Download presentation

Presentation is loading. Please wait.

Published byQuinn Wadlow Modified over 3 years ago

1
Normal Programs with Cardinality Constraints C = L {a 1,..., a n, not b 1,..., not b m } U L lower bound, U upper bound (missing L: 0, missing U: n+m) Lit(C) = {a 1,..., a n, not b 1,..., not b m } intuition: at least L, at most U literals must be true rules built from constraints: C 0 <- C 1,..., C n example: 2{a1, a2, a3}2 <- 1{not b1, not b2}, 2{c1, c2, c3, c4}3 at least one b not in model and 2 or 3 c‘s in => 2 a‘s in. normal programs: all constraints of the form 1{a}1 oder 1{not a}1

2
Satisfaction of constraints in set of literals S S |= a iff a S, S|= not a iff a S. S |= C iff L W(C,S) U, where W(C,S) = | {l Lit(C) : S |= l}| rule C 0 <- C 1,..., C n satisfied in S iff some body constraint not satisfied or C 0 satisfied (integrity constraints: first option only) Example: 1{a,b}1 <- 1 {a, not b, not c} 2 S = {a,b}. body satisfied, head not => rule not satisfied S = {a, c}. body and head constraint and thus rule satisfied

3
Stable models Standard approach: guess S, evaluate „not“ wrt S => reduced program, check whether S minimal model of reduced program Here: reduction to Horn constraint rules: no „not“, no upper bounds, single head atom => Cn(P) smallest set closed under rules Example: a <- 1 {a} b <- c <- 2 {b,d}, 1 {b,a} d <- 1 {a,b,c} Cn(P) = {b}Cn(P) = {b, d, c}

4
Constraint reduction C = L {a1,..., an, not b1,..., not bm} U C S = L´{a1, …, an} where L´ = L - |{not b Lit(C) : S |= not b}| Example: S = {q}, C = 3 {not q, not r, p} 4, reduct: 2 {p} P program, S set of atoms. Reduct P S : {p <- C 1 S,..., C n S : C 0 <- C 1,..., C n P, p Lit(C 0 ) S, for all C i = L i {a1,…}U i, i {1, …, n}, W(C i,S) U i } Def.: S stable model of P iff 1. S satisfies all rules in P 2. S = Cn(P S )

5
Examples 1 {a1, a2, a3} 1 <- stable models: {a1}, {a2}, {a3} 1 {a1, a2, a3} 1 <- 1 {a1,b} 2 single stable model: empty set {a1, a2, a3} <- stable models: alle subsets of the three a‘s rules of this form are also called choice rules

6
Weight Constraints C = L {a 1 = w a1,..., a n = w an, not b 1 = w b1, …, not b m = w bm } U arbitrary weights (card. constraints weight 1 for all literals) implemented in Smodels: integers weight of l in C: w(C)(l) necessary changes: W(C,S) = l Lit(C),S |= l w(C)(l) and in definition of constraint reduct: L’ = L - not p Lit(C) and S |= not p w(C)(l)

7
Smodels optimization constructs minimize{a 1 = w a1,..., a n = w an, not b 1 = w b1, …, not b m = w bm } stable model computed by Smodels minimizes l in L and S |= l w(C)(l) (L = {a1,..., an, not b1,..., not bm}) maximize analogously multiple optimization statements: optimize first one, among optimal models optimize second one etc.

8
Smodels optimization example {a1, a2, a3} <- minimize {a1 = 1} minimize {a2 = 1} minimize {a3 = 1} 1.{ } 2.{a3} 3.{a2 } 4.{a2a3} 5.{a1 } 6.{a1a3} 7.{a1a2 } 8.{a1a2a3}

9
Notation 1{teaches(L,C) : lecturer(L)}1 <- course(C) to obtain ground instantiation: instatiate C, replace each instatiation of the form 1{teaches(L,c1) : lecturer(L)}1 <- course(c1) with 1{teaches(l1,c1),..., teaches(ln,c1)}1 <- course(c1) where l1,..., ln are the known lecturers

Similar presentations

OK

CSE 211- Discrete Structures1 Relations Ch 2 schaums, Ch 7 Rosen.

CSE 211- Discrete Structures1 Relations Ch 2 schaums, Ch 7 Rosen.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on different types of computer softwares list Ppt on indian culture and tradition free download Blood vessel anatomy and physiology ppt on cells Ppt on rich heritage of india Ppt on modes of transport in india Ppt online open course Ppt on infosys narayana murthy Ppt on carbon and its compounds worksheet Ppt on intelligent manufacturing in industrial automation Ppt on levels of management