Formal Concept Analysis Franz Baader 1, Bernhard Ganter 1, Ulrike Sattler 2 and Barış Sertkaya 1 1 TU Dresden, Germany 2 The University of Manchester,
More informationPropositional and Predicate Logic - IV
Propositional and Predicate Logic - IV Petr Gregor KTIML MFF UK ZS 2015/2016 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - IV ZS 2015/2016 1 / 19 Tableau method (from the previous lecture)
More informationComputational Logic Chapter 2. Propositional Logic
Computational Logic Chapter 2. Propositional Logic Pedro Cabalar Dept. Computer Science University of Corunna, SPAIN December 14, 2010 P. Cabalar ( Dept. Ch2. Computer Propositional ScienceLogic University
More informationDescription Logics. Deduction in Propositional Logic. franconi. Enrico Franconi
(1/20) Description Logics Deduction in Propositional Logic Enrico Franconi franconi@cs.man.ac.uk http://www.cs.man.ac.uk/ franconi Department of Computer Science, University of Manchester (2/20) Decision
More informationModular Reuse of Ontologies: Theory and Practice
Journal of Artificial Intelligence Research 31 (2008) 273-318 Submitted 07/07; published 02/08 Modular Reuse of Ontologies: Theory and Practice Bernardo Cuenca Grau Ian Horrocks Yevgeny Kazakov Oxford
More informationPropositional Logic: Part II - Syntax & Proofs 0-0
Propositional Logic: Part II - Syntax & Proofs 0-0 Outline Syntax of Propositional Formulas Motivating Proofs Syntactic Entailment and Proofs Proof Rules for Natural Deduction Axioms, theories and theorems
More informationPropositional Calculus - Deductive Systems
Propositional Calculus - Deductive Systems Moonzoo Kim CS Division of EECS Dept. KAIST moonzoo@cs.kaist.ac.kr http://pswlab.kaist.ac.kr/courses/cs402-07 1 Deductive proofs (1/3) Suppose we want to know
More informationCHAPTER 10. Gentzen Style Proof Systems for Classical Logic
CHAPTER 10 Gentzen Style Proof Systems for Classical Logic Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. By humans, not mentioning
More information02 The Axiomatic Method
CAS 734 Winter 2005 02 The Axiomatic Method Instructor: W. M. Farmer Revised: 11 January 2005 1 What is Mathematics? The essence of mathematics is a process consisting of three intertwined activities:
More informationDynamic Epistemic Logic Displayed
1 / 43 Dynamic Epistemic Logic Displayed Giuseppe Greco & Alexander Kurz & Alessandra Palmigiano April 19, 2013 ALCOP 2 / 43 1 Motivation Proof-theory meets coalgebra 2 From global- to local-rules calculi
More informationPropositional Calculus - Natural deduction Moonzoo Kim CS Dept. KAIST
Propositional Calculus - Natural deduction Moonzoo Kim CS Dept. KAIST moonzoo@cs.kaist.ac.kr 1 Review Goal of logic To check whether given a formula Á is valid To prove a given formula Á ` Á Syntactic
More informationPrefixed Tableaus and Nested Sequents
Prefixed Tableaus and Nested Sequents Melvin Fitting Dept. Mathematics and Computer Science Lehman College (CUNY), 250 Bedford Park Boulevard West Bronx, NY 10468-1589 e-mail: melvin.fitting@lehman.cuny.edu
More informationAutomated Synthesis of Tableau Calculi
Automated Synthesis of Tableau Calculi Renate A. Schmidt 1 and Dmitry Tishkovsky 1 School of Computer Science, The University of Manchester Abstract This paper presents a method for synthesising sound
More informationFirst-Degree Entailment
March 5, 2013 Relevance Logics Relevance logics are non-classical logics that try to avoid the paradoxes of material and strict implication: p (q p) p (p q) (p q) (q r) (p p) q p (q q) p (q q) Counterintuitive?
More informationMeaning and Reference INTENSIONAL AND MODAL LOGIC. Intensional Logic. Frege: Predicators (general terms) have
INTENSIONAL AND MODAL LOGIC Meaning and Reference Why do we consider extensions to the standard logical language(s)? Requirements of knowledge representation / domain modelling Intensional expressions:
More informationPropositional Logic Arguments (5A) Young W. Lim 11/8/16
Propositional Logic (5A) Young W. Lim Copyright (c) 2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationA Resolution Method for Modal Logic S5
EPiC Series in Computer Science Volume 36, 2015, Pages 252 262 GCAI 2015. Global Conference on Artificial Intelligence A Resolution Method for Modal Logic S5 Yakoub Salhi and Michael Sioutis Université
More informationProving Completeness for Nested Sequent Calculi 1
Proving Completeness for Nested Sequent Calculi 1 Melvin Fitting abstract. Proving the completeness of classical propositional logic by using maximal consistent sets is perhaps the most common method there
More informationA Crisp Representation for Fuzzy SHOIN with Fuzzy Nominals and General Concept Inclusions
A Crisp Representation for Fuzzy SHOIN with Fuzzy Nominals and General Concept Inclusions Fernando Bobillo Miguel Delgado Juan Gómez-Romero Department of Computer Science and Artificial Intelligence University
More informationThe Method of Socratic Proofs for Normal Modal Propositional Logics
Dorota Leszczyńska The Method of Socratic Proofs for Normal Modal Propositional Logics Instytut Filozofii Uniwersytetu Zielonogórskiego w Zielonej Górze Rozprawa doktorska napisana pod kierunkiem prof.
More informationDefeasible Inference with Circumscriptive OWL Ontologies
Wright State University CORE Scholar Computer Science and Engineering Faculty Publications Computer Science and Engineering 6-1-2008 Defeasible Inference with Circumscriptive OWL Ontologies Stephan Grimm
More informationA tableaux calculus for ALC + T min R
A tableaux calculus for ALC + T min R Laura Giordano Valentina Gliozzi Adam Jalal Nicola Olivetti Gian Luca Pozzato June 12, 2013 Abstract In this report we introduce a tableau calculus for deciding query
More informationSemantics and Pragmatics of NLP
Semantics and Pragmatics of NLP Alex Ewan School of Informatics University of Edinburgh 28 January 2008 1 2 3 Taking Stock We have: Introduced syntax and semantics for FOL plus lambdas. Represented FOL
More informationReferences A CONSTRUCTIVE INTRODUCTION TO FIRST ORDER LOGIC. The Starting Point. Goals of foundational programmes for logic:
A CONSTRUCTIVE INTRODUCTION TO FIRST ORDER LOGIC Goals of foundational programmes for logic: Supply an operational semantic basis for extant logic calculi (ex post) Rational reconstruction of the practice
More informationPropositional Calculus - Soundness & Completeness of H
Propositional Calculus - Soundness & Completeness of H Moonzoo Kim CS Dept. KAIST moonzoo@cs.kaist.ac.kr 1 Review Goal of logic To check whether given a formula Á is valid To prove a given formula Á `
More informationA Journey through the Possible Worlds of Modal Logic Lecture 1: Introduction to modal logics
A Journey through the Possible Worlds of Modal Logic Lecture 1: Introduction to modal logics Valentin Goranko Department of Philosophy, Stockholm University ESSLLI 2016, Bolzano, August 22, 2016 Outline
More informationINSTANTIAL NEIGHBOURHOOD LOGIC
INSTANTIAL NEIGHBOURHOOD LOGIC JOHAN VAN BENTHEM, NICK BEZHANISHVILI, SEBASTIAN ENQVIST, JUNHUA YU Abstract. This paper explores a new language of neighbourhood structures where existential information
More informationA SIMPLE AXIOMATIZATION OF LUKASIEWICZ S MODAL LOGIC
Bulletin of the Section of Logic Volume 41:3/4 (2012), pp. 149 153 Zdzis law Dywan A SIMPLE AXIOMATIZATION OF LUKASIEWICZ S MODAL LOGIC Abstract We will propose a new axiomatization of four-valued Lukasiewicz
More informationNested Sequent Calculi for Normal Conditional Logics
Nested Sequent Calculi for Normal Conditional Logics Régis Alenda 1 Nicola Olivetti 2 Gian Luca Pozzato 3 1 Aix-Marseille Université, CNRS, LSIS UMR 7296, 13397, Marseille, France. regis.alenda@univ-amu.fr
More informationRefutability and Post Completeness
Refutability and Post Completeness TOMASZ SKURA Abstract The goal of this paper is to give a necessary and sufficient condition for a multiple-conclusion consequence relation to be Post complete by using
More informationJustification logic for constructive modal logic
Justification logic for constructive modal logic Sonia Marin With Roman Kuznets and Lutz Straßburger Inria, LIX, École Polytechnique IMLA 17 July 17, 2017 The big picture The big picture Justification
More informationPropositional and Predicate Logic. jean/gbooks/logic.html
CMSC 630 February 10, 2009 1 Propositional and Predicate Logic Sources J. Gallier. Logic for Computer Science, John Wiley and Sons, Hoboken NJ, 1986. 2003 revised edition available on line at http://www.cis.upenn.edu/
More informationOn the Logic and Computation of Partial Equilibrium Models
On the Logic and Computation of Partial Equilibrium Models Pedro Cabalar 1, Sergei Odintsov 2, David Pearce 3 and Agustín Valverde 4 1 Corunna University (Corunna, Spain), cabalar@dc.fi.udc.es 2 Sobolev
More informationNatural Deduction for Propositional Logic
Natural Deduction for Propositional Logic Bow-Yaw Wang Institute of Information Science Academia Sinica, Taiwan September 10, 2018 Bow-Yaw Wang (Academia Sinica) Natural Deduction for Propositional Logic
More informationESSLLI 2007 COURSE READER. ESSLLI is the Annual Summer School of FoLLI, The Association for Logic, Language and Information
ESSLLI 2007 19th European Summer School in Logic, Language and Information August 6-17, 2007 http://www.cs.tcd.ie/esslli2007 Trinity College Dublin Ireland COURSE READER ESSLLI is the Annual Summer School
More informationTechnische Universität Dresden. Fakultät Informatik EMCL Master s Thesis on. Hybrid Unification in the Description Logic EL
Technische Universität Dresden Fakultät Informatik EMCL Master s Thesis on Hybrid Unification in the Description Logic EL by Oliver Fernández Gil born on June 11 th 1982, in Pinar del Río, Cuba Supervisor
More informationCanonical Calculi: Invertibility, Axiom expansion and (Non)-determinism
Canonical Calculi: Invertibility, Axiom expansion and (Non)-determinism A. Avron 1, A. Ciabattoni 2, and A. Zamansky 1 1 Tel-Aviv University 2 Vienna University of Technology Abstract. We apply the semantic
More informationGödel Negation Makes Unwitnessed Consistency Crisp
Gödel Negation Makes Unwitnessed Consistency Crisp Stefan Borgwardt, Felix Distel, and Rafael Peñaloza Faculty of Computer Science TU Dresden, Dresden, Germany [stefborg,felix,penaloza]@tcs.inf.tu-dresden.de
More informationThe Skolemization of existential quantifiers in intuitionistic logic
The Skolemization of existential quantifiers in intuitionistic logic Matthias Baaz and Rosalie Iemhoff Institute for Discrete Mathematics and Geometry E104, Technical University Vienna, Wiedner Hauptstrasse
More informationProof-theoretic Approach to Deciding Subsumption and Computing Least Common Subsumer in EL w.r.t. Hybrid TBoxes
Technical University Dresden Department of Computer Science Master s Thesis on Proof-theoretic Approach to Deciding Subsumption and Computing Least Common Subsumer in EL w.r.t. Hybrid TBoxes by Novak Novaković
More informationLOGIC PROPOSITIONAL REASONING
LOGIC PROPOSITIONAL REASONING WS 2017/2018 (342.208) Armin Biere Martina Seidl biere@jku.at martina.seidl@jku.at Institute for Formal Models and Verification Johannes Kepler Universität Linz Version 2018.1
More informationExtended Decision Procedure for a Fragment of HL with Binders
Extended Decision Procedure for a Fragment of HL with Binders Marta Cialdea Mayer Università di Roma Tre, Italy This is a draft version of a paper appearing on the Journal of Automated Reasoning. It should
More informationSystems of modal logic
499 Modal and Temporal Logic Systems of modal logic Marek Sergot Department of Computing Imperial College, London utumn 2008 Further reading: B.F. Chellas, Modal logic: an introduction. Cambridge University
More informationDecidability of Description Logics with Transitive Closure of Roles in Concept and Role Inclusion Axioms
Proc. 23rd Int. Workshop on Description Logics (DL2010), CEUR-WS 573, Waterloo, Canada, 2010. Decidability of Description Logics with Transitive Closure of Roles in Concept and Role Inclusion Axioms Chan
More informationLTCS Report. Decidability and Complexity of Threshold Description Logics Induced by Concept Similarity Measures. LTCS-Report 16-07
Technische Universität Dresden Institute for Theoretical Computer Science Chair for Automata Theory LTCS Report Decidability and Complexity of Threshold Description Logics Induced by Concept Similarity
More informationA generalization of modal definability
A generalization of modal definability Tin Perkov Polytechnic of Zagreb Abstract. Known results on global definability in basic modal logic are generalized in the following sense. A class of Kripke models
More information