Turner`s Logic of Universal Causation, Propositional Logic, and
... Every logic program with nested expressions can be equivalently translated to disjunctive logic programs with disjunctive rules of the form l1 ∨ · · · ∨ lk ← lk+1 , . . . , lt , not lt+1 , . . . , not lm , not not lm+1 , . . . , not not ln , where n ≥ m ≥ t ≥ k ≥ 0 and l1 , . . . , ln are propositio ...
... Every logic program with nested expressions can be equivalently translated to disjunctive logic programs with disjunctive rules of the form l1 ∨ · · · ∨ lk ← lk+1 , . . . , lt , not lt+1 , . . . , not lm , not not lm+1 , . . . , not not ln , where n ≥ m ≥ t ≥ k ≥ 0 and l1 , . . . , ln are propositio ...
On the Notion of Coherence in Fuzzy Answer Set Semantics
... [2] M. Gelfond and V. Lifschitz. The stable model semantics for logic programming. In Proc. of ICLP88, pages 1070–1080, 1988. ...
... [2] M. Gelfond and V. Lifschitz. The stable model semantics for logic programming. In Proc. of ICLP88, pages 1070–1080, 1988. ...
Skeptical Reasoning in FC-Normal Logic Programs is Π1 1
... of the collection of ground atomic statements of L is a model of the above clause if whenever it contains the ai ’s, it also contains c. It is a model of the program if it is a model of every clause of the program.) A program clause is an expression of the form c ← a1 , . . . , am , not b1 , . . . , ...
... of the collection of ground atomic statements of L is a model of the above clause if whenever it contains the ai ’s, it also contains c. It is a model of the program if it is a model of every clause of the program.) A program clause is an expression of the form c ← a1 , . . . , am , not b1 , . . . , ...
Temporal Logic Theorem Proving and its Application to the Feature
... | L1 | ♦L1 | L1 U L2 | L1 W L2 | XL1 . These formulas include the usual connectives of propositional logic, as well as the temporal operators “always” (), “eventually” (♦), “until” (U), “weak until” (W), and “next” (X). We do not include quantifiers here; we introduce them in Section 4 when we beg ...
... | L1 | ♦L1 | L1 U L2 | L1 W L2 | XL1 . These formulas include the usual connectives of propositional logic, as well as the temporal operators “always” (), “eventually” (♦), “until” (U), “weak until” (W), and “next” (X). We do not include quantifiers here; we introduce them in Section 4 when we beg ...
Fixed-parameter complexity in AI and nonmonotonic reasoning
... We also study the complexity of circumscriptive inference from a general propositional theory when the attention is restricted to models of size k. This problem, referred-to as small model circumscription (SMC), is easily seen to be fixed-parameter intractable, but it does not seem to be complete fo ...
... We also study the complexity of circumscriptive inference from a general propositional theory when the attention is restricted to models of size k. This problem, referred-to as small model circumscription (SMC), is easily seen to be fixed-parameter intractable, but it does not seem to be complete fo ...
Guarded negation
... as a syntactic fragment of first-order logic, it is also natural to ask for syntactic explanations: what syntactic features of modal formulas (viewed as first-order formulas) are responsible for their good behavior? And can we generalize modal logic, preserving these features, while at the same tim ...
... as a syntactic fragment of first-order logic, it is also natural to ask for syntactic explanations: what syntactic features of modal formulas (viewed as first-order formulas) are responsible for their good behavior? And can we generalize modal logic, preserving these features, while at the same tim ...
Towards a Logic-programming System to Debug ASP Knowledge Bases
... “|”; ← (derivation, also denoted as →); propositional constants ⊥ (falsum); (verum); “¬” (default negation or weak negation, also denoted with the not word); “∼” (strong negation, equally denoted as “−”); auxiliary symbols: “(”, “)” (parentheses). The propositional symbols are also called atoms or ...
... “|”; ← (derivation, also denoted as →); propositional constants ⊥ (falsum); (verum); “¬” (default negation or weak negation, also denoted with the not word); “∼” (strong negation, equally denoted as “−”); auxiliary symbols: “(”, “)” (parentheses). The propositional symbols are also called atoms or ...
Sample pages 2 PDF
... Structural induction is used to prove that a property holds for all formulas. This form of induction is similar to the familiar numerical induction that is used to prove that a property holds for all natural numbers (Appendix A.6). In numerical induction, the base case is to prove the property for 0 ...
... Structural induction is used to prove that a property holds for all formulas. This form of induction is similar to the familiar numerical induction that is used to prove that a property holds for all natural numbers (Appendix A.6). In numerical induction, the base case is to prove the property for 0 ...
Dynamic logic of propositional assignments
... this, decidability of the satisfiability problem follows. Our result contrasts with both Miller and Moss’s undecidability result for the extension of PAL by the PDL program connectives and with Tiomkin and Makowsky’s undecidability result for the extension of PDL by local assignments. But the decida ...
... this, decidability of the satisfiability problem follows. Our result contrasts with both Miller and Moss’s undecidability result for the extension of PAL by the PDL program connectives and with Tiomkin and Makowsky’s undecidability result for the extension of PDL by local assignments. But the decida ...
Strongly equivalent temporal logic programs
... (17) and (18) correspond to the De Morgan axioms between operators U and B. It is easy to see that, together with (13) and (14) they directly imply the corresponding De Morgan axioms (19) and (20) for and ♦. An important difference with respect to LTL is that, when using these De Morgan axioms, so ...
... (17) and (18) correspond to the De Morgan axioms between operators U and B. It is easy to see that, together with (13) and (14) they directly imply the corresponding De Morgan axioms (19) and (20) for and ♦. An important difference with respect to LTL is that, when using these De Morgan axioms, so ...
Representing Synonymity in Causal Logic and in
... complete (that is, contain one member of every complementary pair of literals), provided that we identify a complete set of literals with the corresponding truth assignment. In this paper, McCain’s theorem is extended to causal theories consisting of any number of rules of the form (4) and one “syno ...
... complete (that is, contain one member of every complementary pair of literals), provided that we identify a complete set of literals with the corresponding truth assignment. In this paper, McCain’s theorem is extended to causal theories consisting of any number of rules of the form (4) and one “syno ...
CHAPTER 14 Hilbert System for Predicate Logic 1 Completeness
... Step 2 We show that every maximal finitely consistent set is consistent by constructing its model. Step 3 We show that every finitely consistent set S can be extended to a maximal finitely consistent set S ∗ . I.e we show that for every finitely consistent set S there is a set S ∗ , such that S ⊂ S ...
... Step 2 We show that every maximal finitely consistent set is consistent by constructing its model. Step 3 We show that every finitely consistent set S can be extended to a maximal finitely consistent set S ∗ . I.e we show that for every finitely consistent set S there is a set S ∗ , such that S ⊂ S ...
A New Fixpoint Semantics for General Logic Programs Compared
... This result has been shown independently by [12] in the case where P is a propositional program. A propositional program like for example P2 = {p → p, ¬p → q} has two supported minimal models, {p} and {q}, which are both models of comp(P2 ), but only one well-supported model {q} (called a grounded m ...
... This result has been shown independently by [12] in the case where P is a propositional program. A propositional program like for example P2 = {p → p, ¬p → q} has two supported minimal models, {p} and {q}, which are both models of comp(P2 ), but only one well-supported model {q} (called a grounded m ...
Planning with Different Forms of Domain
... and the best propositional solvers tend to be faster than the best answer set generators. The rest of this paper is organized as follows: we will review the basics of action language and answer set planning in the next section. We then introduce different constructs for domain-dependent control knowl ...
... and the best propositional solvers tend to be faster than the best answer set generators. The rest of this paper is organized as follows: we will review the basics of action language and answer set planning in the next section. We then introduce different constructs for domain-dependent control knowl ...
First-Order Extension of the FLP Stable Model
... can be equivalently rewritten without second-order variables as ¬(p ↔ q) ∧ (r ↔ p ∨ q). Though rule arrows (←) are treated like usual implications in the FOL-representation of Π, they are distinguished in the definition of Π4 (u) because of the presence of ’B∧’ in (7). If we modify Π4 (u) by droppin ...
... can be equivalently rewritten without second-order variables as ¬(p ↔ q) ∧ (r ↔ p ∨ q). Though rule arrows (←) are treated like usual implications in the FOL-representation of Π, they are distinguished in the definition of Π4 (u) because of the presence of ’B∧’ in (7). If we modify Π4 (u) by droppin ...
Nelson`s Strong Negation, Safe Beliefs and the - CEUR
... a rather wide range of such reducts has been proposed. Alternative approaches have been considered like proof theoretic characterizations [9] or inference in different logics [6,8]. However, in contrast to reductions, they are often seen more as theoretical tools than as definitions of the semantics ...
... a rather wide range of such reducts has been proposed. Alternative approaches have been considered like proof theoretic characterizations [9] or inference in different logics [6,8]. However, in contrast to reductions, they are often seen more as theoretical tools than as definitions of the semantics ...
Computing Default Extensions by Reductions on OR
... C (co-belief) 1, M (possibility), O (only knowing), O R and . M and O R were introduced in (Lakemeyer and Levesque 2005); M is a possibility operator that may, or may not, be dual of B. It should not be confused with ¬B¬, which is always the dual of B (and for which one could invent a defined symbo ...
... C (co-belief) 1, M (possibility), O (only knowing), O R and . M and O R were introduced in (Lakemeyer and Levesque 2005); M is a possibility operator that may, or may not, be dual of B. It should not be confused with ¬B¬, which is always the dual of B (and for which one could invent a defined symbo ...
slides (modified) - go here for webmail
... A proof uses a given set of inference rules and axioms. This is called the proof system. Let H be a proof system. ` H φ means: there is a proof of φ in system H whose premises are included in `H is called the provability relation. ...
... A proof uses a given set of inference rules and axioms. This is called the proof system. Let H be a proof system. ` H φ means: there is a proof of φ in system H whose premises are included in `H is called the provability relation. ...
Logic seminar
... Propositional logic • If there are n distinct atoms in a formula, then there will be 2n distinct interpretations for the formula. • Sometimes, if A1, ..., An are all atoms occurring in a formula, it may be more convenient to represent an interpretation by a set {m1, ..., mn}, where mi is either Ai ...
... Propositional logic • If there are n distinct atoms in a formula, then there will be 2n distinct interpretations for the formula. • Sometimes, if A1, ..., An are all atoms occurring in a formula, it may be more convenient to represent an interpretation by a set {m1, ..., mn}, where mi is either Ai ...
Automated Deduction Techniques for the Management of
... We are using a sufficiently powerful logic, such that the queries can be stated in a comfortable way. In particular, including a default negation principle and disjunctions in our logic turned out to facilitate the formalization (cf. Section 1.2 above). As a restriction, we insist on stratified spec ...
... We are using a sufficiently powerful logic, such that the queries can be stated in a comfortable way. In particular, including a default negation principle and disjunctions in our logic turned out to facilitate the formalization (cf. Section 1.2 above). As a restriction, we insist on stratified spec ...
Quick recap of logic: Propositional Calculus - clic
... • ~p: It is not the case that the University of Trento has a Faculty of Medicine • ~p: The University of Trento does not have a Faculty of Medicine • ~p: There is no Faculty of Medicine at the University of Trento ...
... • ~p: It is not the case that the University of Trento has a Faculty of Medicine • ~p: The University of Trento does not have a Faculty of Medicine • ~p: There is no Faculty of Medicine at the University of Trento ...
How to Prove Properties by Induction on Formulas
... A few comments may be helpful. First, the propositional logic meaning of implies is crucial for making this proof work in case (ii) for each of the connectives. As soon as the hypothesis is false, the truth of the implication “comes for free.” Second, in the induction, I’ve tried to make it clear wh ...
... A few comments may be helpful. First, the propositional logic meaning of implies is crucial for making this proof work in case (ii) for each of the connectives. As soon as the hypothesis is false, the truth of the implication “comes for free.” Second, in the induction, I’ve tried to make it clear wh ...
Connecting First-Order ASP and the Logic FO(ID) Through Reducts
... u ∈ |I|. From now on, we use the same symbol for an interpretation I of σ and for its (unique) extension to σ |I| defined above. We represent an interpretation I of σ (and its extension to σ |I| ) by a pair hI f , I r i, where I f is an interpretation of the part of σ (or equivalently, of σ |I| ) t ...
... u ∈ |I|. From now on, we use the same symbol for an interpretation I of σ and for its (unique) extension to σ |I| defined above. We represent an interpretation I of σ (and its extension to σ |I| ) by a pair hI f , I r i, where I f is an interpretation of the part of σ (or equivalently, of σ |I| ) t ...
Answer Sets for Propositional Theories
... (ii) Γ1 is equivalent to Γ2 in the logic of here-and-there, and (iii) for each set X of atoms, Γ1X is equivalent to Γ2X in classical logic. The equivalence between (i) and (ii) is a generalization of the main result of [Lifschitz et al., 2001], and it is an immediate consequence of Lemma 4 from that ...
... (ii) Γ1 is equivalent to Γ2 in the logic of here-and-there, and (iii) for each set X of atoms, Γ1X is equivalent to Γ2X in classical logic. The equivalence between (i) and (ii) is a generalization of the main result of [Lifschitz et al., 2001], and it is an immediate consequence of Lemma 4 from that ...