Splitting a Default Theory. Hudson Turner. University of Texas at Austin.
|
|
- Abel Camron Washington
- 5 years ago
- Views:
Transcription
1 Splitting a Default Theory Hudson Turner Department of Computer Sciences University of Texas at Austin Austin, TX , USA hudson@cs.utexas.edu Abstract This paper presents mathematical results that can sometimes be used to simplify the task of reasoning about a default theory, by \splitting it into parts." These so-called Splitting Theorems for default logic are related in spirit to \partial evaluation" in logic programming, in which results obtained from one part of a program are used to simplify the remainder of the program. In this paper we focus primarily on the statement and proof of the Splitting Theorems for default logic. We illustrate the usefulness of the results by applying them to an example default theory for commonsense reasoning about action. Introduction This paper introduces so-called Splitting Theorems for default logic, which can sometimes be used to simplify the task of reasoning about a default theory, by \splitting it into parts." These Splitting Theorems are related somewhat, in spirit, to \partial evaluation" in logic programming, in which results obtained from one part of a program are used to simplify the remainder of the program. In fact, the Splitting Theorems for default logic closely resemble the Splitting Theorems for logic programming introduced in (Lifschitz & Turner 994), despite complications due to the presence of arbitrary formulas in default theories. 2 Similar results for autoepistemic logic can be found in (Gelfond & Przymusinska 992). Related, independently obtained, results for default logic, restricted to the - nite case, appear in (Cholewinski 995). In this paper we focus on the statement and proof of the Splitting Theorems for default logic. 3 We illustrate See, for example, (Komorowski 99). 2 In (Lifschitz & Turner 994) we presented without proof Splitting Theorems for logic programs with classical negation and disjunction, under the answer set semantics (Gelfond & Lifschitz 99). The results for nondisjunctive logic programs follow from the Splitting Theorems for default logic. The denitions and proofs in this paper can be adapted to the more general case of disjunctive default logic (Gelfond et al. 99), from which the Splitting Theorems for disjunctive logic programs would follow as well. 3 We do not address the possible application of these the usefulness of the results by applying them to an example default theory for reasoning about action. An extended application of the Splitting Theorems to default theories for representing commonsense knowledge about actions will appear in (Turner 997). The paper is organized as follows. We present preliminary denitions. We then introduce the Splitting Set Theorem, followed by Splitting Sequence Theorem, which generalizes it. We exercise the Splitting Sequence Theorem on an example default theory for reasoning about action. We present an abridged proof of the Splitting Set Theorem, followed by a detailed proof of the Splitting Sequence Theorem. Preliminary Denitions Given a set U of atomic symbols (not including the special constants > and?), we denote by L(U) the language of propositional logic with exactly the atoms U f>;?g. 4 We say a set of formulas from L(U) is logically closed if it is closed under propositional logic. We write inference rules over L(U) as expressions of the form where and are formulas from L(U). When convenient, we identify a formula with the inference rule >. We say a set of formulas is closed under a set R of inference rules if for all 2 R, if 2 then 2. By Cn U (R) we denote the least logically closed set of formulas from L(U) that is closed under R. A default rule over L(U) is an expression of the form ; ; n () where ; ; ; n ; are formulas from L(U) (n ). Let r be a default rule of form (). We call the prerequisite of r, and denote it by pre(r). The formulas ; ; n are the justications of r; we write just(r) to denote the set f ; ; n g. We call the consequent of r, and denote it by cons(r). If pre(r) is > we often ideas to the problem of automated default reasoning. 4 Thus, L(;) consists of all formulas in which the only atoms are the constants > and?.
2 omit it, writing ; ; n instead. If just(r) = ;, we identify r with the corresponding inference rule.5 A default theory over L(U) is a set of default rules over L(U). Let D be a default theory over L(U), and E a set of formulas from L(U). The reduct of D by E, denoted by D E, is the following set of inference rules. pre(r) cons(r) r 2 D and for all 2 just(r); =2 E We say E is an extension of D if E = Cn U (D E ) Default logic is due to Reiter (Reiter 98). The (essentially equivalent) denitions given above follow (Gelfond et al. 99). Although the denitions in this section are stated for the propositional case, they are taken, in the standard way, to apply in the rst-order, quantier-free case as well, by taking each non-ground expression to stand for all of its ground instances. Splitting Sets Let D be a default theory over L(U) such that, for every rule r 2 D, pre(r) is in conjunctive normal form. (Of course any default theory can be easily transformed into an equivalent default theory, over the same language, satisfying this condition.) For any rule r 2 D, a formula is a constituent of r if at least one of the following conditions holds (i) is a conjunct of pre(r); (ii) 2 just(r); (iii) = cons(r). A splitting set for D is a subset A of U such that for every rule r 2 D the following two conditions hold. Every constituent of r belongs to L(A) L(U n A). If cons(r) does not belong to L(U n A), then r is a default rule over L(A). If A is a splitting set for D, we say that A splits D. The base of D relative to A, denoted by b A (D), is the default theory over L(A) that consists of all members of D that are default rules over L(A). Let U 3 = fa; b; c; dg. Consider the following default theory D 3 over L(U 3 ). b a a b a _ b a; b c _ d a ^ (c _ d) d d b ^ (c _ d) c Take A 3 = fa; bg. It's easy to verify that A 3 splits D 3, with b b A3 (D 3 ) = a ; a b Notice that default theory b A3 (D 3 ) over L(A 3 ) has two consistent extensions Cn A3 (fag) and Cn A3 (fbg). Given a splitting set A for D, and a set X of formulas from L(A), the partial evaluation of D by X with respect to A, denoted by e A (D; X), is the default theory over L(U n A) obtained from D in the following manner. For each rule r 2 D n b A (D) such that 5 Allowing the empty set of justications is mathematically convenient (Brewka 99; Marek & Truszczynski 993), although it seems Reiter (Reiter 98) may have meant to prohibit it. c every conjunct of pre(r) that belongs to L(A) also belongs to Cn A (X), and no member of just(r) has its complement in Cn A (X) there is a rule r 2 e A (D; X) such that pre(r ) is obtained from pre(r) by replacing each conjunct of pre(r) that belongs to L(A) by >, and just(r ) = just(r) \ L(U n A), and cons(r ) = cons(r). For example, it is easy to verify that > > ^ (c _ d) d e A3 (D 3 ; Cn A3 (fag)) = ; c _ d d and that > > ^ (c _ d) c e A3 (D 3 ; Cn A3 (fbg)) = ; c _ d c Let A be a splitting set for D. A solution to D with respect to A is a pair hx; i of sets of formulas satisfying the following two properties. X is a consistent extension of the default theory b A (D) over L(A). is a consistent extension of the default theory e A (D; X) over L(U n A). For example, given our previous observations, it is easy to verify that D 3 has two solutions with respect to A 3 h Cn A3 (fag) ; Cn U3nA 3 (fc; dg) i and h Cn A3 (fbg) ; Cn U3nA 3 (fc; dg) i Splitting Set Theorem. Let A be a splitting set for a default theory D over L(U). A set E of formulas is a consistent extension of D i E = Cn U (X ) for some solution hx; i to D with respect to A. Thus, for example, it follows from the Splitting Set Theorem that default theory D 3 has exactly two consistent extensions Cn U3 (fa; c; dg) and Cn U3 (fb; c; dg). Splitting Set Corollary. Let A be a splitting set for a default theory D over L(U). If E is a consistent extension of D, then the pair h E \ L(A) ; E \ L(U n A) i is a solution to D with respect to A. Splitting Sequences A (transnite) sequence is a family whose index set is an initial segment of ordinals f g. We say that a sequence ha i of sets is monotone if A A whenever, and continuous if, for each limit ordinal, A = S A. A splitting sequence for a default theory D over L(U) is a nonempty, monotone, continuous sequence ha i of splitting sets for D s.t. S A = U.
3 Holds(Up(Left); S ) ^ Holds(Up(Right); S ) ^ Holds(Spilled; S ) Precedes(S ; S ) ^ Occurs(Raise(Left); S ) ^ Occurs(Raise(Right); S ) Precedes(S ; S 2 ) ^ Occurs(Lower(Left); S ) Occurs(Raise(x); s) ^ Precedes(s; s ) Holds(Up(x); s ) Holds(f; s) ^ Precedes(s; s ) Holds(f; s ) Holds(f; s ) Occurs(Lower(x); s) ^ Precedes(s; s ) Holds(Up(x); s ) Holds(Up(Left); s) 6 Holds(Up(Right); s) Holds(Spilled ; s) Figure Default theory D Holds(f; s) ^ Precedes(s; s ) Holds(f; s ) Holds(f; s ) The notion of a solution with respect to a splitting set is extended to splitting sequences as follows. Let A = ha i be a splitting sequence for D. A solution to D with respect to A is a sequence he i of sets of formulas that satises the following three conditions. is a consistent extension of the default theory b A (D) over L(A ). E For any such that +, E + is a consistent extension of the default theory over L(A + n A ). e ba+ (D); E A For any limit ordinal, E = Cn ; (;). We generalize the Splitting Set Theorem as follows. Splitting Sequence Theorem. Let A = ha i be a splitting sequence for a default theory D over L(U). A set E of formulas is a consistent extension of D if and only if E = Cn U E for some solution he i to D with respect to A. The proof of the Splitting Sequence Theorem, which appears in later in the paper, relies on the Splitting Set Theorem. We also have the following counterpart to the Splitting Set Corollary. Splitting Sequence Corollary. Let A = ha i be a splitting sequence for a default theory D over L(U). Let hu i be the sequence of pairwise disjoint subsets of U such that for all U = A n A If E is a consistent extension of D, then the sequence h E \ L(U ) i is a solution to D with respect to A. Example Splitting a Default Theory for Reasoning About Action We will illustrate the use of the Splitting Sequence Theorem by applying it to a default theory that formalizes commonsense knowledge about actions. The action domain we formalize is based on the \Soup Bowl" example from (Baral & Gelfond 993), which involves reasoning about the eects of concurrent actions. 6 The default logic formalization we consider relies, implicitly, on the notion of \static causal laws," recently investigated in (McCain & Turner 995; Turner 996). Default Theory D We will formalize the following action domain in default logic. There is a bowl of soup. There are actions that raise and lower its left and right sides. If one side of the bowl is up while the other is not up, the soup is spilled. In the initial situation S, both sides of the bowl are not up, and the soup is not spilled. We let S be the situation that would result from raising both sides of the bowl simultaneously. Notice that, intuitively speaking, it follows that in situation S both sides of the bowl are up and the soup is not spilled. We also consider an additional situation S 2 that would result from lowering the left side of the bowl, when in situation S. Intuitively speaking, we can conclude that in situation S 2 the left side of the bowl is not up, the right side is up, and the soup is spilled. We will use the Splitting Sequence Theorem to demonstrate that the default theory D, shown in Figure, is a correct formalization this action domain. More precisely, what we will show is that the literals Holds(Up(Left); S ), Holds(Up(Right); S ), and Holds(Spilled; S ) are among the consequences of default theory D, which establishes that D correctly describes the values of the uents in situation S. 6 We will not attempt in this paper a general analysis of the problem of representing commonsense knowledge about concurrent actions. Nor will we attempt a general justication of the style of representation employed in the example default theory itself. Such analysis and justication is an important problem that is beyond the scope of this paper.
4 Holds(Up(Left); S ) ^ Holds(Up(Right); S ) ^ Holds(Spilled; S ) Precedes(S ; S ) ^ Occurs(Raise(Left); S ) ^ Occurs(Raise(Right); S ) Precedes(S ; S 2 ) ^ Occurs(Lower(Left); S ) Holds(Up(Left); S ) 6 Holds(Up(Right); S ) Holds(Spilled; S ) Figure 2 Default theory b A (D ) Holds(Up(Left); S ) Holds(Up(Right); S ) Holds(Up(Left); S ) 6 Holds(Up(Right); S ) Holds(Spilled ; S ) Holds(Up(Left); S ) Holds(Up(Right); S ) Holds(Up(Left); S ) Holds(Up(Right); S ) Holds(Spilled; S ) Holds(Spilled ; S ) Figure 3 Default theory e A (b A (D ); E ) Holds(Up(Left); S 2 ) Holds(Up(Left); S 2 ) 6 Holds(Up(Right); S 2 ) Holds(Spilled; S 2 ) Holds(Up(Left); S 2 ) Holds(Up(Right); S 2 ) Holds(Up(Left); S 2 ) Holds(Up(Right); S 2 ) Holds(Spilled; S 2 ) Holds(Spilled; S 2 ) Figure 4 Default theory e A (b A2 (D ); E E ) Similarly, we will also show that D entails the literals Holds(Up(Left); S 2 ), Holds(Up(Right); S 2 ), and Holds(Spilled; S 2 ), which establishes that D also correctly describes situation S 2. Splitting Default Theory D Before applying the Splitting Sequence Theorem, we must be more precise about the language of D Let U consist of all ground atoms of the following many-sorted, rst-order language L. The sorts of L are situation, action, uent and side. There are object constants S, S and S 2 of sort situation, Left and Right of sort side, and Spilled of sort uent. There is a unary function symbol Up of sort side uent, and unary function symbols Raise and Lower of sort side action. There is a binary predicate symbol Holds of sort uent situation, a binary predicate symbol Occurs of sort action situation, and a binary predicate symbol Precedes of sort situation situation. We take D to be the (propositional) default theory over L(U ) consisting of all ground instances in L of the rules shown in Figure. Now we can specify a splitting sequence for D, which will allow us to break D into simpler parts. To begin, let A consist of all ground instances in L of the following atoms Holds(f; S ) ; Occurs(a; s) ; Precedes(s; s ) Notice that A splits D, with b A (D ) the default theory over L(A ) shown in Figure 2. We obtain A by adding to A all ground instances in L of Holds(f; S ). Finally, let A 2 = U. One easily checks that ha ; A ; A 2 i is a splitting sequence for D. Let X consist of the following literals Holds(Up(Left); S ) ; Holds(Up(Right); S ) ; Holds(Spilled; S ) ; Precedes(S ; S ) ; Occurs(Raise(Left); S ) ; Occurs(Raise(Right); S ) ; Precedes(S ; S 2 ) ; Occurs(Lower(Left); S ) Take E = Cn A (X ) Notice that E is the unique extension of b A (D ). The default theory e A (b A (D ); E ) over L(A na ) is (essentially) as shown in Figure 3. Thus, if we take X to consist of the literals Holds(Up(Left); S ) ; Holds(Up(Right); S ) ; Holds(Spilled; S ) ; and let E = Cn AnA (X ) we nd that E is the unique extension of default theory e A (b A (D ); E ). Finally, observe that the default theory e A (b A2 (D ); E E ) over L(A 2 n A ) is essentially as shown in Figure 4. Let X 2 consist of the literals Holds(Up(Left); S 2 ) ; Holds(Up(Right); S 2 ) ; Holds(Spilled ; S 2 ) Take E 2 = Cn A2nA (X 2 ) It is not dicult to verify that E 2 is the unique extension of e A (b A2 (D ); E E ). We have shown that he ; E ; E 2 i is a solution to default theory D with respect to ha ; A ; A 2 i. In fact it is the unique solution. It follows by the Splitting Sequence Theorem that Cn U (E E E 2 ) is the unique consistent extension of D. This shows that the literals Holds(Up(Left); S ), Holds(Up(Right); S ), and Holds(Spilled; S ) are among the consequences of D, which, as we discussed previously, is intuitively correct. We have similarly established that the literals Holds(Up(Left); S 2 ), Holds(Up(Right); S 2 ), and Holds(Spilled; S 2 ) are among the consequences of D, which is again the intuitively correct result.
5 Proof of the Splitting Set Theorem Due to space constraints, we present an abridged proof of the Splitting Set Theorem, omitting several intermediate lemmas, and proofs of the remaining lemmas. Lemma Let U; U be disjoint sets of atoms. Let R be a set of inference rules over L(U), and R a set of inference rules over L(U ). Let X = Cn UU (R R ). If X is consistent, then X \ L(U) = Cn U (R). X = Cn UU (Cn U (R) R ). In proving the Splitting Set Theorem it is convenient to introduce a set of alternative denitions, diering very slightly from those used in stating the theorem. (We nonetheless prefer the original denitions, because they are more convenient in applications of the Splitting Theorems.) Let D be a default theory over L(U) split by A. We dene the following. t A (D) = f r 2 D cons(r) 2 L(U n A) g b (D) = A D n t (D) A The advantage of these alternative denitions is captured in the following key lemma, which fails to hold for their counterparts b A (D) and D n b A (D). Lemma 2 Let D be a default theory over L(U) with splitting set A. For any set X of formulas from L(U), b A (D)X = b A (DX ) and t A (D)X = t A (DX ). The proof will also make use of the following additional alternative denitions. Given a set X of formulas from L(A), let e A (D; X) be the default theory over L(U n A) obtained from D in the following manner. For each rule r 2 t A (D) s.t. every conjunct of pre(r) that belongs to L(A) also belongs to Cn A (X), and no member of just(r) has its complement in Cn A (X) there is a rule r 2 e A (D; X) such that pre(r ) is obtained from pre(r) by replacing each conjunct of pre(r) that belongs to L(A) by >, and just(r ) = just(r) \ L(U n A), and cons(r ) = cons(r). Notice that e diers from A e A only in starting with the rules in t A (D) instead of the rules in D n b A(D). Finally, let s A (D) be the set of all pairs hx; i s.t. X is a consistent extension of b A (D), and is a consistent extension of e A (D; X). The following lemma shows that these alternative denitions are indeed suitable for our purpose. Lemma 3 If a default theory D over L(U) is split by A, then s A (D) is precisely the set of solutions to D with respect to A. Now we present the two main lemmas, and then use them to prove the Splitting Set Theorem. Lemma 4 Let D be a default theory over L(U) with splitting set A. Let E be a consistent set of formulas from L(U). E = Cn U (D E ) if and only if E \ L(A) = Cn A b A (D)E\L(A) ] and E = Cn U (E \ L(A)) t A (D)E ]. Lemma 5 Let D be a default theory over L(U) split by A. Let E be a logically closed set of formulas from L(U), with X = E \ L(A) and = E \ L(U n A). We have Cn U X e A (D; X) ] = Cn U X t A (D)E ] Proof of Splitting Set Theorem. Given a default theory D over L(U) with splitting set A, we know by Lemma 3 that s A (D) is precisely the set of solutions to D with respect to A. We will show that E is a consistent extension of D if and only if E = Cn U (X ) for some hx; i 2 s A (D). ()) Assume E is a consistent extension of D. Let X = E \ L(A) and = E \ L(U n A). By Lemma 4, X = Cn A (b A (D)X ) and E = Cn U (X t A (D)E ). By Lemma 5, E = Cn U (X e A (D; X) ). By Lemma, we can conclude that = Cn UnA (e A (D; X) ). So we have established that hx; i 2 s A (D). We can also conclude by Lemma that E = Cn U X Cn UnA (e A (D; X) )]. And since = Cn UnA (e A (D; X) ), we have E = Cn U (X ). (() Assume E = Cn U (X ) for some hx; i 2 s A (D). Since hx; i 2 s A (D), we have = Cn UnA (e A (D; X) ). Hence E = Cn U X Cn UnA (e A (D; X) )]. By Lemma, we can conclude that E = Cn U (X e A (D; X) ). Thus, by Lemma 5, E = Cn U (X t A (D)E ). By Lemma, E \ L(A) = Cn A (X), and since X is logically closed, Cn A (X) = X. So E \ L(A) = X. Since hx; i 2 s A (D), we have X = Cn A (b A (D)X ). We can conclude by Lemma 4 that E = Cn U (D E ). 2 Proof of Splitting Set Corollary. Assume that E is a consistent extension of D. By the Splitting Set Theorem, there is a solution hx; i to D with respect to A such that E = Cn U (X ). Since X L(A) and L(U n A), we can conclude by Lemma that E \ L(A) = Cn A (X). And since X is logically closed, Cn A (X) = X. So E \ L(A) = X. A symmetric argument shows that E \ L(U n A) =. 2 Proof of Splitting Sequence Theorem Lemma 6 Let D be a default theory over L(U) with splitting sequence A = ha i. Let E be a set of formulas from L(U). Let X = hx i be a sequence of sets of formulas from L(U) such that X = E \ L(A ), for all s.t. +, X + = E \ L(A + n A ), for any limit ordinal, X = Cn ; (;). If E is a consistent extension of D, then X is a solution to D with respect to A.
6 Proof. There are three things to check. First, by the Splitting Set Corollary, we can conclude that E \ L(A ) is a consistent extension of b A (D). Second, choose such that +. We must show that X + is a consistent extension of e ba+ (D); X A (2) Let = +. By the Splitting Set Corollary, E \ L(A ) is a consistent extension of b A (D). Let D = b A (D) and let E = E \ L(A ). By the Splitting Set Corollary, since A splits D, E \ L(A n A ) is a consistent extension of e A (D ; E \ L(A )). It is easy to verify that X + = E \ L(A n A ). It is not dicult to verify also that e A (D ; E \ L(A )) is the same as (2). Third, for any limit ordinal, X = Cn ; (;). 2 Lemma 7 Let D be a default theory over L(U) with splitting sequence A = ha i. Let he i be a solution to D with respect to A. For all Cn E A is a consistent extension of b A (D). Proof. For all, let X = Cn E A Proof is by induction on. Assume that for all, X is a consistent extension of b A (D). We'll show that X is a consistent extension of b A (D). There are two cases to consider. Case is not a limit ordinal. Choose such that + =. By the inductive hypothesis, X is a consistent extension of b A (D). We also know that E is a consistent extension of S e A (b A (D); E ). Let D = b A (D). It is clear that b A (D) = b A (D ). It is not dicult to verify that S e A (b A (D); E ) is the same as e A (D ; X ). So we've shown that X is a consistent extension of b A (D ) and that E is a consistent extension of e A (D ; X ). By the Splitting Set Theorem, it follows that Cn A (X E ) is a consistent extension of D. And since it's easy to check that Cn A (X E ) = X, we're done with the rst case. Case 2 is a limit ordinal. First we show that X is closed under b A (D) X. So suppose the contrary. Thus there is an r 2 b A (D) X such that pre(r) 2 X and cons(r) =2 X. Since A is continuous and is a limit ordinal, we know there must be a such that r 2 b A (D) X. Since b A (D) is a default theory over L(A ), we have b A (D) X = b A (D) X. So r 2 b A (D) X. Furthermore, it follows that pre(r) 2 X and cons(r) =2 X. This shows that X is not closed under b A (D) X, which contradicts the fact that, by the inductive hypothesis, X is a consistent extension of b A (D). So we have shown that X is closed under b A (D) X. Now, let E = Cn A (b A (D) X ). We will show that E = X, from which it follows that X is a consistent extension of b A (D). Since X is logically closed and closed under b A (D) X, we know that E X. Suppose E 6= X, and consider any formula 2 X n E. Since A is continuous and is a limit ordinal, there must be a such that is from L(A ) and therefore 2 X. Thus, X is a proper superset of E \ L(A ). By the inductive hypothesis, we know that X is a consistent extension of b A (D). Thus, X = Cn A (b A (D) X ). And since b A (D) X = b A (D) X, we have X = Cn A (b A (D) X ). Since E = Cn A (b A (D) X ) and b A (D) X b A (D) X, we know that E is closed under b A (D) X. Moreover, since b A (D) X is a default theory over L(A ), E \ L(A ) is closed under b A (D) X. But X is the least logically closed set closed under b A (D) X, so X E \L(A ), which contradicts the fact that X is a proper superset of E \ L(A ). We can conclude that E = X, which completes the second case. 2 Let D be a default theory over L(U) with splitting sequence A = ha i. The standard extension of A is the sequence B = hb i + such that for all, B = A, and B = U. Notice that the standard extension of A is itself a splitting sequence for D. Lemma 8 Let D be a default theory over L(U) with splitting sequence A = ha i. Let B = hb i + be the standard extension of A. Let X = hx i be a sequence of sets of formulas from L(U). Let = h i + be dened as follows. For all, = X. = Cn ; (;). If X is a solution to D with respect to A, then is a solution to D with respect to B. Proof. First, it's clear that is a consistent extension of b B (D), since = X, b B (D) = b A (D), and X is a consistent extension of b A (D). Similarly, it's clear that for any such that +, + is a consistent extension of e bb+ (D); A We also know that for any limit ordinal, = Cn ; (;). It remains to show that we handle correctly. There are two cases to consider. Case is a limit ordinal. In this case we must show that = Cn ; (;), which it does.
7 Case 2 is not a limit ordinal. In this case, choose such that + =. We must show that is a consistent extension of the default theory e bb (D); A (3) over L(B n B ). Since A is a splitting sequence for a default theory over L(U), we know that S A = U. Moreover, since A is monotone and is not a limit ordinal, it follows that A = U. And since B = A, we know that b B (D) = D. It follows that default theory (3) is empty. It also follows that B n B = ;, so the language of (3) is L(;). Since = Cn ; (;), we have shown that is a consistent extension of (3). 2 Proof of Splitting Sequence Theorem. ()) Assume that E is a consistent extension of D. By Lemma 6, there is a solution he i to D with respect to ha i for which it is not dicult to verify that E = Cn U E (() Assume that X = hx i is a solution to D with respect to ha i. Let E = Cn U X Let B = hb i + be the standard extension of ha i. By Lemma 8, we know there is a solution h i + to D with respect to B such that E = Cn U + Moreover, we know there is an + such that B = U. Thus b B (D) = D and E = Cn A It follows by Lemma 7 that E is a consistent extension of D. 2 Proof of Splitting Sequence Corollary. Assume that E is a consistent extension of D. By the Splitting Sequence Theorem, there is a solution hx i to D S with respect to A such that E = Cn U X. We will show that for all, E \ L(U ) = X. Let X = S X. Consider any. We have X L(U ), X n X L(U n U ), and E = Cn U (X X n X ). Thus, by Lemma we can conclude that E \ L(U ) = Cn U (X ). And since X is logically closed, we have Cn U (X ) = X. 2 Acknowledgments Many thanks to Vladimir Lifschitz and Norm McCain. This work was partially supported by National Science Foundation grant IRI References Baral, C., and Gelfond, M Representing concurrent actions in extended logic programming. In Proc. of IJCAI-93, 866{87. Besnard, P An Introduction to Default Logic. Springer-Verlag. Brewka, G. 99. Nonmonotonic Reasoning Logical Foundations of Commonsense. Cambridge University Press. Cholewinski, P Reasoning with stratied default theories. In Proc. of 3rd Int'l Conf. on Logic Programming and Nonmonotonic Reasoning, 273{286. Gelfond, M., and Lifschitz, V. 99. Classical negation in logic programs and disjunctive databases. New Generation Computing 9365{385. Gelfond, M., and Przymusinska, H On consistency and completeness of autoepistemic theories. Fundamenta Informaticae XVI59{92. Gelfond, M.; Lifschitz, V.; Przymusinska, H.; and Truszczynski, M. 99. Disjunctive defaults. In Allen, J.; Fikes, R.; and Sandewall, E., eds., Principles of Knowledge Representation and Reasoning Proc. of the 2nd Int'l Conference, 23{237. Komorowski, J. 99. Towards a programming methodology founded on partial deduction. In Proc. of ECAI-9. Lifschitz, V., and Turner, H Splitting a logic program. In Van Hentenryck, P., ed., Proc. Eleventh Int'l Conf. on Logic Programming, 23{37. Marek, W., and Truszczynski, M Nonmonotonic Logic Context-Dependent Reasoning. Springer- Verlag. McCain, N., and Turner, H A causal theory of ramications and qualications. In Proc. of IJCAI- 95, 978{984. Poole, D Default logic. In Gabbay, D.; Hogger, C.; and Robinson, J., eds., The Handbook of Logic in AI and Logic Programming, volume 3. Oxford University Press. 89{25. Reiter, R. 98. A logic for default reasoning. Arti- cial Intelligence 3(,2)8{32. Turner, H Representing actions in default logic A situation calculus approach. In Working Papers of the Third Symposium on Logical Formalizations of Commonsense Reasoning. Turner, H Representing actions in logic programs and default theories A situation calculus approach. Journal of Logic Programming. Forthcoming.
2 Transition Systems Denition 1 An action signature consists of three nonempty sets: a set V of value names, a set F of uent names, and a set A of act
Action Languages Michael Gelfond Department of Computer Science University of Texas at El Paso Austin, TX 78768, USA Vladimir Lifschitz Department of Computer Sciences University of Texas at Austin Austin,
More informationSplitting a efault Theory
From: AAAI-96 Proceedings. Copyright 1996, AAAI (www.aaai.org). All rights reserved. Splitting a efault Theory Hudson Turner Department of Computer Sciences University of Texas at Austin Austin, TX 78712-1188,
More informationTwo-Valued Logic Programs
Two-Valued Logic Programs Vladimir Lifschitz University of Texas at Austin, USA Abstract We define a nonmonotonic formalism that shares some features with three other systems of nonmonotonic reasoning
More informationA Preference Semantics. for Ground Nonmonotonic Modal Logics. logics, a family of nonmonotonic modal logics obtained by means of a
A Preference Semantics for Ground Nonmonotonic Modal Logics Daniele Nardi and Riccardo Rosati Dipartimento di Informatica e Sistemistica, Universita di Roma \La Sapienza", Via Salaria 113, I-00198 Roma,
More informationThis second-order avor of default logic makes it especially useful in knowledge representation. An important question is, then, to characterize those
Representation Theory for Default Logic V. Wiktor Marek 1 Jan Treur 2 and Miros law Truszczynski 3 Keywords: default logic, extensions, normal default logic, representability Abstract Default logic can
More informationLoop Formulas for Circumscription
Loop Formulas for Circumscription Joohyung Lee Department of Computer Sciences University of Texas, Austin, TX, USA appsmurf@cs.utexas.edu Fangzhen Lin Department of Computer Science Hong Kong University
More informationExtremal problems in logic programming and stable model computation Pawe l Cholewinski and Miros law Truszczynski Computer Science Department Universi
Extremal problems in logic programming and stable model computation Pawe l Cholewinski and Miros law Truszczynski Computer Science Department University of Kentucky Lexington, KY 40506-0046 fpaweljmirekg@cs.engr.uky.edu
More informationNested Epistemic Logic Programs
Nested Epistemic Logic Programs Kewen Wang 1 and Yan Zhang 2 1 Griffith University, Australia k.wang@griffith.edu.au 2 University of Western Sydney yan@cit.uws.edu.au Abstract. Nested logic programs and
More informationAn assumption-based framework for. Programming Systems Institute, Russian Academy of Sciences
An assumption-based framework for non-monotonic reasoning 1 Andrei Bondarenko 2 Programming Systems Institute, Russian Academy of Sciences Pereslavle-Zalessky, Russia andrei@troyka.msk.su Francesca Toni,
More informationLoop Formulas for Disjunctive Logic Programs
Nineteenth International Conference on Logic Programming (ICLP-03), pages 451-465, Mumbai, India, 2003 Loop Formulas for Disjunctive Logic Programs Joohyung Lee and Vladimir Lifschitz Department of Computer
More informationTrichotomy Results on the Complexity of Reasoning with Disjunctive Logic Programs
Trichotomy Results on the Complexity of Reasoning with Disjunctive Logic Programs Mirosław Truszczyński Department of Computer Science, University of Kentucky, Lexington, KY 40506, USA Abstract. We present
More informationDecision procedure for Default Logic
Decision procedure for Default Logic W. Marek 1 and A. Nerode 2 Abstract Using a proof-theoretic approach to non-monotone reasoning we introduce an algorithm to compute all extensions of any (propositional)
More informationYet Another Proof of the Strong Equivalence Between Propositional Theories and Logic Programs
Yet Another Proof of the Strong Equivalence Between Propositional Theories and Logic Programs Joohyung Lee and Ravi Palla School of Computing and Informatics Arizona State University, Tempe, AZ, USA {joolee,
More informationSyntactic Conditions for Antichain Property in CR-Prolog
This version: 2017/05/21 Revising my paper before submitting for publication. Syntactic Conditions for Antichain Property in CR-Prolog Vu Phan Texas Tech University Abstract We study syntactic conditions
More informationDisjunctive Signed Logic Programs
Disjunctive Signed Logic Programs Mohamed A. Khamsi 1 and Driss Misane 2 1 Department of Mathematical Sciences University of Texas at El Paso, El Paso, TX 79968. 2 Département de Mathématiques, Faculté
More informationsignicant in view of the fact that in many applications existential queries are of main interest. It also plays an important role in the problem of nd
On the Relationship Between CWA, Minimal Model and Minimal Herbrand Model Semantics Michael Gelfond Halina Przymusinska Teodor Przymusinski March 3, 1995 Abstract The purpose of this paper is to compare
More informationDefault Reasoning and Belief Revision: A Syntax-Independent Approach. (Extended Abstract) Department of Computer Science and Engineering
Default Reasoning and Belief Revision: A Syntax-Independent Approach (Extended Abstract) Dongmo Zhang 1;2, Zhaohui Zhu 1 and Shifu Chen 2 1 Department of Computer Science and Engineering Nanjing University
More informationKRIPKE S THEORY OF TRUTH 1. INTRODUCTION
KRIPKE S THEORY OF TRUTH RICHARD G HECK, JR 1. INTRODUCTION The purpose of this note is to give a simple, easily accessible proof of the existence of the minimal fixed point, and of various maximal fixed
More informationparticular formulas are entailed by this theory. Autoepistemic logic (as well as other nonmonotonic systems) can be used in this mode, too. Namely, un
Fixpoint 3-valued semantics for autoepistemic logic Marc Denecker Department of Computer Science K.U.Leuven Celestijnenlaan 200A, B-3001 Heverlee, Belgium marcd@cs.kuleuven.ac.be Victor Marek and Miros
More informationThe Modal Logic S4F, the Default Logic, and the Logic Here-and-There
The Modal Logic S4F, the Default Logic, and the Logic Here-and-There Mirosław Truszczyński Department of Computer Science, University of Kentucky, Lexington, KY 40506-0046, USA Abstract The modal logic
More informationComputing the acceptability semantics. London SW7 2BZ, UK, Nicosia P.O. Box 537, Cyprus,
Computing the acceptability semantics Francesca Toni 1 and Antonios C. Kakas 2 1 Department of Computing, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK, ft@doc.ic.ac.uk 2 Department of Computer
More information1 Introduction In this paper we propose a framework for studying the process of database revision. Revisions that we have in mind are specied by means
Revision programming Victor W. Marek Miros law Truszczynski Department of Computer Science University of Kentucky Lexington, KY 40506-0027 Abstract In this paper we introduce revision programming a logic-based
More information(A 3 ) (A 1 ) (1) COMPUTING CIRCUMSCRIPTION. Vladimir Lifschitz. Department of Computer Science Stanford University Stanford, CA
COMPUTING CIRCUMSCRIPTION Vladimir Lifschitz Department of Computer Science Stanford University Stanford, CA 94305 Abstract Circumscription is a transformation of predicate formulas proposed by John McCarthy
More informationChange, Change, Change: three approaches
Change, Change, Change: three approaches Tom Costello Computer Science Department Stanford University Stanford, CA 94305 email: costelloqcs.stanford.edu Abstract We consider the frame problem, that is,
More informationOn 3-valued paraconsistent Logic Programming
Marcelo E. Coniglio Kleidson E. Oliveira Institute of Philosophy and Human Sciences and Centre For Logic, Epistemology and the History of Science, UNICAMP, Brazil Support: FAPESP Syntax Meets Semantics
More informationINTRODUCTION TO NONMONOTONIC REASONING
Faculty of Computer Science Chair of Automata Theory INTRODUCTION TO NONMONOTONIC REASONING Anni-Yasmin Turhan Dresden, WS 2017/18 About the Course Course Material Book "Nonmonotonic Reasoning" by Grigoris
More informationNew insights on the intuitionistic interpretation of Default Logic
New insights on the intuitionistic interpretation of Default Logic Pedro Cabalar David Lorenzo 1 Abstract. An interesting result found by Truszczyński showed that (non-monotonic) modal logic S4F can be
More informationSemantic Forcing in Disjunctive Logic Programs
Semantic Forcing in Disjunctive Logic Programs Marina De Vos and Dirk Vermeir Dept. of Computer Science Free University of Brussels, VUB Pleinlaan 2, Brussels 1050, Belgium Abstract We propose a semantics
More informationLogic as The Calculus of Computer Science
1 Ottobre, 2007 1 Università di Napoli Federico II What is a Logic? A Logic is a formalism with a sintax a semantics an inference mechanism for reasoning Historical Diagram The First Age of Logic: Symbolic
More informationrules strictly includes the class of extended disjunctive programs. Interestingly, once not appears positively as above, the principle of minimality d
On Positive Occurrences of Negation as Failure 3 Katsumi Inoue Department of Information and Computer Sciences Toyohashi University of Technology Tempaku-cho, Toyohashi 441, Japan inoue@tutics.tut.ac.jp
More informationAn Alternative To The Iteration Operator Of. Propositional Dynamic Logic. Marcos Alexandre Castilho 1. IRIT - Universite Paul Sabatier and
An Alternative To The Iteration Operator Of Propositional Dynamic Logic Marcos Alexandre Castilho 1 IRIT - Universite Paul abatier and UFPR - Universidade Federal do Parana (Brazil) Andreas Herzig IRIT
More informationWarm-Up Problem. Is the following true or false? 1/35
Warm-Up Problem Is the following true or false? 1/35 Propositional Logic: Resolution Carmen Bruni Lecture 6 Based on work by J Buss, A Gao, L Kari, A Lubiw, B Bonakdarpour, D Maftuleac, C Roberts, R Trefler,
More informationIntroduction to Metalogic
Philosophy 135 Spring 2008 Tony Martin Introduction to Metalogic 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: Remarks: (i) sentence letters p 0, p 1, p 2,... (ii)
More information330 SU Kaile Vol.16 Truszczyński [12] constructed an example of denumerable family of pairwise inconsistent theories which cannot be represented by no
Vol.16 No.4 J. Comput. Sci. & Technol. July 2001 Constraints on Extensions of a Default Theory SU Kaile (±ΞΠ) Department of Computer Science, Zhongshan University, Guangzhou 510275, P.R. China State Key
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 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 informationAN ANALYSIS OF SYSTEMATIC APPROACHES TO REASONING ABOUT ACTIONS AND CHANGE
AN ANALYSIS OF SYSTEMATIC APPROACHES TO REASONING ABOUT ACTIONS AND CHANGE MICHAEL THIELSCHER Intellektik, Informatik, TH Darmstadt Alexanderstraße 10, D 64283 Darmstadt, Germany In: P. Jorrand, ed., Proc.
More informationOverview. I Review of natural deduction. I Soundness and completeness. I Semantics of propositional formulas. I Soundness proof. I Completeness proof.
Overview I Review of natural deduction. I Soundness and completeness. I Semantics of propositional formulas. I Soundness proof. I Completeness proof. Propositional formulas Grammar: ::= p j (:) j ( ^ )
More informationA theory of modular and dynamic knowledge representation
A theory of modular and dynamic knowledge representation Ján Šefránek Institute of Informatics, Comenius University, Mlynská dolina, 842 15 Bratislava, Slovakia, phone: (421-7) 6029 5436, e-mail: sefranek@fmph.uniba.sk
More informationUltimate approximation and its application in nonmonotonic knowledge representation systems
Ultimate approximation and its application in nonmonotonic knowledge representation systems Marc Denecker Department of Computer Science, K.U.Leuven Celestijnenlaan 200A, B-3001 Heverlee Département d
More informationBoolean Algebra and Propositional Logic
Boolean Algebra and Propositional Logic Takahiro Kato September 10, 2015 ABSTRACT. This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a
More informationBoolean Algebra and Propositional Logic
Boolean Algebra and Propositional Logic Takahiro Kato June 23, 2015 This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a more direct connection
More informationApplications of the Situation Calculus To Formalizing. Control and Strategic Information: The Prolog Cut Operator. Fangzhen Lin
Applications of the Situation Calculus To Formalizing Control and Strategic Information: The Prolog Cut Operator Fangzhen Lin (in@csusthk) Department of Computer Science The Hong Kong University of Science
More informationGuarded resolution for Answer Set Programming
Under consideration for publication in Theory and Practice of Logic Programming 1 Guarded resolution for Answer Set Programming V.W. Marek Department of Computer Science, University of Kentucky, Lexington,
More informationMathematical Logic Propositional Logic - Tableaux*
Mathematical Logic Propositional Logic - Tableaux* Fausto Giunchiglia and Mattia Fumagalli University of Trento *Originally by Luciano Serafini and Chiara Ghidini Modified by Fausto Giunchiglia and Mattia
More informationInfinitary Equilibrium Logic and Strong Equivalence
Infinitary Equilibrium Logic and Strong Equivalence Amelia Harrison 1 Vladimir Lifschitz 1 David Pearce 2 Agustín Valverde 3 1 University of Texas, Austin, Texas, USA 2 Universidad Politécnica de Madrid,
More informationOn the Progression of Situation Calculus Basic Action Theories: Resolving a 10-year-old Conjecture
On the Progression of Situation Calculus Basic Action Theories: Resolving a 10-year-old Conjecture Stavros Vassos and Hector J Levesque Department of Computer Science University of Toronto Toronto, ON,
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 informationReasoning: From Basic Entailments. to Plausible Relations. Department of Computer Science. School of Mathematical Sciences. Tel-Aviv University
General Patterns for Nonmonotonic Reasoning: From Basic Entailments to Plausible Relations Ofer Arieli Arnon Avron Department of Computer Science School of Mathematical Sciences Tel-Aviv University Tel-Aviv
More informationis a model, supported model or stable model, respectively, of P. The check can be implemented to run in linear time in the size of the program. Since
Fixed-parameter complexity of semantics for logic programs Zbigniew Lonc? and Miros law Truszczynski?? Department of Computer Science, University of Kentucky Lexington KY 40506-0046, USA flonc, mirekg@cs.engr.uky.edu
More informationPearl s Causality In a Logical Setting
Pearl s Causality In a Logical Setting Alexander Bochman Computer Science Department Holon Institute of Technology, Israel bochmana@hit.ac.il Vladimir Lifschitz Department of Computer Science University
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter 1 The Real Numbers 1.1. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {1, 2, 3, }. In N we can do addition, but in order to do subtraction we need
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter The Real Numbers.. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {, 2, 3, }. In N we can do addition, but in order to do subtraction we need to extend
More informationComplexity of computing with extended propositional logic programs
Complexity of computing with extended propositional logic programs V. Wiktor Marek 1 Arcot Rajasekar 2 Miros law Truszczyński 1 Department of Computer Science University of Kentucky Lexington, KY 40506
More information2 1. INTRODUCTION We study algebraic foundations of semantics of nonmonotonic knowledge representation formalisms. The algebraic framework we use is t
Chapter 1 APPROXIMATIONS, STABLE OPERATORS, WELL-FOUNDED FIXPOINTS AND APPLICATIONS IN NONMONOTONIC REASONING Marc Denecker Department of Computer Science, K.U.Leuven Celestijnenlaan 200A, B-3001 Heverlee,
More informationLloyd-Topor Completion and General Stable Models
Lloyd-Topor Completion and General Stable Models Vladimir Lifschitz and Fangkai Yang Department of Computer Science The University of Texas at Austin {vl,fkyang}@cs.utexas.edu Abstract. We investigate
More informationEncoding formulas with partially constrained weights in a possibilistic-like many-sorted propositional logic
Encoding formulas with partially constrained weights in a possibilistic-like many-sorted propositional logic Salem Benferhat CRIL-CNRS, Université d Artois rue Jean Souvraz 62307 Lens Cedex France benferhat@criluniv-artoisfr
More informationGeneral Patterns for Nonmonotonic Reasoning: From Basic Entailments to Plausible Relations
General Patterns for Nonmonotonic Reasoning: From Basic Entailments to Plausible Relations OFER ARIELI AND ARNON AVRON, Department of Computer Science, School of Mathematical Sciences, Tel-Aviv University,
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 information2 Z. Lonc and M. Truszczynski investigations, we use the framework of the xed-parameter complexity introduced by Downey and Fellows [Downey and Fellow
Fixed-parameter complexity of semantics for logic programs ZBIGNIEW LONC Technical University of Warsaw and MIROS LAW TRUSZCZYNSKI University of Kentucky A decision problem is called parameterized if its
More informationProduction Inference, Nonmonotonicity and Abduction
Production Inference, Nonmonotonicity and Abduction Alexander Bochman Computer Science Department, Holon Academic Institute of Technology, Israel e-mail: bochmana@hait.ac.il Abstract We introduce a general
More informationAdding Modal Operators to the Action Language A
Adding Modal Operators to the Action Language A Aaron Hunter Simon Fraser University Burnaby, B.C. Canada V5A 1S6 amhunter@cs.sfu.ca Abstract The action language A is a simple high-level language for describing
More informationReducts of propositional theories, satisfiability relations, and generalizations of semantics of logic programs
Reducts of propositional theories, satisfiability relations, and generalizations of semantics of logic programs Miroslaw Truszczyński Department of Computer Science, University of Kentucky, Lexington,
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 informationOn Modelling of Inertia in Action Languages.
On Modelling of Inertia in Action Languages. Mikhail Prokopenko CSIRO Division of Information Technology, Locked Bag 17, North Ryde, NSW 2113, Australia E-mail: mikhail@syd.dit.csiro.au Pavlos Peppas Department
More informationEquivalence for the G 3-stable models semantics
Equivalence for the G -stable models semantics José Luis Carballido 1, Mauricio Osorio 2, and José Ramón Arrazola 1 1 Benemérita Universidad Autóma de Puebla, Mathematics Department, Puebla, México carballido,
More informationCharacterizing Causal Action Theories and Their Implementations in Answer Set Programming: Action Languages B, C and Beyond
Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015) Characterizing Causal Action Theories and Their Implementations in Answer Set Programming: Action
More informationArgumentation and rules with exceptions
Argumentation and rules with exceptions Bart VERHEIJ Artificial Intelligence, University of Groningen Abstract. Models of argumentation often take a given set of rules or conditionals as a starting point.
More informationAbductive Disjunctive Logic Programming. Ullrich Hustadt
Abductive Disjunctive Logic Programming Ullrich Hustadt Max-Planck-Institut fur Informatik, Im Stadtwald, W-6600 Saarbrucken, Germany Phone: (+9 681) 302-531, Fax: (+9 681) 302-530, E-mail: Ullrich.Hustadt@mpi-sb.mpg.de
More informationA Strong Relevant Logic Model of Epistemic Processes in Scientific Discovery
A Strong Relevant Logic Model of Epistemic Processes in Scientific Discovery (Extended Abstract) Jingde Cheng Department of Computer Science and Communication Engineering Kyushu University, 6-10-1 Hakozaki,
More informationBelief revision: A vade-mecum
Belief revision: A vade-mecum Peter Gärdenfors Lund University Cognitive Science, Kungshuset, Lundagård, S 223 50 LUND, Sweden Abstract. This paper contains a brief survey of the area of belief revision
More informationA Theory of Forgetting in Logic Programming
A Theory of Forgetting in Logic Programming Kewen Wang 1,2 and Abdul Sattar 1,2 and Kaile Su 1 1 Institute for Integrated Intelligent Systems 2 School of Information and Computation Technology Griffith
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 informationDuality in Logic Programming
Syracuse University SURFACE Electrical Engineering and Computer Science Technical Reports College of Engineering and Computer Science 3-1991 Duality in Logic Programming Feng Yang Follow this and additional
More informationA Fixed Point Representation of References
A Fixed Point Representation of References Susumu Yamasaki Department of Computer Science, Okayama University, Okayama, Japan yamasaki@momo.cs.okayama-u.ac.jp Abstract. This position paper is concerned
More informationCompleteness in the Monadic Predicate Calculus. We have a system of eight rules of proof. Let's list them:
Completeness in the Monadic Predicate Calculus We have a system of eight rules of proof. Let's list them: PI At any stage of a derivation, you may write down a sentence φ with {φ} as its premiss set. TC
More informationTableau Calculus for Local Cubic Modal Logic and it's Implementation MAARTEN MARX, Department of Articial Intelligence, Faculty of Sciences, Vrije Uni
Tableau Calculus for Local Cubic Modal Logic and it's Implementation MAARTEN MARX, Department of Articial Intelligence, Faculty of Sciences, Vrije Universiteit Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam,
More informationQuantiers and Partiality. Jan van Eijck. May 16, Quantication can involve partiality in several ways. Quantiers loaded
Quantiers and Partiality Jan van Eijck May 16, 1995 1 Varieties of Partiality Quantication can involve partiality in several ways. Quantiers loaded with presuppositions give rise to partiality by introducing
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 informationatoms should became false, i.e., should be deleted. However, our updates are far more expressive than a mere insertion and deletion of facts. They can
Dynamic Updates of Non-Monotonic Knowledge Bases J. J. Alferes Dept. Matematica Univ. Evora and A.I. Centre Univ. Nova de Lisboa, 2825 Monte da Caparica Portugal H. Przymusinska Computer Science California
More informationLogic Part I: Classical Logic and Its Semantics
Logic Part I: Classical Logic and Its Semantics Max Schäfer Formosan Summer School on Logic, Language, and Computation 2007 July 2, 2007 1 / 51 Principles of Classical Logic classical logic seeks to model
More informationA MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ
A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ NICOLAS FORD Abstract. The goal of this paper is to present a proof of the Nullstellensatz using tools from a branch of logic called model theory. In
More informationDept. of Computer Science, University of Texas at El Paso, El Paso, Texas
Formalizing Narratives using Nested Circumscription Chitta Baral, Alfredo Gabaldon Dept. of Computer Science, University of Texas at El Paso, El Paso, Texas 79968 http://cs.utep.edu/csdept/krgroup.html
More informationExtending Logic Programs with Description Logic Expressions for the Semantic Web
Extending Logic Programs with Description Logic Expressions for the Semantic Web Yi-Dong Shen 1 and Kewen Wang 2 1 State Key Laboratory of Computer Science, Institute of Software Chinese Academy of Sciences,
More informationRepresenting First-Order Causal Theories by Logic Programs
Under consideration for publication in Theory and Practice of Logic Programming 1 Representing First-Order Causal Theories by Logic Programs Paolo Ferraris Google, Inc. (e-mail: otto@cs.utexas.edu) Joohyung
More informationA note on fuzzy predicate logic. Petr H jek 1. Academy of Sciences of the Czech Republic
A note on fuzzy predicate logic Petr H jek 1 Institute of Computer Science, Academy of Sciences of the Czech Republic Pod vod renskou v 2, 182 07 Prague. Abstract. Recent development of mathematical fuzzy
More informationOn the Relationship of Defeasible Argumentation and Answer Set Programming
On the Relationship of Defeasible Argumentation and Answer Set Programming Matthias Thimm a Gabriele Kern-Isberner a a Information Engineering Group, Department of Computer Science University of Dortmund,
More informationArgumentative Characterisations of Non-monotonic Inference in Preferred Subtheories: Stable Equals Preferred
Argumentative Characterisations of Non-monotonic Inference in Preferred Subtheories: Stable Equals Preferred Sanjay Modgil November 17, 2017 Abstract A number of argumentation formalisms provide dialectical
More informationRepresenting First-Order Causal Theories by Logic Programs
Representing First-Order Causal Theories by Logic Programs Paolo Ferraris Google, Inc. Joohyung Lee School of Computing, Informatics and Decision Systems Engineering, Arizona State University Yuliya Lierler
More information{},{a},{a,c} {},{c} {c,d}
Modular verication of Argos Programs Agathe Merceron 1 and G. Michele Pinna 2 1 Basser Department of Computer Science, University of Sydney Madsen Building F09, NSW 2006, Australia agathe@staff.cs.su.oz.au
More informationModel Generation and State Generation for Disjunctive Logic. Programs. University of Tubingen. Sand 13, D { Tubingen, Germany
Model Generation and State Generation for Disjunctive Logic rograms Dietmar Seipel 1, Jack Minker 2, Carolina Ruiz 2 1 Department of Computer Science. University of Tubingen. Sand 13, D { 72076 Tubingen,
More informationChapter 2. Assertions. An Introduction to Separation Logic c 2011 John C. Reynolds February 3, 2011
Chapter 2 An Introduction to Separation Logic c 2011 John C. Reynolds February 3, 2011 Assertions In this chapter, we give a more detailed exposition of the assertions of separation logic: their meaning,
More informationAn Action Description Language for Iterated Belief Change
An Action Description Language for Iterated Belief Change Aaron Hunter and James P. Delgrande Simon Fraser University Burnaby, BC, Canada {amhunter, jim}@cs.sfu.ca Abstract We are interested in the belief
More informationClausal Presentation of Theories in Deduction Modulo
Gao JH. Clausal presentation of theories in deduction modulo. JOURNAL OF COMPUTER SCIENCE AND TECHNOL- OGY 28(6): 1085 1096 Nov. 2013. DOI 10.1007/s11390-013-1399-0 Clausal Presentation of Theories in
More informationFuzzy Answer Set semantics for Residuated Logic programs
semantics for Logic Nicolás Madrid & Universidad de Málaga September 23, 2009 Aims of this paper We are studying the introduction of two kinds of negations into residuated : Default negation: This negation
More informationIntroduction to Metalogic 1
Philosophy 135 Spring 2012 Tony Martin Introduction to Metalogic 1 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: (i) sentence letters p 0, p 1, p 2,... (ii) connectives,
More informationPropositional Logic: Syntax
4 Propositional Logic: Syntax Reading: Metalogic Part II, 22-26 Contents 4.1 The System PS: Syntax....................... 49 4.1.1 Axioms and Rules of Inference................ 49 4.1.2 Definitions.................................
More informationcurrent set (database, belief set) and none of the elements b l, 1 l n, belongs to the current set then, in the case of rule (1), the item a should be
Annotated revision programs Victor Marek Inna Pivkina Miros law Truszczynski Department of Computer Science, University of Kentucky, Lexington, KY 40506-0046 marek inna mirek@cs.engr.uky.edu Abstract Revision
More information3. Only sequences that were formed by using finitely many applications of rules 1 and 2, are propositional formulas.
1 Chapter 1 Propositional Logic Mathematical logic studies correct thinking, correct deductions of statements from other statements. Let us make it more precise. A fundamental property of a statement is
More informationTemporal Logic and reasoning about actions
Temporal Logic and reasoning about actions Gisela Mendez Depto. de Matematicas Universidad Central de Venezuela Caracas, VENEZUELA gmendez@dino.conicit.ve Jorge Lobo Department of EECS Univ. of Illinois
More information