### Table 2. The 25 hardest tests for poSAT

2006

"... In PAGE 9: ... The current implementation does not decompose partial order constraints to their SCC-components (Lemma 3). The experimental results indicate that the implementation would not benefit from that: (a) Most of the tests are very fast without this decomposition; and (b) It is typical for hard cases of LPO termina- tion (see Table2 ) to have a large strongly connected component including the majority of the symbols. For experimentation we have taken all 751 term rewrite systems from the Termination Problem Data Base [18] which do not specify a theory or a strat- egy .... In PAGE 10: ... In contrast, poSAT demonstrates similar performance for both LPO and quasi-LPO. Table2 presents a detailed analysis for the 25 most challenging examples for poSAT chosen by maximum total time for strict- and quasi- LPO analysis. The two parts of the table present the respective results for strict- and quasi- LPO termination analyses.... In PAGE 10: ... The columns labeled poSAT and TTT indicate run times (in seconds) for the poSAT and TTT solvers. All of the tests in Table2 are not strict- nor quasi-LPO terminating. This is not surprising for the 25 hardest tests, as proving unsatisfiability is typi- cally harder than finding a solution for a satisfiable formula.... In PAGE 11: ... The differ- ence is due to the fact that in the case of poSAT the generation of a partial order formula never introduces trivial sub-formula ( true or false ), these are evaluated on-the-fly. Another observation based on the results of Table2 is that the partial or- der constraints derived from the tests typically have domain graphs with large strongly-connected components. Almost every test in the table has a core com- ponent including the majority of the symbols.... In PAGE 11: ... Therefore, it is unlikely that the performance of poSAT for the presented tests can be improved by using the SCC-based decomposition of the formula. As Table2 shows, the maximum CNF instance solved in our tests includes 12827 propositional variables and 18205 CNF clauses. This is well below the ca- pacity limits of MiniSat, which is reported to handle benchmarks with hundreds... In PAGE 12: ... However, in view of Lemma 3 we may assume that we are testing satisfiability for partial order constraints which have strongly connected domain graphs. Moreover, as indicated by our experimental evaluation (see Table2 ), the domain graphs for some of the more challenging examples have... ..."

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### Table 2. Rewriting Rules for the Application of Updates

2006

"... In PAGE 46: ... Program Entities: \program t There is a large number of di erent kinds of program entities that can be rep- resented using program schema variables. Table2 contains the most important ones of the types that existed when this chapter was written, for a list that is more complete and up-to-date we refer to the homepage6 accompanying [13]. In KeY, the name of a program schema variable always has to start with a hash (like #se), mostly for the purpose of parsing schematic programs.... In PAGE 47: ...Schema Variables Table2 . A selection of the kinds of schema variables for program entities Expressions Expression Arbitrary JAVA expressions SimpleExpression Side-e ect free expressions: 1.... In PAGE 55: ... Modi ers for Schema Variables Modi er Applicable to rigid \term A \formula Terms or formulae that can syntactically be identi ed as rigid strict \term A Terms of type A (and not of proper subtypes of A) list \program t Sequences of program entities. The type t can be any of the types of Table2 apart from Label, Type and NonPrimitiveType. Sequences of expressions can be used to represent arguments of method invocations.... In PAGE 103: ... The non-recursive op- erator enables us to separate the syntactic propagation of updates to subterms and subformulas from the syntactic evaluation of updates. The actual syntactic application of updates is described by the rewriting rules in Table2 .... ..."

### Table 2. Basic priority rewrite system.

1996

"... In PAGE 11: ... In our PRS, these rules will be given the highest priority. For the kernel objects of Fig 1, this gives the rewrite rules (1-3) in Table2 . Because the invocation of the continuation K will discard the current continuation, for simplicity we chose this one to be IK, the identity continuation.... In PAGE 12: ... We need three di erent priorities to cope with the previous requirements. The Table2 uses labels l; m; h to suggest lower, medium and higher priority respectively. We now give a simple example of the rewriting process where the current program con guration contains an object o that has a method m, which... In PAGE 15: ... We distinguish two subsets in C: S is the set of objects representing slot names and O = CnS contains the remaining objects. F should rst contain the prede ned functors used as the set F0 in Table2 . For all other objects o 2 O and s 2 S not represented in F0, F can be completed with functors such as o1, o2, : : : and s1, s2, : : : respectively.... In PAGE 17: ... Consider the program con guration depicted in Figure 2. The following table gives the rewrite rules added to the one of Table2 by (we omit certain rules which are not used for the moment). h : eval(o; ;[];K) ! eval(K; ;[mo];ik) (19) h : eval(mo; ;[];K) ! eval(K; ;[BMO];ik) (20) h : eval(lmo; ;[];K) ! eval(K; ;[BMO];ik) (21) h : blf( ;mo) ! lmo (22) h : blf( ;lmo) ! BA (23) The Table 4 illustrates the reduction process for the lookup phase assuming a message (o apos;s a) and current continuation k.... In PAGE 26: ... Terms of the form blf(s; o) are I- reduced either to a term m representing a method if o represents an object in C, or to the term doesNotUnderstand, and in both cases no further I-reduction is possible. Terms of the form eval(o; s; a; k) can match many rules in (C), but in the most general case, they are I-reduced using rule (14) ( Table2 ) to eval(o; ; []; c( ; [s; o]; c( ; [o; a; k]; IK))). Since o is an object in the kernel, by de nition its meta-object is BMO and by Def.... In PAGE 26: ... Proof: The proof proceeds by induction on the number of objects in the program con guration C. Basic case: applied to the kernel produces a PRS whose rewrite rules are rules (4) to (14) in Table2 augmented with meta-object fetching rules and rules whose... ..."

Cited by 7

### Table 3 Rewrite rules according to a program

1995

"... In PAGE 5: ...An execution state maps each mail address to a process { specifying the behavior of the corresponding Actor { and a mail queue (or ; if the mail address in not yet used). Each rewrite rule in Table3 changes the mapping for one or two mail addresses. The rewrite rules expect most processes to be of the speci c form p p1 + p2 jj p3, where p is an action.... In PAGE 6: ... The execution of a program according to De nition 2 may unintentionally stop because of errors. For example, no rewrite rule in Table3 deals with sending messages to variables or process names; and an Actor cannot proceed if its process does not specify how to handle the next message in the mail queue. Nonetheless, this formalism is the basis for a better formalism introduced in the next section.... In PAGE 12: ... Theorem 1 Let be a program conforming to a type system , and h 0; : : : ; i; : : :i an execution according to . There exists a rewrite rule (in Table3 ) applicable to i (for 0 i) if there exists an (a 7! p; hr1; : : : ; rki) in i with 1 k. Proof.... ..."

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### Table 3. Five hard examples: SAT solving increases termination proving power

2006

"... In PAGE 13: ... However, for the SAT-based analyses, the overall runtimes are still extremely fast in comparison to the non-SAT-based configurations. Table3 highlights 5 examples which could not be solved by any tool in the termination competition 2005, whereas the SAT-based configuration proves ter- mination for all 5 in a total of 4.3 seconds.... ..."

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### Table 1. Results on Termination Benchmarks

2006

"... In PAGE 5: ...art is written in Objective CAML (CAML 3.0.9). We have tested the perfor- mance of the system on a number of examples, including benchmarks for LP in Termination Problems Database [1] (Tables 1(b), 2(a), 2(b), 3(a) and 3(b)), and examples collected from other sources ( Table1 (a))2. The domain of all variables in the generated Diophantine constraints is xed to the set D = f0; 1; 2g.... In PAGE 6: ... The claim could come from the fact that the approach based on polynomial interpretations used in Polytool can be considered as a generalization of the semi-linear norm based approach used in Hasta La Vista. The results in Table1 (a) show that there is a class of examples (e.g.... In PAGE 7: ... Examples from TerminWeb and Taboch Example 6. Consider the program normal with the query pattern norm(g,f) in Table1 (b): norm(F; N) : rewrite(F;F1); norm(F1; N): norm(a;a): rewrite(op(op(A;B); C); op(A; op(B;C))): rewrite(op(A;op(B;C)); op(A; L)) : rewrite(op(B;C); L): For this example, both Polytool and Hasta La Vista produce nonlinear Diophan- tine constraints, but only Polytool succeeds. If we take the constraints generated by Hasta La Vista as an input for CiME 2.... ..."

Cited by 2

### Table 2: Term rewriting system, used for the elimination of the pre x operator.

1996

"... In PAGE 9: ... Note that kk and kk are allowed (that is, k and k without communication). The complete pre x expression can be rewritten using the term rewriting system in Table2 . This term rewriting system is basically an implementation of the axioms for the free (left) merge, encapsu- lation, hiding, renaming, and process pre xing.... In PAGE 9: ... After interpretation, atomic elements will always end up in the set A . Because the rewrite rules from Table2 remain valid when we substitute elements of this set for , this does not lead to inconsistencies. In rewrite rules M3g, M5 and PP7, some reservation has been made in case that the variable d should occur free in y or z.... In PAGE 13: ...ariable later. Next, the function bpa performs one rewrite step on Pre;Pre2. The pre x operator is no longer the main operator and the result Pre apos; is given back to trans-expr. The function bpa performs one rewrite step from the term rewriting system in Table2 . The function normalize is nothing more than a sequence of bpa applications, as long as the result is not yet a pCRL expression.... In PAGE 13: ...xpression. For example, see the equation that implements the rule M3c. atomic(Pre1) = true bpa(Globvar; Locvar; Pre1 kk Pre2; Newvar) = lt; Pre1 : Pre2; Newvar gt; [bpa-39] If Pre1 is an atomic element, then Pre1 kk Pre2 is rewritten into Pre1 Pre2. The variable Newvar in the equations above has to do with the reservation that has been made towards rules M3g, M5 and PP7 in Table2... In PAGE 14: ... It will only be executed if none of the other equations can be executed. Because the free left-merge is the main operator, this will be the case if none of the rules M2a{M6 from Table2 apply. In the condition of equation [bpa-54], bpa performs one rewrite step on Pre1, resulting in Pre apos;.... In PAGE 14: ... Next, Pre apos; kk Pre2 is returned for further rewriting. Finally, we present the equation modeling the most important rewrite rule from Table2 : rule PP4 that eliminates the early read and the pre x operator: bpa(Globvar; Locvar; er(Trm; Cns); Pre; Newvar) [bpa-58] = lt; put-sums(Cns; Trm; Cns; Pre; r); Newvar gt; The actual transformation is done in the function put-sums. Trms = transform(Cns) put-sums(N; Trm; Cns; Pre; N0) = sum(N; N0(Trm; Trms) : Pre)... ..."

Cited by 3

### Table 1: Rewrite Rules for Combinational PE

2004

"... In PAGE 1: ... Any of these techniques are potentially capable of PE of combinational circuits, though for clarity and generality, we have opted for a simple approach based upon term rewriting. Table1 shows a simple set of rewrite rules that are sufficient to implement (sub-optimal) combinational PE in time and space that is linear with respect to the original circuit. The software analogue of a combinational circuit, from the point of view of PE, would be a program consisting only of assignment statements, if-then and if-then-else constructs, but strictly no loops.... ..."

Cited by 1

### Table 2: Term Rewriting Process

1998

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### Table 1 Programmed Search vs. Classical Constraint Solving for OPL and T OY

"... In PAGE 9: ... We have obtained running times for these parameters as the average of four runs. Table1 shows these results and has several columns: The column Size represent the size of the problem in terms of the number of months of the timetable. The column TO stands for the labeling strategy equally specified in both systems that assigns values to variables using their textual (static) order, and its possible values in ascending order.... In PAGE 10: ... The columns Posting and Propagation and Labeling show the time for these processes, whereas the last column shows the total time. The cells in the table follow the same data format as Table1 . Note that there are times shown as 0.... In PAGE 11: ... OY is less than before. The gain of the labeling is, in the average, about 23.5, with small deviations. The last column shows the same data as Table1 and is kept for reference. Next, Table 3 shows the impact of propagation alone over the total computation time for both systems.... ..."