### Table 1: Description of our runs. The step in temperatures was 0:1 in all runs. (*) means that for the L = 48 lattice in the measure of the staggered overlap we only have simulated 100 samples. In the next paragraphs we will describe the observables that we have measured in our numerical simulations. We have measured the global (m) and staggered (ms) magnetizations : m 1 L2 Xi Si ; ms 2 L2 Xi1 Si1 ; (7) where the sum runs over all the lattice sites (i) and over a sub-lattice (i1) respectively. In order to calculate spin glass quantities we have simulated two replicas and in parallel with the same disorder. The overlaps between the replicas, global (q) and staggered (qs), are:

"... In PAGE 5: ... In all the runs we have used an annealing procedure from higher temperatures to the lower ones in order to thermalize the system. In Table1 we report the statistics we have used. We have performed in the annealing procedure for all the temperatures the same number of thermalization steps (NT), that we have written in Table 1.... In PAGE 6: ...(i1) respectively. The magnetic global and staggered susceptibilities (without the factor) are de ned as: L2 hm2i ? hjmji 2 ; s L2 2 hm2 si ? hjmsji 2 : (9) The spin glass or overlap susceptibilities ( q, s q) are de ned by: q L2 hq2i ? hjqji 2 ; s q L2 2 hq2 si ? hjqsji 2 : (10) We have measured also the Binder cumulants of the magnetization: global gm and staggered gs m, gm 12 2 6 43 ? hm4i hm2i 2 3 7 5 ; gs m 12 2 6 43 ? hm4 si hm2 si 2 3 7 5 (11) and Binder parameters of the overlaps: global gq and staggered gs q, gq 1 2 2 6 43 ? hq4i hq2i 2 3 7 5 ; gs q 12 2 6 43 ? hq4 si hq2 si 2 3 7 5 : (12) Finally the speci c heat is de ned by: CV 1 L2 hH2i ? hHi2 : (13) In order to decide a safe thermalization time, NT , (written in Table1 ) we have used the method proposed by Bhatt and Young [5] which consists in running, at the lowest temperature, with an ordered (all spins up) and high temperature initial con gurations and monitoring the behavior of the susceptibilities (in our case the non connected overlap susceptibility: L2hq2i) with the Monte Carlo time. When the two curves reach the same plateau we can say that the system has thermalized.... ..."

### Table 2. Statistics for SI.

### Table 2. Statistics for SI.

### Table 1: Totally Parallel Addition at Digit Position i

1997

"... In PAGE 4: ... The computation of the intermediate values si and ci has to be done in such a way that the subsequently addition of the intermediate sum digit si with the intermediate carry ci?1 do not generate a carry out. One truth table that describes this approach is given in the Table1 . Based on... In PAGE 5: ...+ ? x?, and unique representation for 0, i.e., the combination x? = x+ = 1 is not allowed. For each digit position i; i = 0; 1; : : : ; n ? 1 we have to compute the digits ci and si as specified in Table1 . In order to achieve this calculation we produce in the first layer of the network the following quantities: [1]+ = i = sgnfx+ i ? x? i + y+ i ? y? i ? 1g; [2]+ = i = sgnfx+ i ? x? i + y+ i ? y? i ? 2g (10) [?1]? = i = sgnfx? i ? x+ i + y? i ? y+ i ? 1g; [?2]? = i = sgnfx? i ? x+ i + y? i ? y+ i ? 2g (11) BNNi?1 = sgnf?x? i?1 ? y? i?1g; BNNi?1 = sgnfx? i?1 + y? i?1 ? 1g (12) With these signals we can compute in the second layer the digits ci and si as follow: c+ i = sgnfx+ i ? x? i + y+ i ? y? i + BNNi?1 ? 2g (13) c? i = sgnfx? i ? x+ i + y? i ? y+ i + BNNi?1 ? 2g (14) s+ i = sgnf[1]+ = i ? [2]+ = i + [?1]? = i ? [?2]? = i + BNNi?1 ? 2g (15) s? i = sgnf[1]+ = i ? [2]+ = i + [?1]? = i ? [?2]? = i + BNNi?1 ? 2g (16) Finally, in the third layer of the network we compute the sum digit zi as: z+ i = sgnfs+ i ? s? i + c+ i?1 ? c? i?1 ? 1g; z? i = sgnfs? i ? s+ i + c? i?1 ? c+ i?1 ? 1g (17) The detailed proofs of the previous equations can be found in [4].... ..."

Cited by 7

### Table 2: Continuous Integrator from smin to smax MI(smin; smax; imax) = ( I; SI; I) I = f(i o) j i 2 R[?imax; imax] ^ o 2 R[smin; smax]g SI = R[smin; smax]

"... In PAGE 23: ...b w = b w1 b w2 b wk ^ 8i 2 f1; :::; k ? 1g jwij = 1 ^ j b wij = 1 ^ jwkj 1 ^ j b wkj = 1 8i 2 f1; ::; kg (wi; b wi) 2 e That is, a trace w is related to the sequence b w by ?(e ) if and only if each can be partitioned in to traces, such that all but the last is of length 1, the last is of length less than or equal to 1, and corresponding partitions are related by e . For example, consider an input trace w 2 I to the integrator de ned in Table2 . Since elements of I are ordered pairs, we can view w = iko as the parallel combination of two traces, the integrand, i, and the integral, o.... ..."

### Table 6: Execution time (S): Sicstus vs. amp;-Prolog on Balance

"... In PAGE 14: ... Benchmarks have been parallelized so that only unconditional parallelism (both strict and non-strict { \si quot;/\nsi quot;) is exploited. Table6 provides similar results for the Sequent Balance. Timings from the execution of SICStus0.... ..."

### Table 10. Comparison of cluster identification CPU time: FSM sequential implementation vs. the FSM parallel implementation on the FORD map.

1997

"... In PAGE 18: ... Future research concerning the combination process is discussed in the next and final section. The SI and PI results for the 31 map classes of the FORD map, which are detailed in Table10 (see Section 6 in the Appendix) and Figure 12, demonstrate a constant speedup... ..."

Cited by 3

### Table 1. Dual elements in CC-SI and RD-SI

2000

"... In PAGE 5: ... After CC-SI encoding, CG B7 CB, CG BR CB; after RD-SI decoding, CM CG BC BP AN B7B4AQ A3 B7AD A3 B5CH BC , AN BR CH BC .For exact duality to hold, it is necessary that AQ A3 B7 AD A3 BPBD;this equation is satisfied only for CP BP CP BR BP AR BE BPB4AR BE B7 C6 BC B5BM (15) Table1 lists the dual elements when (15) is satisfied.4 CC-SI with C8, C9,andC6 is the dual of RD-SI via CP BR BP C9 C8B7C9B7C6 BN AR BE BP C8 B7 C9 B7 C6BN C6 BC BP B4C8B7C9B7C6B5B4C8B7C6B5 C9 BN BW BP C6BM (16) Likewise, RD-SI with AR BE , C6 BC ,andBW (and CP BP CP BR )isthe dual of CC-SI via C8 BP C6 BC AR BE AR BE B7C6 BC A0 BWBN C9 BP AR BG AR BE B7C6 BC BN C6 BP BWBM (17) 4The RD-SI noise CI BC does not correspond directly to a CC-SI entity but forms part of the CC-SI state CB.... ..."

Cited by 8

### Table 1. Dual elements in CC-SI and RD-SI

"... In PAGE 5: ... After CC-SI encoding, CG B7 CB, CG BR CB; after RD-SI decoding, CM CGBC BP AN B7B4AQA3B7ADA3B5CHBC, AN BR CHBC.For exact duality to hold, it is necessary that AQA3 B7 ADA3 BPBD;this equation is satisfied only for CP BP CPBR BP ARBEBPB4ARBE B7 C6BCB5BM (15) Table1 lists the dual elements when (15) is satisfied.4 CC-SI with C8, C9,andC6 is the dual of RD-SI via CPBR BP C9 C8B7C9B7C6 BN ARBE BP C8 B7 C9 B7 C6BN C6BC BP B4C8B7C9B7C6B5B4C8B7C6B5 C9 BN BW BP C6BM (16) Likewise, RD-SI with ARBE, C6BC,andBW (and CP BP CPBR)isthe dual of CC-SI via C8 BP C6BCARBE ARBEB7C6BC A0 BWBN C9 BP ARBG ARBEB7C6BC BN C6 BP BWBM (17) 4The RD-SI noise CIBC does not correspond directly to a CC-SI entity but forms part of the CC-SI state CB.... ..."