### Table 1. Algorithms for Constraint Solving

1996

"... In PAGE 2: ...Table 1. Algorithms for Constraint Solving Table1 gives an overview of some methods and compares them with Parcon. Due to its high performance Parcon is well applicable for interactive applications.... ..."

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### Table 6: Algorithm for Solving Constraints.

1997

"... In PAGE 13: ... The fourth and nal step is to record the solution in a more familiar form. The following result shows that the algorithm does indeed compute the solution we want: Proposition 15 Given input C [[PP ]] the algorithm of Table6 terminates and the result ( b R; b C; b ;b ) produced by the algorithm satis- es (b R; b C; b ;b ) = uf( b R1; b C1; b 1;b 1) j ( b R1; b C1; b 1;b 1) j=c C [[PP ]]g and hence is the least solution to C [[PP ]]. 2 Proof It is immediate that Steps 1, 2 and 4 terminate, and this leaves us with Step 3.... ..."

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### Table 1: Computational complexity of constraint planning problems. Cycles in the constraint

"... In PAGE 5: ... Indeed, a general computational result states that a restriction imposed on a solution does not necessarily make the corresponding problem easier [Papadimitriou 94]. Table1 shows the computational complexity of some existing constraint planning problems. 4 An NP-complete planning problem with method restriction De nition 3 The method restriction imposes that every constraint method of a given problem must use all of the variables in the constraint either as an input or an output variable.... ..."

### Table 3: Constraints on the duration and the start time of the plan

2005

"... In PAGE 54: ... Again values in brackets represent the results obtained from a second CSP resolution. Table3 represents a restriction on the plan duration plus a restriction on the start time of the plan. First column de- note that the makespan must be below 250 seconds and the start time of the plan at any point later than time 10.... In PAGE 54: ...tart time of the plan at any point later than time 10.00. The missing data for the start/end time rows correpond to those cases where a plan is not found. We can also observe that CPU time is shorter in Table 1 than Table3 . In general, if only restrictions on the duration are considered then appropriate timing intervals combina- tions are found faster.... In PAGE 54: ...258 t.u.) is very close to the limit and the CPU time is very high (543 seconds) which indicates that almost all possible combinations of timing intervals have been searched until nding a solution. From comparing results in Table 2 and Table3 we can deduce that when there are restrictions on the start and end time of the plan the computation is faster ICAPS 2005 50... In PAGE 67: ... The set j100. Table3 and Table 4 show the results ob- tained in the case of the benchmark j100. In this case the difference of the two problem are smoother than in the pre- vious benchmark.... In PAGE 68: ...17 413.47 Table3 : ISESiC j100 j exj j dtj cpu npc mk GRASPiC+MINIDflex 0.207 0.... In PAGE 83: ...10 75.80 Table3 . Efficiency and stability on add STC restrictions.... ..."

### Table 3: Comparison of the number of constraints, unit prop- agations (UP), decisions, and solution time of SATPLAN04, Blackbox, and SATPLAN04 with londex, for solving the truck3 problem from IPC5 at time step 15. 1. procedure SATPLAN04+londex (INPUT: A propositional planning problem P) 2. generate the DTGs and londex constraints for P; 3. set k to be an lower bound of the optimal plan length; 4. repeat

"... In PAGE 3: ... Empirically, our prepro- cessor takes less than 30 seconds to generate all londex con- straints for each of the problems in IPC5, which is negligible comparing to the planning time that londex can help reduce, often in thousands of seconds for large problems. Table3 illustrates the use of londex in reducing planning time. We can compare the size of mutex constraints derived by Blackbox and SATPLAN04 (two SAT-based planners us- ing mutex) and the londex constraints.... In PAGE 3: ... The amount of londex constraints is 2 to 100 times larger than that of mutex, de- pending on planning domains. It is evident from Table3 that incorporating londex in a SAT planner, although largely in- creases the size of the SAT instance, can significantly reduce the speed of SAT solving because of the much stronger con- straint propagation and pruning. IJCAI07... ..."

### Table 2: An example of constraint solving on lt; C

"... In PAGE 20: ... This is represented by the triples: fc1; c2; c3; c4g as deflned in Table 1. Ta- ble 1 also gives the interval constraints used by Table2 to show the constraint solving process. Table 1: Triples and interval constraints in the example of lt; Triples Simple Interval constraints (c1) (::0; fSg; fd1 Sg) (d1 S) S 2 [?Real; gt;Real] (c2) (2; fSg; fd2 Sg) (d2 S) S 2 [0; 4] (c3) ( 0; fSg; fd3 Sg) (d3 S) S 2 [1; gt;Real] (c4) ( 0; fSg; fd4 Sg) (d4 S) S 2 [3:75; gt;Real] (d5 S) S 2 [1; 4] (d6 S) S 2 [3:75; 4] The values ?Real and gt;Real are added to the Real domain to ensure that Real has glb and lub elements.... ..."

### Table 2: An example of constraint solving on lt; C

"... In PAGE 20: ... This is represented by the triples: fc1; c2; c3; c4g as de ned in Table 1. Ta- ble 1 also gives the interval constraints used by Table2 to show the constraint solving process. Table 1: Triples and interval constraints in the example of lt; Triples Simple Interval constraints (c1) (::0; fSg; fd1 Sg) (d1 S) S 2 [?Real; gt;Real] (c2) (2; fSg; fd2 Sg) (d2 S) S 2 [0; 4] (c3) ( 0; fSg; fd3 Sg) (d3 S) S 2 [1; gt;Real] (c4) ( 0; fSg; fd4 Sg) (d4 S) S 2 [3:75; gt;Real] (d5 S) S 2 [1; 4] (d6 S) S 2 [3:75; 4]... ..."

### Table 1 gives some of the example systems with which we tested our proof plan. PT stands for the total elapsed planning time, given in seconds. Space constraints do not allow us to provide a full description of each veri cation problem. These example veri cation problems are all taken from [7]. They all involve the use of generalise, UFI and equation solving. The success rate of the proof plan was

"... In PAGE 11: ... Table1 . Some example veri cation conjectures 83%, with an average total elapsed planning time of 750 seconds, and standard deviation of 345.... ..."

### Table 2. An example of constraint solving on sets of integers

"... In PAGE 14: ... For example, fS;; S1:: 0 Set Int;; S 2 [f1g;; f1;; 2;; 3;; 4g];;S1 2 [f3g;; f1;; 2;; 3g];; S 0 S1g produces the re ned domains S 2 [f1g;; f1;; 2;; 3g]andS1 2 [f1;; 3g;; f1;; 2;; 3g]. Table2 details the process fol- lowed. The constraints are labeled as c1;;c2;;c3;;c4andc5 respectively and trans- lated to the following triples: (c1) (:: 0 ;; fSg;; fS 2 [? Set Int ;; gt; Set Int ]g) (c2) (:: 0 ;; fS1g;; fS1 2 [? Set Int ;; gt; Set Int ]g) (c3) (2;; fSg;; fS 2 [f1g;; f1;; 2;; 3;; 4g]g) (c4) (2;; fS1g;; fS1 2 [f3g;; f1;; 2;; 3g]g) (c5) ( 0 ;; fS;; S1g;; fS 2 [? Set Int ;;max(S1)];;S1 2 [min(S);; gt; Set Int ]g) Table 2.... ..."

### Table 1. An example of constraint solving on lt; C

1998

"... In PAGE 11: ...3. Table1 shows the process followed by the system. Columns 2, 3 and 4 show the current value of the state hC; Sd; SIi.... ..."

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