我的台灣, 看見心靈的故鄉 =2009林磐聳藝術與設計展

During the year 2006, the work at the Excimer Laser Laboratory was mainly focused on the development and applications of a laser-driven plasma source emitting in the extreme ultraviolet (EUV) spectrum, on laser irradiation of textiles, on the design and test of optical systems, and on the know-how transfer of the ENEA-patented laser beam homogeniser. Since early ’90, we have developed an Laser-Plasma source emitting in the spectral range EUV/soft X-rays which has been applied to atomic spectroscopy, radiobiology, in vivo microscopy and photonics. More recently the source has been addressed for use in EUV lithography, integrating it in the EGERIA (Extreme-ultraviolet-radiation Generation for Experimental Research and Industrial Applications) facility, which was described in previous Annual Reports. The development of a 14nm Exposure Tool is the main objective of a National project (FIRB-EUVL) ending 2007, while an efficient debris mitigation system has been developed within the frame of the European Integrated Project More Moore.

Annual Report for 2006

59.4 T h e C u t Elimination Theorem for Second Order Logic 171 (respectively, v . ) . In particular, (a) guarantees t h a t *"-PIND or c ~ ' ~ I N D , and C ~ ' ~ C R inferences are still valid after the substitution of V for a. However, P ( V ) may fail t o be a proof in t h a t there may be initial sequents of P ( V ) of the form where V={Z')A(Z'). However, sequents of this form are easy t o prove without free cuts. S o we merely tack onto P ( V ) free cut free proofs of these initial sequents and thus obtain a proof Q of I'(V)+A(V). Q is not necessarily free cu t free, as Q may contain free cu ts with principal formula A(3). B u t since V is a first order abstract, every free cut inference in Q has a first order principal formula. Hence, by (b), there is a free cut free proof of I'(V)--+A(V). Theorem 10: (Cut Elimination Theorem). Let P be a firproof o r a VTproof. Then there is a proof P * in the same theory such tha t P * has the same endsequent as P and there are no free cu ts in P * . Furthermore, each principal formula of an induction inference in P * is a substitution instance of a principal formula of an induction inference in P and each principal abstract of a comprehension inference in P * is a substitution instance of a principal abstract of a comprehension inference in P. Hence, for all i 2 0 , if P is a fii(or vi-) proof then so is P*. Proofi We shall modify Takeuti 's exposition on pages 22-29, 112, 143-144 of [28]. T h e reader should have [28] available as he reads the proof. Following [28], we define the grade of a formula A t o be the number of logical symbols in A . T h e level of A is the number of second order quantifiers in A . A mix inference with principal formula A is of the form where l l* and A * are obtained from rI and A by removing all occurrences of A . A mix inference is free iff all of the occurrences of A in A and II are free. Since a mix inference and a cut inference are so similar, i t suffices t o prove Theorem 10 for proofs which use mix inferences instead of cu t inferences. Suppose P is a proof whose last inference is a mix with principal formula A as shown above. Define the distance of a sequent in P t o be the number of inferences separating i t from the endsequent of P. T h e right rank of P is defined t o be the maximum distance of a sequent containing a direct ancestor of an occurrence of A in the cedent ll. T h e left rank of P is the maximum distance of a sequent containing a direct ancestor of an occurrence of A in the cedent 172 Foundations of Second Order Bounded Arithmetic A . T h e rank of P is the sum of the right rank and left rank. I t suffices t o consider P with a single mix inference as the last inference. T h e proof is by ordinal induction on where level(P) and grade(P) are the level and grade of the principal formula of t he final mix of P, and w is the first infinite ordinal. T h u s it suffices t o show tha t if P is a proof with no free mixes except for the final inference of P and if Theorem 10 is satisfied for all proofs P' with ord(P' )<ord(P) , then P satisfies Theorem 10. W e modify the proof of Lemma 5.4 of Takeut i [28]: Case (1): rank(P)=2. Cases (1 .I)-(1.5. ii): Similar t o pages 2 4 2 7 of [28]. Case (1.5.iii): Suppose A=(V+)B(q5) and the last inferences of P are where V is a ~ d ~ ~ a b s t r a c t , and since rank(P)=f the indicated occurrences of (V+)B(+) are the only ones. By Lemma 9(c), we can obtain a free mix free proof of rA,B( V) from the free mix free proof of I'-+A,B(cr). T h u s we have a proof Q such t h a t the only free mix in Q is i ts last inference: where II# and A# are II and A minus all occurrences of B(V). By the induction hypothesis, there is a free mix free proof Q* of ~,II#-A#,A since ord(Q)<ord(P). By adding weak inferences t o the end of Q* we obtain the desired proof P* . Case (1.5. i~) : Suppose A=(34)B(4). This case is very similar t o Case (1.5.iii). Case (2): rank(P) >2. Case (2.1): T h e right rank of P is >l. Cases (2.1.1)-(2.1.9. ii): Similar t o 1281. T h e C u t Elimination Theorem for Second Order Logic 173 Case (2.1.J.iii): Suppose A=(V4)B(4), V is a ~:p~-abstract and the last inferences of P are: where now A and II, b u t not r , may contain occurrences of (Vc$)B(4). T h e cedents TI* and A* are II and A minus all occurrences of (V+)B(c$). Modify the end of P t o obtain a proof P1 which ends as T h e right rank of P1 is one less than the right rank of P so by the induction hypothesis there is a free mix free proof Pg of the endsequent of P I . Now consider the proof which ends T h e right rank of this is one, so by the induction hypothesis and some exchanges and contractions we obtain a free mix free proof of r , n * + A*,A. T h e rest of the cases are similar. Q.E.D. 9.5. C:yb-Defined Functions and A:yb-Defined Predicates. T h e second order theories of Bounded Arithmetic are in many respects analogous t o the first order theories s~~ and T;. One of the most fundamental properties of second order Bounded Arithmetic is t ha t new function and predicate symbols may be introduced into the language of Bounded Arithmetic; under certain conditions, these new function and predicate symbols may be used freely in the principal formulae of induction axioms and comprehension axioms. 174 Foundations of Second Order Bounded Arithmetic Definition: Let R be a second order theory of Bounded Arithmetic. Suppose A(2,y) is a formula with all free variables indicated and tha t Then we say t h a t R can define the function j such t h a t N t= ('d$)A(Z',j($)). T h e ~ t ) ~ d e f i n e d function symbols and the Atpb-defined predicate symbols play the same role in the second order theories of Bounded Arithmetic as the C:-defined function symbols and the A/-defined predicate symbols d o in the first order theories Si and T;'. In particular, the analogues of Theorems 2.2, 2.3 and 2.4 hold for second order Bounded Arithmetic. Definition: Let 7 and ji be new function and predicate symbols. T h e sets C,'tb(7,ji) and ll,'*b(7,j?) are sets of bounded formulae in the language of second order Bounded Arithmetic plus the symbols f' and ji. These sets are defined by counting alternations of second order quantifiers ignoring the first order, bounded quantifiers. Theorem 11: Let R be a second order theory of Bounded Arithmetic. Suppose R can C;lb-define each of the functions f' and can define each of the predicates ji. Let R * be the theory obtained from R by adjoining the new symbols 7 and j3 and their defining axioms. Then, if i > O and B is a C;'lb(7,j?)(or a Il,'lb@,j?)-) formula, then there is a formula B*Ec;'~~ (or II,'pb, respectively) such t h a t R * t B** B. T h e proof of Theorem 11 is similar t o the proofs of Theorems 2.2 and 2.4. Definition: Let R be a theory of Bounded Arithmetic and let 7 be a vector of defined function symbols of R and j3 be a vector of defined predicate symbols. Then ~(7, j i ) is the conservative extension of R obtained by enlarging the language t o include 7 and 3 and including the defining axioms for these symbols. Corollary 12: ( i 2 1): (a) Let f' be a vector of ~ t * ~ d e f i n e d function symbols of U; (respectively, v;) and let ji be a vector of At*b-defined predicate symbol-. of u;' (respectively, v;' ). Then U;@,j3) (respectively, vfl,j?) ) has as theorems the E : ~ ~ ~ , ~ ~ ) P I N D axioms (respectively, the c,"~(~!,~~)-IND axioms). (b) Let 7 be a vector of C:rb-defined function symbols of fi; (respectively, F;' ) and let ji be a vector of A:.b-defined predicate symbols of 6; (respectively, ) Then fi;'(],j?) (respectively, ?i@,j?)) has as theorems the E:I*(~!,~I)-PIND axioms (respectively, the E,'P~@,~~)-IND axioms). (c) Let f' be a vector of C:lb-defined function symbols of 8; (respectively, 8;:) and let j? be a vector of A:rb-defined predicate symbols of 8; (respectively, ) Then 8i(],jI) s9.5 zipb-Defined Functions and airb-Defined Predicates 175 (respectively, ) has as theorems the E:V~-PIND v , a ) axioms (respectively, the c~'~~-IND (7,jJ) axioms) and the i :9b(7,$)-~~ axioms. Corollary 12 tells us t h a t C:~~-defined function symbols and a:1b-defined predicate symbols may be used freely in the principal formulae of induction inferences. Furthermore, if we are working in the theory 82 or 82 we may use such function and predicate symbols freely in principal abstracts of comprehension inferences. T h e next two theorems give an application of ~ : ~ ~ c o m ~ r e h e n s i o n t show t h a t 82 and 8: can define the iteration of defined predicates. I t is an open question whether these theorems hold for the theories U; and v;. Theorem 18: Let ~(a,i?,?) and ~ ( a , b,t,a:,q) be formulae of 8:, where a: is a unary predicate variable. Let t (b , t ) be a term which contains only the free variables b and t. Then the predicate ~ ( a , b,i?,;) which satisfies if b=O and a < t ( b , t ) ~ ( a , b , t , < ) if a > t ( b , t ) B ( ~ , b , t , {z}K(z, [ ;b ~, t , ; j ) ,3 otherwise Proofi T h e idea, of course, is t o define K ( a , b , t , 3 by induction on the length of b . Let B*(a, b , t , c u ~ , ~ ) be the formula and let D(u ,4) be the formula I t is easy t o see t h a t

Bounded Arithmetic と計算量の根本問題

Thank you very much for downloading steps to an ecology of mind collected essays in anthropology psychiatry evolution and epistemology. Maybe you have knowledge that, people have search hundreds times for their chosen readings like this steps to an ecology of mind collected essays in anthropology psychiatry evolution and epistemology, but end up in malicious downloads. Rather than enjoying a good book with a cup of tea in the afternoon, instead they juggled with some infectious bugs inside their desktop computer.

Steps To An Ecology Of Mind Collected Essays In Anthropology Psychiatry Evolution And Epistemology

Internet of things for smart factories in industry 4.0, a review

In the context of Industry 4.0, smart factories use advanced sensing and data analytic technologies to understand and monitor the manufacturing processes. To enhance production efficiency and reliability, statistical Artificial Intelligence (AI) technologies such as machine learning and data mining are used to detect and predict potential anomalies within manufacturing processes. However, due to the heterogeneous nature of industrial data, sometimes the knowledge extracted from industrial data is presented in a complex structure. This brings the semantic gap issue which stands for the lack of interoperability among different manufacturing systems. Furthermore, as the Cyber-Physical Systems (CPS) are becoming more knowledge-intensive, uniform knowledge representation of physical resources and real-time reasoning capabilities for analytic tasks are needed to automate the decision-making processes for these systems. These requirements highlight the potential of using symbolic AI for predictive maintenance. To automate and facilitate predictive analytics in Industry 4.0, in this paper, we present a novel Knowledge-based System for Predictive Maintenance in Industry 4.0 (KSPMI). KSPMI is developed based on a novel hybrid approach that leverages both statistical and symbolic AI technologies. The hybrid approach involves using statistical AI technologies such as machine learning and chronicle mining (a special type of sequential pattern mining approach) to extract machine degradation models from industrial data. On the other hand, symbolic AI technologies, especially domain ontologies and logic rules, will use the extracted chronicle patterns to query and reason on system input data with rich domain and contextual knowledge. This hybrid approach uses Semantic Web Rule Language (SWRL) rules generated from chronicle patterns together with domain ontologies to perform ontology reasoning, which enables the automatic detection of machinery anomalies and the prediction of future events’ occurrence. KSPMI is evaluated and tested on both real-world and synthetic data sets.

KSPMI: A Knowledge-based System for Predictive Maintenance in Industry 4.0

/pdf/logical-approaches-to-computational-barriers-1jpf4z84fs.pdf

Logical Approaches to Computational Barriers

We determine the exact bounds for the length of an arbitrary reduction sequence of a term in the typed λ-calculus with β-, ξ- and η-conversion. There will be two essentially different classifications, one depending on the height and the degree of the term and the other depending on the length and the degree of the term.

Exact Bounds for Lengths of Reductions in Typed $\lambda$-Calculus

A natural term rewriting framework for the Bellantoni Cook schemata of predicative recursion, which yields a canonical definition of the polynomial time computable functions, is introduced. In terms of an exponential function both, an upper bound and a lower bound are proved for the resulting derivation lengths of the functions in question. It is proved that any natural reduction strategy yields an algorithm which runs in exponential time. We give an example in which this estimate is tight. It is proved that the resulting derivation lengths become polynomially bounded in the lengths of the inputs if the rewrite rules are only applied to terms in which the safe arguments – no restrictions are assumed for the normal arguments – consist of values, i.e. numerals, and not of names, i.e. non numeral terms. It is proved that in the latter situation any inside first reduction strategy and any head reduction strategy yield algorithms, for the function under consideration, for which the running time is bounded by an appropriate polynomial in the lengths of the input. A feasible rewrite system for predicative recursion with predicative parameter substitution is defined. It is proved that the derivation lengths of this rewrite system are polynomially bounded in the lengths of the inputs. As a corollary we reobtain Bellantoni’s result stating that predicative recursion is closed under predicative parameter recursion.

A term rewriting characterization of the polytime functions and related complexity classes

This article concerns the second-order systems U12 and V12 of bounded arithmetic, which have proof-theoretic strengths corresponding to polynomial-space and exponential-time computation. We formulate improved witnessing theorems for these two theories by using S12 as a base theory for proving the correctness of the polynomial-space or exponential-time witnessing functions. We develop the theory of nondeterministic polynomial-space computation, including Savitch's theorem, in U12. Kolodziejczyk et al. [2011] have introduced local improvement properties to characterize the provably total NP functions of these second-order theories. We show that the strengths of their local improvement principles over U12 and V12 depend primarily on the topology of the underlying graph, not the number of rounds in the local improvement games. The theory U12 proves the local improvement principle for linear graphs even without restricting to logarithmically many rounds. The local improvement principle for grid graphs with only logarithmically-many rounds is complete for the provably total NP search problems of V12. Related results are obtained for local improvement principles with one improvement round and for local improvement over rectangular grids.

Improved witnessing and local improvement principles for second-order bounded arithmetic

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