Semantic Reachability. Richard Mayr. Institut fur Informatik. Technische Universitat Munchen. Arcisstr. 21, D Munchen, Germany E. N. T. C. S.
|
|
- Kelley Gardner
- 6 years ago
- Views:
Transcription
1 URL: pges Semntic Rechbility Richrd Myr Institut fur Informtik Technische Universitt Munchen Arcisstr. 21, D Munchen, Germny e-mil: E. N. T. C. S. Elsevier Science B. V. Abstrct This pper is n pproch to combine the rechbility problem with semntic notions like bisimultion equivlence. It dels with questions of the following form: Is there rechble stte tht is bisimultion equivlent to given stte? Here we show some decidbility results for process lgebrs nd Petri nets. 1 Introduction The rechbility problem plys n importnt role in the theory of concurrent systems. The question is if given stte is rechble from the initil stte by sequence of ctions. The complexity of this problem hs been extensively studied (for exmple it is decidble nd EXPSPACE-hrd for generl Petri nets nd NP-complete for Bsic Prllel Processes (BPP) [4]). Here we generlize the rechbility problem by regrding clsses of semnticlly equivlent sttes insted of single sttes. The question is now if stte is rechble (from the initil stte) tht is member of given clss. In other words: Is it possible to rech stte tht is t lest semnticlly equivlent to given stte? It is nturl to choose strong bisimultion equivlence s semntic equivlence, s it hs become one of the most successful equivlence notions in concurrency theory. We will cll this new problem the bisimultion-rechbility problem. The question is now for which clsses of concurrent systems this problem is decidble, nd if ecient lgorithms cn be found. In section 2 we dene strong bisimultion, the bisimultion-rechbility problem nd severl process lgebrs. Section 3 contins some hrdness results for the bisimultion-rechbility problem. In sections 4, 5 nd 6 we study the bisimultion-rechbility problem for Bsic Prllel Processes (BPP), contextfree processes nd normed PA-processes. Section 7 is bout relted problem c1997 Elsevier Science B. V.
2 clled the non-bisimultion-rechbility problem nd section 8 describes some open problems. 2 Preliminries Denition 2.1 A binry reltion R over the sttes of lbelled trnsition system (LTS) is strong bisimultion (often simply clled bisimultion) i 8(s1; s2) 2 R 8 2 Act: (s1! s 0 1 ) 9s2! s 0 2 : s0 1 Rs0 2) ^ (s2! s 0 2 ) 9s 1! s 0 1 : s0 1 Rs0 2 ) There is lwys lrgest bisimultion, which is n equivlence reltion, denoted by. Two sttes s1 nd s2 re clled strongly bisimultion equivlent (or strongly bisimilr) i s1 s2. The most generl form of the bisimultion-rechbility problem cn be formulted for lbelled trnsition systems (LTS): Instnce: An LTS R with initil stte s0 nd stte s in R. Question: Is there stte s 0 nd sequence of trnsitions s.t. s0! s 0 nd s s 0? In the sequel we consider trnsition systems described by process lgebrs or Petri nets. In process lgebrs processes re described by process terms nd set of dynmic rules of the form t! t 0, mening tht process t cn perform ction nd become process t 0. Processes dene n LTS, whose nodes re mrked with process terms. The semntics of the processes is given by dening n equivlence reltion over the term lgebr. The equivlence clsses then represent the intended processes. It follows tht the dynmic rules describe unmbiguously the dynmics of the quotient lgebr, only if the chosen equivlence on the term lgebr is bisimultion. This is min reson why bisimultion equivlence is the preferred choice for process equivlence. The Bsic Process Algebr PA is simple model of innite stte concurrent systems. It hs opertors for nondeterministic choice, prllel composition nd sequentil composition. PA-processes nd Petri nets re incomprble, mening tht neither model is more expressive tht the other one. PA is not syntcticl subset of CCS [9], becuse CCS does not hve n explicit opertor for sequentil composition. However, s CCS cn simulte sequentil composition by prllel composition nd synchroniztion, PA is still weker model for concurrent systems thn CCS. The denition of PA is s follows: Assume countbly innite set of tomic ctions Act = f; b; c; : : :g nd countbly innite set of process vribles V r = fx; Y; Z; : : :g. The clss of PA expressions is dened by the following bstrct syntx E ::= j X j E j E + E j EkE j E:E A PA is dened by fmily of recursive equtions fx i := E i j 1 i ng, where the X i re distinct nd the E i re PA expressions t most contining 2
3 the vribles fx1; : : : ; X n g. We ssume tht every vrible occurrence in the E i is gurded, i.e. ppers within the scope of n ction prex, which ensures tht PA-processes generte nitely brnching trnsition grphs. This would not be true if ungurded expressions were llowed. For exmple, the process X := + kx genertes n innitely brnching trnsition grph. For every 2 Act the trnsition reltion! is the lest reltion stisfying the following inference rules: E! E! E 0 F! F 0 E! E 0 E := E) E + F! E 0 E + F! F 0 X! E0(X E! E 0 F! F 0 E! E 0 EkF! E 0 kf EkF! EkF 0 E:F! E 0 :F Alterntively, PA-processes cn be described by stte represented by term of the form G ::= j X j G1:G2 j G1kG2 nd set of dynmic rules of the form X! G whose ppliction to sttes must respect sequentil composition. This is described by the following inference rules: X! G if (X! G) 2 EkF E! E 0! E 0 kf F! F 0 E! E 0 EkF! EkF 0 E:F! E 0 :F Bsic Prllel Processes (BPP) re the subset of PA-processes without sequentil composition, while context-free processes re the subset of PA-processes without prllel composition. There is one-to-one correspondence between BPPs nd clss of lbelled Petri nets, the communiction-free nets [4]. In these nets every trnsition hs exctly one input plce with n rc lbelled by 1. The trnsltion of BPP lgebr into communiction-free net is s follows: Introduce plce for ech process vrible nd trnsition for ech trnsition rule. For rule X! Y m 1 1 k kyn mn introduce trnsition t lbelled by, n rc lbelled by 1 leding from plce X to t nd rcs lbelled by m i leding from t to plces Y i. The other direction is nlogous. 3 Generl Hrdness Results How does the computtionl complexity of the bisimultion-rechbility problem compre to the complexities of the rechbility problem nd the problem of deciding strong bisimilrity? For most models of systems the bisimultionrechbility problem is t lest s hrd s the other two problems. Lemm 3.1 For ll clsses of Petri nets tht re t lest s powerful s communiction-free nets the bisimultion-rechbility problem is t lest s hrd s the problem of deciding strong bisimilrity. Proof. The problem of deciding strong bisimilrity cn be reduced to the bisimultion rechbility problem by constructing slightly modied system s.t. the 3
4 only rechble stte tht cn possibly be bisimilr to the given stte is the initil stte itself. This construction is possible for ll models tht llow the cretion of new prllel processes. Without restriction we cn regrd the problem for dierent mrkings 1 ; 2 in the sme Petri net N. Let A = f1; : : : ; n g be the set of ctions occurring in N nd A 0 = f 0 1 ; : : : ; 0 ng new set of ctions s.t. A\A 0 = ;. Now construct new net N 0 in the following wy. For ech trnsition t in N lbelled by i introduce new plce s nd new trnsition t 0 lbelled by 0 i with n rc from t to s nd from s to t 0. Any mrking of N cn be extended to mrking of N 0 in nturl wy by dening (s) = 0 for ll plces s tht re not in N. Let L be the lbelling function tht ssigns ctions to the trnsitions in N 0. We show tht (N; 1 ) (N; 2 ) () 9(N 0 ; 1 )! (N 0 ; 0 1) (N 0 ; 2 ). ) If (N; 1 ) (N; 2 ) then f((n 0 ; ); (N 0 ; 0 )) j (N; jn ) (N; 0 jn ) ^ A 0 : X L(s)= 0 (s) = X L(s)= 0 0 (s)g is strong bisimultion. It follows directly tht (N 0 ; 1 ) (N 0 ; 2 ), becuse (N; 1 ) (N; 2 ) nd P L(s)= 0 (s) = 0 = P L(s)= 0 0 (s) for every 0 2 A 0. So we cn simply choose = nd 0 1 = 1 nd the condition is stised. ( Assume tht 9(N 0 ; 1 )! (N 0 ; 0 1 ):(N 0 ; 0 1 ) (N 0 ; 2 ). It follows tht length() = 0 nd 0 1 = 1, becuse otherwise (N 0 ; 0 1 ) could do n ction from A 0 while (N 0 ; 2 ) cn't. As the sets A nd A 0 re disjoint, it follows tht f((n; jn ); (N; 0 )) j jn (N 0 ; ) (N 0 ; 0 )g is bisimultion nd therefore (N; 1 ) (N; 2 ). This construction is possible for communiction-free nets, s well s for ll models more generl thn them. This is becuse the construction does not exceed the bounds of the model (i.e. if N is communiction-free net then N 0 is communiction-free net s well). Figure 1 illustrtes the construction for Petri net. 2 Theorem 3.2 The bisimultion-rechbility problem is undecidble for Petri nets. Proof. Directly from Lemm 3.1 nd the result from Jncr [7] tht strong bisimilrity is undecidble for generl Petri nets. 2 Unfortuntely Lemm 3.1 yields no complexity bounds for BPPs s, to our knowledge, there is no hrdness result for the problem of deciding strong bisimilrity of BPPs yet. We cn give complexity bounds by showing tht for mny models of systems the bisimultion rechbility problem is t lest s hrd s the rechbility problem. Lemm 3.3 For BPPs the bisimultion-rechbility problem is t lest s hrd s the rechbility problem. Proof. We reduce the rechbility problem to the bisimultion-rechbility 4
5 New in N 0 b b Fig. 1. How to reduce the problem of deciding strong bisimilrity to the bisimultion-rechbility problem. problem. Let N be communiction-free net, A the set of tomic ctions occurring in A, 0 the initil mrking nd mrking of N. Now construct new lrger net N 0 : for ech plce s in N dd new trnsition t s in the postset of s tht is lbelled by unique new ction s. Then dd one new plce ^s nd one new trnsition ^t to the net nd n rc from every ts to ^s nd n rc from ^s to ^t nd from ^t to ^s. Let ^t be lbelled by unique new ction ^. Figure 2 illustrtes the construction. It follows tht if 1 nd 2 re mrkings of N 0 nd 1 2 nd 1 (^s) = 0 then 1 = 2. Now we show tht 9:(N; )! (N; 0 ) () 9 0 :(N 0 ; )! 0 (N 0 ; 00 ) (N 0 ; 0 ) ) If (N; )! (N; 0 ) then (N 0 ; )! (N 0 ; 0 ). Choose 0 = nd 00 = 0 nd the condition is stised. ( As 0 (^s) = 0 it follows tht 00 = 0 nd thus 00 (^s) = 0. Therefore no trnsition t s occurs in 0. So 0 is sequence enbled by (N; ) nd the condition is stised with = 0. 2 Corollry 3.4 The bisimultion-rechbility problem for BPPs is NP-hrd. Proof. Directly from Lemm 3.3 nd the fct tht the rechbility problem for BPPs is NP-compete [4]. 2 Remrk 3.5 The construction used in Lemm 3.3 is possible for mny clsses of Petri nets, but not for T-systems nd free-choice nets. Also the newly constructed net is never normed, so Lemm 3.3 yields no hrdness result for normed BPP. 5
6 x t x x New in N 0 b y z t z z ^s ^ ^t y c d t y Fig. 2. How to reduce the rechbility problem to the bisimultion-rechbility problem. A process t is normed if every process t 0 rechble from t hs terminting computtion. The length of the shortest terminting computtion is clled the norm of t. It is denoted by [t]. A BPP is normed i in the corresponding communiction-free net N with initil mrking 0 it is impossible to rech mrking s.t. mrks trp of N. This property cn be decided in polynomil time, becuse the mximl trp cn be computed in polynomil time nd becuse in these nets tokens cn move independently. Lemm 3.6 The bisimultion-rechbility problem is NP-hrd, even for normed BPP. Proof. The proof is done by reduction of SAT to the bisimultion-rechbility problem. We illustrte the construction by n exmple (see Figure 3): The formul (x1 _ :x2 _ x3) ^ (x2 _ :x3) ^ (:x1 _ x3) is stisble i stte is rechble from fx1; x2; x3g tht is bisimilr to the stte fy1; y2; y3g. (The only such stte is fy1; y2; y3g itself). Note tht the constructed communiction-free net is nite stte nd normed. 2 In the next sections we show tht the bisimultion-rechbility problem is decidble for severl specil clsses of process lgebrs. 6
7 x 1 x 2 x 3 t f t f t f y 1 y 2 y 3 b c Fig. 3. NP-hrdness of the bisimultion-rechbility problem for normed BPP. 4 Bsic Prllel Processes Lemm 4.1 Let t0 be BPP nd t BPP tht hs terminting computtion. It is decidble if there is sequence t0! t 0 s.t. t t 0. Proof. We know tht [t] 2 N. If such t 0 exists tht is rechble from t0 nd bisimilr to t, then [t 0 ] = [t] nd therefore size(t 0 ) [t]. There re only nitely mny cndidtes for such t 0. It is decidble if cndidte t 0 is rechble from t0 nd it is decidble if t 0 t [3]. Check ll cndidtes until correct one is found (nswer \yes") or none is left (nswer \no"). 2 Note tht this lemm is especilly true if t is normed. Lemm 4.2 Let t0 be normed BPP nd t BPP. It is decidble if there is sequence t0! t 0 s.t. t t 0. Proof. As t0 is normed, every t 0 tht is rechble from t0 hs terminting computtion. It cn be decided in polynomil time if t hs terminting computtion. First pply mrking lgorithm tht mrks ll vribles X s.t. 9: X!. Then check if ll vribles in t re mrked. There re two cses: (i) If t hs no terminting computtion, then t 6 t 0 for every t 0 tht is rechble from t0, becuse t 0 is normed. Thus the nswer to the question is \no". (ii) If t hs terminting computtion, then we hve the sme cse s in Lemm
8 The lgorithms used in Lemm 4.1 nd Lemm 4.2 hve non-elementry complexity, s they use the lgorithm from [2] for deciding strong bisimilrity of BPPs. The only known lower bound is NP-hrdness. However, for the specil cse of two normed BPPs we cn give n ccurte complexity mesure. Lemm 4.3 Let t0 nd t be normed BPPs. It is decidble in NP if there is sequence t0! t 0 s.t. t t 0. Proof. It suces to prove the property for normed mrkings 0 (corresponding to t0) nd (corresponding to t) of communiction-free net. As 0 is normed, every 0 rechble from 0 is normed. The norm [] of is t most exponentil. If correct 0 exists, then it must hve the sme norm s. So it cn contin t most exponentilly mny tokens nd cn therefore be described in polynomil spce. Thus it cn be reched by sequence of t most exponentil length [4]. So the Prikh-vector of this sequence of trnsitions cn be described in polynomil spce. As nd 0 re normed, it is decidble in polynomil time if 0 [3]. The lgorithm goes like this: Nondeterministiclly guess Prikh-vector of trnsitions of polynomil size. Then check in polynomil time if there is reble sequence of trnsitions strting t 0 with this Prikh-vector [5], clculte the result 0 (lso in polynomil time) nd check in polynomil time if 0 [3]. 2 Theorem 4.4 For normed BPP the bisimultion-rechbility problem is NPcomplete. Proof. Directly from Lemm 4.3 nd Lemm Context-free Processes Theorem 5.1 The bisimultion-rechbility problem for normed context-free processes is decidble in exponentil spce. Proof. Let t0 be the initil stte nd t the given stte. As t0 is normed, every t 0 tht is rechble from t0 is normed. The norm [t] of t is t most exponentil in size(t). If correct t 0 exists, then it must hve the sme norm s t nd cn therefore be described in exponentil spce. Now for ech of these cndidtes for t 0 with size(t 0 ) O(2 size(t) ) rst check if it is rechble from t0. This requires O(2 size(t) ) time s the rechbility problem for context-free processes is polynomil. Then check if it is bisimilr to t. As both t 0 nd t re normed the time needed for this is polynomil in size(t0) + size(t 0 ) size(t0) + O(2 size(t) ). This is becuse deciding bisimilrity for normed contextfree processes is polynomil [6]. So overll the lgorithm requires t most exponentil spce. 2 It ws shown by Burkrt, Cucl nd Steen in [1] tht for context-free processes nite bsis for bisimultion cn be eectively constructed. The set of sttes tht re bisimilr to given stte form regulr lnguge. 8
9 Theorem 5.2 The Bisimultion-rechbility problem is decidble for contextfree processes. Proof. Let t0 be the initil stte nd t the given stte. By [1] the sttes tht re bisimilr to t cn be described by nite utomton. The set of sttes tht re rechble from t0 is context-free lnguge. The intersection of regulr lnguge nd context-free lnguge is context-free lnguge, whose emptiness cn be decided. 2 6 Normed PA-Processes vs. Finite Stte Systems First we prove decidbility of the rechbility problem for PA-processes. Lemm 6.1 The rechbility problem for PA is decidble in polynomil spce. Proof. Let n be the size of the instnce of the problem. We show tht if t cn be reched from t0, then it cn be reched vi pth s.t. the size of every intermedite stte t 0 is bounded by constnt c O(n 2 ). Every intermedite stte t 0 consists of three prts: A The stble prt. This prt will not chnge in the rest of the sequence nd will be prt of t. B The ctive prt. This prt will chnge in the rest of the sequence nd t lest prt of the result will be prt of t. C The wste prt. This prt will be reduced to in the rest of the sequence. It is vlid strtegy to reduce prt C to rst whenever C isn't empty, before doing nything else. It is cler tht the sum of the sizes of prt A nd B must never exceed size(t). To keep prt C smll we will rst reduce the ccessible vribles in C tht hve the lowest norm. How big cn prt C ever become if we follow this strtegy? Let m be the number of vribles in the PA-lgebr nd l the mximl size of the right hnd side of rule X! G. So the size of the wste descending from vrible X will never exceed (l? 1)(m? 2) + l. The size of the C-prt of t0 is t most size(t0)? 1 nd the wste generted by the ppliction of reduction rule is t most l? 1. Therefore the size of the C-prt of t 0 never exceeds mx(size(t0)? 1; l? 1) + (l? 1)(m? 2) + l. Thus size(t 0 ) size(t)+mx(size(t0)?1; l?1)+(l?1)(m?2)+l O(n 2 ). There re only exponentilly mny such terms t 0. So if t cn be reched t ll, then it cn be reched by sequence of t most exponentil length. The counter for the depth of the serch nd the ctul term both require only polynomil spce nd thus the problem is in PSPACE. 2 Remrk 6.2 The rgument in Lemm 6.1 bout the mximl length of the sequence needed to rech given term is somewht crude. A longer nd more creful nlysis of the structure of PA terms shows tht the problem is in fct NP-complete. However, in the sequel we only need continment in PSPACE. Theorem 6.3 Let t0 be normed PA-process nd R nite stte LTS with initil stte r0. It is decidble in PSPACE if there is sequence t0! t s.t. t r0. 9
10 Proof. It is decidble in polynomil time if r0 is normed. (i) If r0 is not normed then the nswer is \no". This is becuse t is lwys normed nd normed process is never bisimilr to n unnormed one. (ii) If r0 is normed, then [r0] k? 1, where k is the number of sttes in R. So if ny correct t exists, then [t] = [r0] k? 1 nd thus size(t) k? 1. So there re only nitely mny cndidtes for t, ech of which hs only polynomil size. It remins to check for ech cndidte t if it is rechble from t0 nd if t r0. By Lemm 6.1 the rst condition cn be checked in polynomil spce. The second condition cn be checked in polynomil spce s well, s in bisimultion gme the size of child ^t of t must never exceed k? 1, becuse ^t nd r 0 re normed nd R hs k sttes. 2 Remrk 6.4 As the BPPs re subset of PA, the problem of Theorem 6.3 is t lest NP-hrd, becuse of Lemm The Non-Bisimultion-rechbility Problem So fr we hve studied the bisimultion-rechbility problem: \Is there rechble stte tht is bisimilr to given stte?". The opposite question is if there is rechble stte tht is not bisimilr to given stte. This cn be generlized to nite sets of sttes. The non-bisimultion-rechbility problem. Instnce: An LTS with the initil stte s nd nite set of given sttes fs1; : : : ; s n g. Question: Is there sequence s.t. s! s 0 nd 8i 2 f1; : : : ; ng: s 0 6 s i. In section 3 we hve shown tht the bisimultion-rechbility problem is undecidble for Petri nets (Theorem 3.2). On the other hnd the non-bisimultionrechbility problem is decidble. Theorem 7.1 Non-bisimultion-rechbility is decidble for Petri nets. Proof. Let N be Petri net with initil mrking nd 1 ; : : : ; n mrkings of N. Let R() be the set of rechble mrkings. There re two cses: (i) If j[r()] j = k n then there is nite stte LTS U with k sttes nd n initil stte u0 s.t. u0. If 9u 2 U:8i 2 f1; : : : ; ng:u 6 i then the nswer is \yes" else the nswer is \no". (ii) In this cse the system (N; ) hs more thn n dierent sttes w.r.t. strong bisimilrity (possibly even innitely mny). If j[r()] j > n then there is t lest one 0 2 R() s.t. 8i 2 f1; : : : ; ng: 0 6 i nd the nswer to the question is \yes". This yields decision procedure, becuse there re only nitely mny nite stte LTS with n sttes nd it is decidble if Petri net nd nite stte LTS re strongly bisimilr [8]. 2 10
11 8 Conclusion The bisimultion-rechbility problem is decidble for severl simple clsses of process lgebrs, but mny cses re still open. We conjecture tht the problem is decidble even for unnormed BPP. Decidbility for (normed) PAprocesses is lso open. To our knowledge, it isn't even known yet if strong bisimultion equivlence is decidble for (normed) PA-processes. Another interesting eld would be studying the sme problems for wek bisimultion equivlence. Acknowledgement I would like to thnk Didier Cucl for drwing my ttention to his joint work with Burkrt nd Steen in [1]. References [1] O. Burkrt, D. Cucl, nd B. Steen. An elementry bisimultion decision procedure for rbitrry context-free processes. In MFCS'95, number 969 in LNCS. Springer Verlg, [2] S. Christensen, Y. Hirshfeld, nd F. Moller. Bisimultion equivlence is decidble for bsic prllel processes. In E. Best, editor, Proceedings of CONCUR'93, number 715 in LNCS. Springer Verlg, [3] S. Christensen, Y. Hirshfeld, nd F. Moller. Decomposbility, decidbility nd xiomtisbility for bisimultion equivlence on bsic prllel processes. In Proceedings of LICS'93. IEEE Computer Society Press, [4] Jvier Esprz. Decidbility of model checking for innite-stte concurrent systems. To pper in Act Informtic. [5] Jvier Esprz. Petri nets, commuttive context-free grmmrs nd bsic prllel processes. In Horst Reichel, editor, Fundmentls of Computtion Theory, number 965 in LNCS. Springer Verlg, [6] Y. Hirshfeld, M. Jerrum, nd F. Moller. A polynomil lgorithm for deciding bisimultion of normed context free processes. Technicl report, LFCS report series , Edinburgh University, [7] P. Jncr. Undecidbility of bisimilrity for petri nets nd some relted problems. Theoreticl Computer Science, [8] P. Jncr nd F. Moller. Checking regulr properties of petri nets. In Insup Lee nd Scott A. Smolk, editors, Proceedings of CONCUR'95, number 962 in LNCS. Springer Verlg, [9] R. Milner. Communiction nd Concurrency. Prentice Hll,
Semantic reachability for simple process algebras. Richard Mayr. Abstract
Semntic rechbility for simple process lgebrs Richrd Myr Abstrct This pper is n pproch to combine the rechbility problem with semntic notions like bisimultion equivlence. It dels with questions of the following
More informationConcepts of Concurrent Computation Spring 2015 Lecture 9: Petri Nets
Concepts of Concurrent Computtion Spring 205 Lecture 9: Petri Nets Sebstin Nnz Chris Poskitt Chir of Softwre Engineering Petri nets Petri nets re mthemticl models for describing systems with concurrency
More informationStrong Bisimulation. Overview. References. Actions Labeled transition system Transition semantics Simulation Bisimulation
Strong Bisimultion Overview Actions Lbeled trnsition system Trnsition semntics Simultion Bisimultion References Robin Milner, Communiction nd Concurrency Robin Milner, Communicting nd Mobil Systems 32
More informationConvert the NFA into DFA
Convert the NF into F For ech NF we cn find F ccepting the sme lnguge. The numer of sttes of the F could e exponentil in the numer of sttes of the NF, ut in prctice this worst cse occurs rrely. lgorithm:
More informationBisimulation. R.J. van Glabbeek
Bisimultion R.J. vn Glbbeek NICTA, Sydney, Austrli. School of Computer Science nd Engineering, The University of New South Wles, Sydney, Austrli. Computer Science Deprtment, Stnford University, CA 94305-9045,
More informationFinite Automata. Informatics 2A: Lecture 3. John Longley. 22 September School of Informatics University of Edinburgh
Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 22 September 2017 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is
More informationNondeterminism and Nodeterministic Automata
Nondeterminism nd Nodeterministic Automt 61 Nondeterminism nd Nondeterministic Automt The computtionl mchine models tht we lerned in the clss re deterministic in the sense tht the next move is uniquely
More informationDecidability of Weak Bisimilarity for a Subset of Basic Parallel Processes
Decidbility of We Bisimilrity for Subset of Bsic Prllel Processes Colin Stirling Division of Informtics University of Edinburgh emil: cps@dcs.ed.c.u 1 Introduction In the pst decde there hs been vriety
More informationLinear Algebra 1A - solutions of ex.4
Liner Algebr A - solutions of ex.4 For ech of the following, nd the inverse mtrix (mtritz hofkhit if it exists - ( 6 6 A, B (, C 3, D, 4 4 ( E i, F (inverse over C for F. i Also, pick n invertible mtrix
More informationLecture 09: Myhill-Nerode Theorem
CS 373: Theory of Computtion Mdhusudn Prthsrthy Lecture 09: Myhill-Nerode Theorem 16 Ferury 2010 In this lecture, we will see tht every lnguge hs unique miniml DFA We will see this fct from two perspectives
More informationCS 275 Automata and Formal Language Theory
CS 275 Automt nd Forml Lnguge Theory Course Notes Prt II: The Recognition Problem (II) Chpter II.5.: Properties of Context Free Grmmrs (14) Anton Setzer (Bsed on book drft by J. V. Tucker nd K. Stephenson)
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationA Polynomial-Time Algorithm for Checking Consistency of Free-Choice Signal Transition Graphs
Fundment Informtice XX (2004) 1 23 1 IOS Press A Polynomil-Time Algorithm for Checking Consistency of Free-Choice Signl Trnsition Grphs Jvier Esprz Institute for Forml Methods in Computer Science University
More informationPetri Nets and Regular Processes
Uppsl Computing Science Reserch Report No. 162 Mrch 22, 1999 ISSN 1100 0686 Petri Nets nd Regulr Processes Petr Jnčr y Deprtment of Computer Science, Technicl University of Ostrv 17. listopdu 15, CZ-708
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationThe University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER LANGUAGES AND COMPUTATION ANSWERS
The University of Nottinghm SCHOOL OF COMPUTER SCIENCE LEVEL 2 MODULE, SPRING SEMESTER 2016 2017 LNGUGES ND COMPUTTION NSWERS Time llowed TWO hours Cndidtes my complete the front cover of their nswer ook
More informationSummer School Verification Technology, Systems & Applications
VTSA 2011 Summer School Verifiction Technology, Systems & Applictions 4th edition since 2008: Liège (Belgium), Sep. 19 23, 2011 free prticiption, limited number of prticipnts ppliction dedline: July 22,
More informationCSC 473 Automata, Grammars & Languages 11/9/10
CSC 473 utomt, Grmmrs & Lnguges 11/9/10 utomt, Grmmrs nd Lnguges Discourse 06 Decidbility nd Undecidbility Decidble Problems for Regulr Lnguges Theorem 4.1: (embership/cceptnce Prob. for DFs) = {, w is
More information12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2014
CS125 Lecture 12 Fll 2014 12.1 Nondeterminism The ide of nondeterministic computtions is to llow our lgorithms to mke guesses, nd only require tht they ccept when the guesses re correct. For exmple, simple
More information3 Regular expressions
3 Regulr expressions Given n lphet Σ lnguge is set of words L Σ. So fr we were le to descrie lnguges either y using set theory (i.e. enumertion or comprehension) or y n utomton. In this section we shll
More informationAUTOMATA AND LANGUAGES. Definition 1.5: Finite Automaton
25. Finite Automt AUTOMATA AND LANGUAGES A system of computtion tht only hs finite numer of possile sttes cn e modeled using finite utomton A finite utomton is often illustrted s stte digrm d d d. d q
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More information1.4 Nonregular Languages
74 1.4 Nonregulr Lnguges The number of forml lnguges over ny lphbet (= decision/recognition problems) is uncountble On the other hnd, the number of regulr expressions (= strings) is countble Hence, ll
More informationFormal Languages and Automata
Moile Computing nd Softwre Engineering p. 1/5 Forml Lnguges nd Automt Chpter 2 Finite Automt Chun-Ming Liu cmliu@csie.ntut.edu.tw Deprtment of Computer Science nd Informtion Engineering Ntionl Tipei University
More informationDeadlocking States in Context-free Process Algebra
Dedlocking Sttes in Context-free Process Algebr JiříSrb Fculty of Informtics MU, Botnická 68, 60200 Brno, Czech Republic srb@fi.muni.cz Abstrct. Recently the clss of BPA (or context-free) processes hs
More informationCategorical approaches to bisimilarity
Ctegoricl pproches to bisimilrity PPS seminr, IRIF, Pris 7 Jérémy Dubut Ntionl Institute of Informtics Jpnese-French Lbortory for Informtics April 2nd Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity
More information12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2016
CS125 Lecture 12 Fll 2016 12.1 Nondeterminism The ide of nondeterministic computtions is to llow our lgorithms to mke guesses, nd only require tht they ccept when the guesses re correct. For exmple, simple
More informationKleene Theorems for Free Choice Nets Labelled with Distributed Alphabets
Kleene Theorems for Free Choice Nets Lbelled with Distributed Alphbets Rmchndr Phwde Indin Institute of Technology Dhrwd, Dhrwd 580011, Indi Emil: prb@iitdh.c.in Abstrct. We provided [15] expressions for
More informationComplexity and Decidability of Some Equivalence-Checking Problems
Complexity nd Decidbility of Some Equivlence-Checking Problems Zdeněk Sw Ph.D. Thesis Fculty of Electricl Engineering nd Computer Science Technicl University of Ostrv 2005 ii Acknowledgements I would like
More informationHow to simulate Turing machines by invertible one-dimensional cellular automata
How to simulte Turing mchines by invertible one-dimensionl cellulr utomt Jen-Christophe Dubcq Déprtement de Mthémtiques et d Informtique, École Normle Supérieure de Lyon, 46, llée d Itlie, 69364 Lyon Cedex
More informationClosure Properties of Regular Languages
Closure Properties of Regulr Lnguges Regulr lnguges re closed under mny set opertions. Let L 1 nd L 2 e regulr lnguges. (1) L 1 L 2 (the union) is regulr. (2) L 1 L 2 (the conctention) is regulr. (3) L
More informationCSCI FOUNDATIONS OF COMPUTER SCIENCE
1 CSCI- 2200 FOUNDATIONS OF COMPUTER SCIENCE Spring 2015 My 7, 2015 2 Announcements Homework 9 is due now. Some finl exm review problems will be posted on the web site tody. These re prcqce problems not
More informationFinite Automata. Informatics 2A: Lecture 3. Mary Cryan. 21 September School of Informatics University of Edinburgh
Finite Automt Informtics 2A: Lecture 3 Mry Cryn School of Informtics University of Edinburgh mcryn@inf.ed.c.uk 21 September 2018 1 / 30 Lnguges nd Automt Wht is lnguge? Finite utomt: recp Some forml definitions
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationMinimal DFA. minimal DFA for L starting from any other
Miniml DFA Among the mny DFAs ccepting the sme regulr lnguge L, there is exctly one (up to renming of sttes) which hs the smllest possile numer of sttes. Moreover, it is possile to otin tht miniml DFA
More informationProcess Algebra CSP A Technique to Model Concurrent Programs
Process Algebr CSP A Technique to Model Concurrent Progrms Jnury 15, 2002 Hui Shi 1 Contents CSP-Processes Opertionl Semntics Trnsition systems nd stte mchines Bisimultion Firing rules for CSP Model-Checker
More information1.3 Regular Expressions
56 1.3 Regulr xpressions These hve n importnt role in describing ptterns in serching for strings in mny pplictions (e.g. wk, grep, Perl,...) All regulr expressions of lphbet re 1.Ønd re regulr expressions,
More informationC. C^mpenu, K. Slom, S. Yu upper boun of mn. So our result is tight only for incomplete DF's. For restricte vlues of m n n we present exmples of DF's
Journl of utomt, Lnguges n Combintorics u (v) w, x{y c OttovonGuerickeUniversitt Mgeburg Tight lower boun for the stte complexity of shue of regulr lnguges Cezr C^mpenu, Ki Slom Computing n Informtion
More informationBisimilarity of one-counter processes is PSPACE-complete
Bisimilrity of one-counter processes is PSPACE-complete Stnislv Böhm 1, Stefn Göller 2, nd Petr Jnčr 1 1 Techn. Univ. Ostrv (FEI VŠB-TUO), Dept of Computer Science, Czech Republic 2 Universität Bremen,
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationFinite Automata-cont d
Automt Theory nd Forml Lnguges Professor Leslie Lnder Lecture # 6 Finite Automt-cont d The Pumping Lemm WEB SITE: http://ingwe.inghmton.edu/ ~lnder/cs573.html Septemer 18, 2000 Exmple 1 Consider L = {ww
More informationNon-Deterministic Finite Automata. Fall 2018 Costas Busch - RPI 1
Non-Deterministic Finite Automt Fll 2018 Costs Busch - RPI 1 Nondeterministic Finite Automton (NFA) Alphbet ={} q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 2 Nondeterministic Finite Automton (NFA) Alphbet
More informationCMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014
CMPSCI 250: Introduction to Computtion Lecture #31: Wht DFA s Cn nd Cn t Do Dvid Mix Brrington 9 April 2014 Wht DFA s Cn nd Cn t Do Deterministic Finite Automt Forml Definition of DFA s Exmples of DFA
More informationA PROOF OF THE FUNDAMENTAL THEOREM OF CALCULUS USING HAUSDORFF MEASURES
INROADS Rel Anlysis Exchnge Vol. 26(1), 2000/2001, pp. 381 390 Constntin Volintiru, Deprtment of Mthemtics, University of Buchrest, Buchrest, Romni. e-mil: cosv@mt.cs.unibuc.ro A PROOF OF THE FUNDAMENTAL
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationRefined interfaces for compositional verification
Refined interfces for compositionl verifiction Frédéric Lng INRI Rhône-lpes http://www.inrilpes.fr/vsy Motivtion Enumertive verifiction of concurrent systems Prllel composition of synchronous processes
More information1 Nondeterministic Finite Automata
1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you
More informationParse trees, ambiguity, and Chomsky normal form
Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs
More informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018
Finite Automt Theory nd Forml Lnguges TMV027/DIT321 LP4 2018 Lecture 10 An Bove April 23rd 2018 Recp: Regulr Lnguges We cn convert between FA nd RE; Hence both FA nd RE ccept/generte regulr lnguges; More
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationRevision Sheet. (a) Give a regular expression for each of the following languages:
Theoreticl Computer Science (Bridging Course) Dr. G. D. Tipldi F. Bonirdi Winter Semester 2014/2015 Revision Sheet University of Freiurg Deprtment of Computer Science Question 1 (Finite Automt, 8 + 6 points)
More informationCSCI 340: Computational Models. Kleene s Theorem. Department of Computer Science
CSCI 340: Computtionl Models Kleene s Theorem Chpter 7 Deprtment of Computer Science Unifiction In 1954, Kleene presented (nd proved) theorem which (in our version) sttes tht if lnguge cn e defined y ny
More informationCS 275 Automata and Formal Language Theory
CS 275 Automt nd Forml Lnguge Theory Course Notes Prt II: The Recognition Problem (II) Chpter II.6.: Push Down Automt Remrk: This mteril is no longer tught nd not directly exm relevnt Anton Setzer (Bsed
More informationarxiv: v1 [math.ra] 1 Nov 2014
CLASSIFICATION OF COMPLEX CYCLIC LEIBNIZ ALGEBRAS DANIEL SCOFIELD AND S MCKAY SULLIVAN rxiv:14110170v1 [mthra] 1 Nov 2014 Abstrct Since Leibniz lgebrs were introduced by Lody s generliztion of Lie lgebrs,
More informationRegular expressions, Finite Automata, transition graphs are all the same!!
CSI 3104 /Winter 2011: Introduction to Forml Lnguges Chpter 7: Kleene s Theorem Chpter 7: Kleene s Theorem Regulr expressions, Finite Automt, trnsition grphs re ll the sme!! Dr. Neji Zgui CSI3104-W11 1
More informationMath 270A: Numerical Linear Algebra
Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner
More informationFoundations for Timed Systems
Foundtions for Timed Systems Ptrici Bouyer LSV CNRS UMR 8643 & ENS de Cchn 6, venue du Président Wilson 9423 Cchn Frnce emil: bouyer@lsv.ens-cchn.fr Introduction Explicit timing constrints re nturlly present
More informationChapter Five: Nondeterministic Finite Automata. Formal Language, chapter 5, slide 1
Chpter Five: Nondeterministic Finite Automt Forml Lnguge, chpter 5, slide 1 1 A DFA hs exctly one trnsition from every stte on every symol in the lphet. By relxing this requirement we get relted ut more
More information1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.
York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech
More informationSpeech Recognition Lecture 2: Finite Automata and Finite-State Transducers
Speech Recognition Lecture 2: Finite Automt nd Finite-Stte Trnsducers Eugene Weinstein Google, NYU Cournt Institute eugenew@cs.nyu.edu Slide Credit: Mehryr Mohri Preliminries Finite lphet, empty string.
More informationGlobal Types for Dynamic Checking of Protocol Conformance of Multi-Agent Systems
Globl Types for Dynmic Checking of Protocol Conformnce of Multi-Agent Systems (Extended Abstrct) Dvide Ancon, Mtteo Brbieri, nd Vivin Mscrdi DIBRIS, University of Genov, Itly emil: dvide@disi.unige.it,
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationExercises Chapter 1. Exercise 1.1. Let Σ be an alphabet. Prove wv = w + v for all strings w and v.
1 Exercises Chpter 1 Exercise 1.1. Let Σ e n lphet. Prove wv = w + v for ll strings w nd v. Prove # (wv) = # (w)+# (v) for every symol Σ nd every string w,v Σ. Exercise 1.2. Let w 1,w 2,...,w k e k strings,
More informationCentrum voor Wiskunde en Informatica REPORTRAPPORT. Supervisory control for nondeterministic systems
Centrum voor Wiskunde en Informtic REPORTRAPPORT Supervisory control for nondeterministic systems A. Overkmp Deprtment of Opertions Reserch, Sttistics, nd System Theory BS-R9411 1994 Supervisory Control
More information19 Optimal behavior: Game theory
Intro. to Artificil Intelligence: Dle Schuurmns, Relu Ptrscu 1 19 Optiml behvior: Gme theory Adversril stte dynmics hve to ccount for worst cse Compute policy π : S A tht mximizes minimum rewrd Let S (,
More informationGood-for-Games Automata versus Deterministic Automata.
Good-for-Gmes Automt versus Deterministic Automt. Denis Kuperberg 1,2 Mich l Skrzypczk 1 1 University of Wrsw 2 IRIT/ONERA (Toulouse) Séminire MoVe 12/02/2015 LIF, Luminy Introduction Deterministic utomt
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationIntermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4
Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one
More informationExercises with (Some) Solutions
Exercises with (Some) Solutions Techer: Luc Tesei Mster of Science in Computer Science - University of Cmerino Contents 1 Strong Bisimultion nd HML 2 2 Wek Bisimultion 31 3 Complete Lttices nd Fix Points
More informationChapter 5 : Continuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationFirst Midterm Examination
Çnky University Deprtment of Computer Engineering 203-204 Fll Semester First Midterm Exmintion ) Design DFA for ll strings over the lphet Σ = {,, c} in which there is no, no nd no cc. 2) Wht lnguge does
More informationNondeterminism. Nondeterministic Finite Automata. Example: Moves on a Chessboard. Nondeterminism (2) Example: Chessboard (2) Formal NFA
Nondeterminism Nondeterministic Finite Automt Nondeterminism Subset Construction A nondeterministic finite utomton hs the bility to be in severl sttes t once. Trnsitions from stte on n input symbol cn
More informationLearning Moore Machines from Input-Output Traces
Lerning Moore Mchines from Input-Output Trces Georgios Gintmidis 1 nd Stvros Tripkis 1,2 1 Alto University, Finlnd 2 UC Berkeley, USA Motivtion: lerning models from blck boxes Inputs? Lerner Forml Model
More informationUniversität Augsburg. Institut für Informatik. Output-Determinacy and Asynchronous Circuit Synthesis. Victor Khomenko Mark Schaefer Walter Vogler
à ÊÇÅÍÆ ËÀǼ Universität Augsburg Output-Determincy nd Asynchronous Circuit Synthesis Victor Khomenko Mrk Schefer Wlter Vogler Report 2007-02 Jnury 2007 Institut für Informtik D-86135 Augsburg Copyright
More informationHandout: Natural deduction for first order logic
MATH 457 Introduction to Mthemticl Logic Spring 2016 Dr Json Rute Hndout: Nturl deduction for first order logic We will extend our nturl deduction rules for sententil logic to first order logic These notes
More informationNFAs and Regular Expressions. NFA-ε, continued. Recall. Last class: Today: Fun:
CMPU 240 Lnguge Theory nd Computtion Spring 2019 NFAs nd Regulr Expressions Lst clss: Introduced nondeterministic finite utomt with -trnsitions Tody: Prove n NFA- is no more powerful thn n NFA Introduce
More informationSTRUCTURE OF CONCURRENCY Ryszard Janicki. Department of Computing and Software McMaster University Hamilton, ON, L8S 4K1 Canada
STRUCTURE OF CONCURRENCY Ryszrd Jnicki Deprtment of Computing nd Softwre McMster University Hmilton, ON, L8S 4K1 Cnd jnicki@mcmster.c 1 Introduction Wht is concurrency? How it cn e modelled? Wht re the
More information2 Fundamentals of Functional Analysis
Fchgruppe Angewndte Anlysis und Numerik Dr. Mrtin Gutting 22. October 2015 2 Fundmentls of Functionl Anlysis This short introduction to the bsics of functionl nlysis shll give n overview of the results
More informationAutomata Theory 101. Introduction. Outline. Introduction Finite Automata Regular Expressions ω-automata. Ralf Huuck.
Outline Automt Theory 101 Rlf Huuck Introduction Finite Automt Regulr Expressions ω-automt Session 1 2006 Rlf Huuck 1 Session 1 2006 Rlf Huuck 2 Acknowledgement Some slides re sed on Wolfgng Thoms excellent
More informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl
More informationReasoning with Bayesian Networks
Complexity of Probbilistic Inference Compiling Byesin Networks Resoning with Byesin Networks Lecture 5: Complexity of Probbilistic Inference, Compiling Byesin Networks Jinbo Hung NICTA nd ANU Jinbo Hung
More informationDesigning finite automata II
Designing finite utomt II Prolem: Design DFA A such tht L(A) consists of ll strings of nd which re of length 3n, for n = 0, 1, 2, (1) Determine wht to rememer out the input string Assign stte to ech of
More informationCS 275 Automata and Formal Language Theory
CS 275 utomt nd Forml Lnguge Theory Course Notes Prt II: The Recognition Prolem (II) Chpter II.5.: Properties of Context Free Grmmrs (14) nton Setzer (Bsed on ook drft y J. V. Tucker nd K. Stephenson)
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationCS415 Compilers. Lexical Analysis and. These slides are based on slides copyrighted by Keith Cooper, Ken Kennedy & Linda Torczon at Rice University
CS415 Compilers Lexicl Anlysis nd These slides re sed on slides copyrighted y Keith Cooper, Ken Kennedy & Lind Torczon t Rice University First Progrmming Project Instruction Scheduling Project hs een posted
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationTutorial Automata and formal Languages
Tutoril Automt nd forml Lnguges Notes for to the tutoril in the summer term 2017 Sestin Küpper, Christine Mik 8. August 2017 1 Introduction: Nottions nd sic Definitions At the eginning of the tutoril we
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More informationCalculus of Variations
Clculus of Vritions Com S 477/577 Notes) Yn-Bin Ji Dec 4, 2017 1 Introduction A functionl ssigns rel number to ech function or curve) in some clss. One might sy tht functionl is function of nother function
More informationLECTURE. INTEGRATION AND ANTIDERIVATIVE.
ANALYSIS FOR HIGH SCHOOL TEACHERS LECTURE. INTEGRATION AND ANTIDERIVATIVE. ROTHSCHILD CAESARIA COURSE, 2015/6 1. Integrtion Historiclly, it ws the problem of computing res nd volumes, tht triggered development
More informationCS375: Logic and Theory of Computing
CS375: Logic nd Theory of Computing Fuhu (Frnk) Cheng Deprtment of Computer Science University of Kentucky 1 Tble of Contents: Week 1: Preliminries (set lgebr, reltions, functions) (red Chpters 1-4) Weeks
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationCMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature
CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy
More informationCISC 4090 Theory of Computation
9/6/28 Stereotypicl computer CISC 49 Theory of Computtion Finite stte mchines & Regulr lnguges Professor Dniel Leeds dleeds@fordhm.edu JMH 332 Centrl processing unit (CPU) performs ll the instructions
More information