"... In PAGE 15: ...989]. However, application of MDL to other types of model, i.e. feedforward neural networks has not been successful, due to difficulty in developing optimal encoding of the network in a data-dependent fashion. We conclude this section by summarizing properties of various inductive principles (see Table1 ). All inductive principles use a (given) class of approximating functions.... In PAGE 15: ... Meaningful (empirical) comparisons could be certainly helpful, but are not readily available, mainly because each inductive approach comes with its own set of assumptions and terminology. In this respect, the comparison in Table1 may be helpful for developing future comparisons. Each inductive principle, when reasonably applied, usually yields an acceptable solution for practical applications.... ..."

"... In PAGE 7: ... We compared such an implementation with a state-of-the-art BMC implementation that is based on a plain CNF translation of the unfolded formulas [1]. Both implementations utilize the same 0 20 40 60 80 100 2 4 6 8 10 12 14 Relative reduction of vertices compared to first frame in % Time frame Simplifcation of transition relation d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13 d14 d15 d16 d17 d18 d19 d20 d21 Figure 4: Simplification of the transition relation for the industrial benchmarks used in Table1 . At each time frame the size of the transition relation in terms of AND vertices is compared with the... In PAGE 7: ... core SAT solver [4] which makes them to some degree compara- ble. Table1 provides an overview of the results on the set of in- dustrial property checking benchmarks. The table lists the number of state variables and the lengths of the shortest counter-example in columns 2 and 3.... ..."

"... In PAGE 9: ... (EVAL C1 = EVAL C2) == gt; (EVAL (for (LVAR i) (VAL 1) (VAL m) C1) = EVAL (for (LVAR i) (VAL 1) (VAL m) C2)) for LEMMA1 is a loop termination theorem, for LEMMA2 says that assigning the index variable before a for loop has no e ect, for LEMMA3 unrolls a loop from the bottom, for LEMMA4 unrolls a loop from the top and for LEMMA5 is a congruence theorem. To reason about individual commands in a for command body we have de ned the operator FOR as an inductive de nition (See Table2 ) and proved an additional lemma: FOR_LEMMA = |- !m i C. EVAL (for (LVAR i) (VAL 1) (VAL m) C) = FOR (LVAR i) (VAL 1) (VAL m) (EVAL C) To reason about individual commands inside the scope of a local declaration we... ..."

"... In PAGE 10: ... One possible reason is that the recipes generated by the decision tree induction approach were sorted in descending order of the con dence levels of the corresponding induction rules, which may help to have an order close the correct order. Table2 shows the accuracy of classi cation of 456 testing emails using the recipes generated by the naive approach and decision tree induction approach. The overall accuracy has decreased,... ..."

"... In PAGE 13: ... There- fore, we tested the bene t in three domains, the - -proofs from the analysis textbook [1], the residue class domain, and inductive proofs. Table1 gives sam- ple problems from all three domains and the failure-reasoning they require. The numbered colons denote (i) case split introduction, (ii) unblock constraint solv- ing, (iii) unblock by lemma speculation, (iv) analyze variable dependencies.... In PAGE 13: ... Note that x ! a and x ! a+ denote the left-hand limit and the right-hand limit, respectively. The relevance of failure reasoning is not only demonstrated by Table1 . Its gures alone are underestimating because many similar problems can be formu- lated.... In PAGE 14: ...xperiments). Some representative examples occur in Table 1. Inductive Proofs So far, we did not apply Multi to inductive proofs. The induc- tive theorems in Table1 are taken from [9], which describes failure reasoning by so-called critics in the proof planner CLaM. Since the critics employed in CLaM are a special case bound to a particular method (see related work in section 7), our general failure reasoning rules for case-split introduction and lemma spec- ulation are applicable for inductive proofs as well.... ..."

"... In PAGE 5: ...1. Table2 shows the runtime for a few industrial instances. We can see that the induction-based learning can be very powerful, espe- cially for hard UNSAT cases.... ..."

"... In PAGE 5: ...1. Table2 shows the runtime for a few industrial instances. We can see that the induction-based learning can be very powerful, espe- cially for hard UNSAT cases.... ..."

"... In PAGE 7: ... By the middle of the war, the manpower pool was thin, particularly among young men and new registrants were in high demand and at significant risk of induction. Nearly a million men were drafted in 1941, followed by more than 3 million men in 1942 [ Table1 ]. Conscription continued to fill manpower needs in whole or in part through 1946, when more than 180,00 men were drafted in the last calls under the Selective Service Act of 1940.... In PAGE 7: ... Since quotas were issued from the federal level to each state based on the stock of residents 3 Among men ages 19-25 who were in a deferred classification in August of 1945, 54% held deferments for physical or mental unfitness (IV-F) as compared to 43% who were deferred for occupational reasons (II) [Table 94, Selective Service and Victory]. Among those IV-F, a relatively large fraction were deferred for reasons of mental deficiency (roughly 10%) or mental health conditions ( roughly 30%), with the remainder being deferred for a wide range of physical health limitations [ Table1... In PAGE 23: ... We would expect the Korean War experience to have fewer adverse effects on educational attainment, particularly at the secondary level, than service in World War II. At the same time, these men were eligible for generous educational subsidies and many, did, in fact use them (see Appendix Table1 ). Thus, it seems likely that the Korean War service had a positive effect on post secondary attainment for these cohorts.... In PAGE 26: ...arentheses. Some of these men might have continued their education even with the G.I. Bill. Thus these figures would seem to represent upper bounds on the effect of service on educational attainment.14 (As Appendix Table1 shows, later cohorts in the Korean conflict were more likely to use the G.... In PAGE 48: ...Appendix Table1 : Educational attainment and use of G.I.... ..."