2. Elements of the Theory of Computation, Lewis and Papadimitrou,

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1 Introduction Finite Automata DFA, regular languages Nondeterminism, NFA, subset construction Regular Epressions Synta, Semantics Relationship to regular languages Properties of regular languages Pumping lemma Minimizing DFA, Myhill-Nerode theorem Machines Modeling hardware in terms of machines Semantics of interaction, compleity issues References:. Introduction to Automaton Theory, Languages, and Computation, Hopcroft and Ullman, 979, Chapters 2,3 2. Elements of the Theory of Computation, Lewis and Papadimitrou, 98, Chapters,2

2 Finite Automaton Arise in various contets: Tet Editors, Neuron modelling, Hardware Notation: Given an alphabet Σ, Σ is the set of all finite strings on Σ, a language is any L Σ, a b denotes the concatenation of string a with b, ɛ is the empty string Deterministic Finite Automaton : characterised by a 5-tuple (Q, Σ, δ, q 0, F ). Q - finite set of states 2. Σ - finite input alphabet 3. δ - transition function, maps Q Σ Q (can assume total) 4. q 0 - initial state 5. F - subset of S, the set of final states Eample $q_0$ q 0 q$q_ $q_ $q_3$ q 3 q 2

3 Behavior of DFA on input strings: Etend δ to ˆδ : Q Σ Q in natural way. ˆδ(q, ɛ) = q 2. ˆδ(q, wa) = δ(ˆδ(q, w), a) Language of DFA A : L(A) = { Σ ˆδ(q 0, ) F } Language L is said to be regular if DFA M such that L = L(M) EE290H 8/24/93 AA

4 Nondeterminism Less obvious, useful for theoretical analysis, modelling initial stages of hardware designs, can be more compact Nondeterministic Finite Automaton : characterised a 5-tuple (Q, Σ,, q 0, F ). Q - finite set of states 2. Σ - finite input alphabet 3. - transition function, maps Q Σ 2 Q (variously Q Σ Q) 4. q 0 - initial state 5. F - subset of S, the set of final states Eample,0 0 0,0 q$q_0$ 0 q$q_$ q$q 2 q$q_3$ 3 q$q_4$ 4,0

5 Behavior of NFA on input strings: Etend to ˆ : Q Σ 2 Q in natural way. ˆ (q, ɛ) = {q} 2. ˆ (q, wa) = {p r ˆ (q, w), p (r, a)} Language of NFA A : L(A) = { Σ ˆ (q 0, ) F φ} EE290H 8/24/93 AA

6 Nondeterminism - II Obvious fact : Class of languages accepted by NFA Class of languages accepted by DFA ; Less obvious fact : Equality holds Proof : by the subset construction Intuition : Nondeterminism leads to parallel paths simulate NFA to determinize Eample NFA DFA 0 0, {q 0 } $\{q_0\}$ 0 0 $\{q 0, q } q$q_0$ 0 q$q_$ Subset Construction $\{q_ } Remark: Potential blowup; generate subsets incrementally; Average case increase? EE290H 8/24/93 AA

7 Regular Epressions Idea: Formulae that define certain subsets of Σ not commonly used to describe systems; describe properties eg 3 REQs never occur consecutively Notation: Let L, L, L 2 Σ. Concatenation of L and L 2 = { y L y L 2 } 2. Define L 0 = {ɛ} and L i = L L i for i Kleene closure L positive closure L + = i=0 L i = i= L i Eample: Σ - finite alphabet L = { has 3 consecutive 0 s } L 2 = {y y has 2 consecutive s } L L 2 = {z z has 3 consecutive 0 s and 2 consecutive s which follow the 3 0 s } L = { has an even number of s } L = L EE290H 8/24/93 AA

8 Regular Epressions - II Synta and Semantics. φ is a regular epression, denotes the empty set 2. ɛ is a regular epression, denotes the set {ɛ} 3. a Σ, a is a regular epression, denotes the set {a} 4. let r, s be regular epressions denoting the languages R and S, then (r + s), (r s), (r) are regular epressions denoting R S, R S, and R respectively Eample: 0 + (0 0 0 ) Remark: Can etend above definition to include other operators such as - intersection, - complement, () 2 - squaring, () i - eponentiation wjere i is encoded in binary. The resultant epressions do not define any new languages, but can be eponentially more succint. Interesting fact : deciding universality of regular epressions drawn from +,,, () i is EXP-space complete. EE290H 8/24/93 AA

9 Equivalence of FA and Regular Epressions Theorem: Given any regular epression r, there eists an FA accepting L(r) Proof: by construction, use ɛ transitions Theorem: [Kleene, 956] Given a regular language L, there is a regular epression r such that L = L(r). Proof: Let L be a regular set. There is a DFA which acepts L. Let the DFA be M = ({q,..., q n }, Σ, δ, q, F ) We will inductively construct regular epressions that drive the DFA from one state to another. Let R k ij denote all strings such that δ(q, ) = q j and if δ(q i, y) = q l for some y which is a prefi of, other than or ɛ, then l k. Clearly : R 0 ij R k ij = {a δ(q, ) = q j }if i j = {a δ(q, ) = q j } {ɛ} if i = j = Rij k Rik k (Rkk k ) Rkj k, k

10 Claim: i, j, k R k ij is defined by a regular epression r k ij Proof: By induction on k Base case: R 0 ij is a finite set r 0 ij = (a +... a p )ifi ) = (a +... a p + ɛ)ifi = j Induction: r k ij = r k ij r k ik (rkk k ) rkj k, k Conclusion: L(M) = q j FR n ij r L = r n ij + + r n ij p where F = {q j,..., q ip } EE290H 8/24/93 AA

11 Equivalence of FA and Regular Epressions Theorem: Given any regular epression r, there eists an FA accepting L(r) Proof: by construction, use ɛ transitions Theorem: [Kleene, 956] Given a regular language L, there is a regular epression r such that L = L(r). Proof: Let L be a set accepted by the DFA M = ({q,..., q n }, Σ, δ, q, F ) Let R k ij denote all strings such that δ(q, ) = q j and if δ(q i, y) = q l for some y which is a prefi of, other than or ɛ, then l k. Clearly : R 0 ij R k ij = {a δ(q, ) = q j }if i j = {a δ(q, ) = q j } {ɛ} if i = j = Rij k Rik k (Rkk k ) Rkj k, k Lemma: i, j, k Rij k is defined by a regular epression rij k Proof: By induction on k Base case: Rij 0 is a finite set rij 0 = (a +... a p ) or = (a +... a p + ɛ)ifi = j Induction: rij k = rij k, k Conclusion: r k kj r k ik (r k kk ) L(M) = q j FR n ij r L = r n ij + + r n ij p where F = {q j,..., q ip } EE290H 8/24/93 AA

12 Equivalence of FA and Regular Epressions - II Eample: 0 0 $q_$ $q q q 2 Step r 0 = ɛ + 0, r 0 22 = ɛ + 0, r 0 2 = r 0 2 = φ Step 2 r = 0, r 22 = 0 + 0, r 2 = r 2 = Step 3 r 2 = 0 + (0 + 0 ) Remark: Associating formula with states is a common theme in verification, eg can charcterize states in a Kripke structure upto bisimulation using a CTL formula EE290H 8/24/93 AA

13 Regular Language Properties - The Pumping Lemma Let L be a regular language defined by DFA M = (Q, Σ, δ, q 0, F ) Consider an input of n or more symbols a, a 2,..., a m for i =, 2,..., m let δ(q 0, a,..., a i ) = q i Observe: Cant have q 0, q,..., q n all distinct j, k 0 j < k n q j = q k Eample $q_j = q_k$ q $q_0$ q j = q k 0 q m m n $q_m a... a j $a_ \ldots a_j$ a k... a m $a_k \ldots a_m$ a j+... a k $a_{j+} \ldots a_k$ a m F iff a a 2... a m L(M) so is the string a... a j a k+... a m in fact can loop around as many times as desired: a... a j (a j+... a k ) i a k+... a m L(M) i 0 If FA accepts arbitrarily long strings must have a string s with the property a substring of s that can be repeated

14 Pumping Lemma: Let L be a regular set. Then there eists a constant n such that if z L and z n, can write z = u v w in such a way that u v n, v and i 0 u v i w L. Also n # number of states in smallest DFA for L EE290H 8/24/93 AA

15 Regular Language Poperties - Closure, Minimization Closure Closure under union, concatenation, and Kleene closure : follows from equivalence of regular epression and FA Closure under complementation, intersection : trivial to complement DFA Deciding Emptiness, Universality, Equivalence of DFA Deciding Emptiness, Universality, Equivalence of NFA Minimizing DFA The Myhill-Nerode theorem : Given a regular language L, there is a unique minimium state DFA accepting the language. Proof based on partitioning Σ, under the equivalence relation R L y iff z z L y z L Simple Algorithm : Merge states in a DFA if there is no input string that can differentiate them EE290H 8/24/93 AA

16 Hardware Scenario: Two processors, single disk drive Only one processor should be allowed to access the drive at any time Design arbiter to decide which processor gets access Processors and arbiter communicate through Σ P = {Idle, Req, Busy}, Σ P 2 = {Idle2, Req2, Busy2}, Σ Arbit = {Stop, Go, Go2} Block Diagram Proc Proc 2 Arbiter ${Output}_{Proc} \{Idle, Request, Busy2 \}$ P roc = Request, Busy} ${Output}_{Proc2} =\{ Request2, Busy2\}$ P roc2 = {Idle2, Request2, Busy2} ${Output}_{Arbiter} \{Stop, Go, Go2\}$ Arbiter = Go, Go2}

17 Controller State Diagram Observe: Nondeterminism in the controller Inputs, Outputs, States : Symbolic Transition Conditions ( Predicates ) are described by formulae Outputs on state : Moore machine Properties: Safety : Never give both processors access to drive at same time Liveness : Processor makes request eventually gets access Relationship to Regular Languages: View behavor of arbiter as map from (Σ P Σ P 2 ) to (Σ Arbit ). Can naturally associate a regular language with the behavior. EE290H 8/24/93 AA

18 Interacting Machines Game of Life Single cellular organisms, atmost one on each square of a chess board. Only two states: { Dead, Live }, simple transition rule Eample: Petri Dish Transition rule: LIVE DEAD

19 Definition: The product machine M = M M 2... M n is the machine on state space S = S S 2... S n and output space O = O... O n characterized by Output relation Transition relation T (, y) = ( i)[ n Θ(, o) = i= n i= Θ i ( i, o i ) T i ( i, i, y i ) Θ i ( i, o i ) (i = [o... o n ])] where the present state variable s = [s s 2... s n ], and the net state variable t = [t t 2... t n ]. Compleity : The computational compleity of verifying systems of interacting machines is very high; for eample the following problem is P-space complete: Reached State Analysis: Given a system of interacting machines, and two states s and p, is a s reachable from p? EE290H 8/24/93 AA

20 Modeling Hardware Recall can go from netlist to FSM w2 0/0 0/,/ 00 0 /0 0/0,/0 0/,/ w L L2 0 Problem is output behavior of N at state s 0 equal to output behavior of N 2 at state s? Problem 2 is N a safe replacement for N 2? EE382m

21 Language Containment Suggests another verification paradigm Language Containment System FA Property FA/RE Check S contained in P What properties can we epress in this paradigm? D s0 0/a /a s s2 0/b,/b 0/b,/b D2 t0 0/a,/a 0/b t /b 0/c,/c s3 s4 0/d,/d 0/c,/c t3 t4 0/d,/d EE382m

22 ω Automata Accept infinite sequences rather than finite sequences!!! Contrast with Finite Automata (FA) Motivation: Would like to be able to epress properties like AG (REQ= (AF ACK= )) Intuitively, such a property is a function of input/output behavior of design, but can not be captured using Finite Automata. Would like to capture designs which have fairness constraints How to describe processor with fairness constraint which ensures REQ= infinite often? Terminology. Recall ω = {0,, 2,...}; an ω sequence f over a set Σ is a function from ω to Σ. f(0), f(), f(2),..., 2. Given set Σ (alphabet), Σ ω is set of all infinite sequences over Σ. 3. Subsets of Σ ω are referred to as ω-languages. How to build automata accepting ω-sequences???

23 Defn. Büchi automata synta and semantics Syntactically identical to Finite Automata Same dichotomy between deterministic and nondeterminitic Semantics Accept infinite input sequence eactly when there is a resulting path on which an accepting state occurs infinitely often. A (0,0),(,0),(0,) (,0),(0,),(,) A2 a,b a,b c s s2 (,) (0,0) s3 T u u2 u3 c T u4 c a,b Σ = {(0,0),(0,),(,0),(,)} Σ 2 = {a,b,c} A3 0 A4 0, 0 t 0 t2 t 0 t2 Σ = {0,} 3 Σ 4 = {0,} t3 0, Defn. An ω-language L is said to be ω-regular if it is accepted (variously defined, recognized) by some Büchi automaton.

24 Closure Properties of ω-regular languages. Closed under union could do eactly as for FA Alternate approach product construction. Let L and L 2 be ω-regular languages over alphabet Σ. ω-regular must be accepted by BA. Let (S, α, T, Σ, F ) and (S 2, α, T 2, Σ, F 2 ) be BA for L and L 2. Consider the following BA: States = S S 2 Initial state = (α, β) Transition Relation T = {((s, s 2 ), a, (t, t 2 )) (s, a, t ) T (s 2, a, t 2 ) T 2 } Alphabet = Σ Accepting States = F S 2 S F 2 Why does this work?

25 2. Closed under intersection for FA, could use complementation, union, and DeMorgans s laws. Can we do directly? First try: build product automata (a) Accepting States = F F 2 Q. Does this catch everything? o.w. Product Transition Structure o.w. F F2 T Figure : Monitor for Intersection 3. Closure under projection eraser construction, eactly as before (May go from deterministic to nondeterministic.)

26 Closure of ω-regular languages under Complement How did we achieve this for -regular languages?. L is -regular has an NFA N L. 2. From N L built DFA D L for L through subset construction. 3. Complementing D L trivial. First attempt for ω-regular languages mimic above.. Problem: how to complement D L? 2. Not possible in general!!! Consider L = { {0, } ω 0 occurs a.a. in } Fundamental result: [Büchi 969] ω-regular languages are closed under complement Idea: go from NBA to deterministic Müller automaton which are easy to complement; then go back to NBA

27 General Knowledge Other types of ω-automata: states, transtion structure remains the same. Differ in acceptance condition. Let σ be a path, inf(σ) set of states occurring infinitely often in σ.. Müller automata: F = set of subset of state set. σ is accepted iff inf(σ) is a member of F 2. Streett automata: F = set of pairs of subsets of state space. σ is accepted iff for every pair (A i, B i ) in F, A i inf(σ) = or B i inf(σ) in the path 3. Rabin automata: F = set of pairs of subsets of state space. σ is accepted iff for some pair (A i, B i ) in F, A i inf(σ) or B i inf(σ) in the path All these accept eactly the class of ω-regular languages (even in the deterministic case). Differ in terms of succintness refer to Safra FOCS 988. Language containment paradigm for ω-automata basis for COSPAN.

28 Relationship to Temporal Logic Generally can t convert CTL formula to Automata ( or ω). For the logic PLTL, can automatically build Büchi automata To analyse CTL (and other branching time logics) from automata point of view, need automata which accept trees. Similar issues complementation, epressiveness, etc. Strong analogy to language containment