Reachability in Petri nets with Inhibitor arcs

Transcription

1 Reachability in Petri nets with Inhibitor arcs Klaus Reinhardt Universität Tübingen

2 Overview Multisets, New operators Q and Q on multisets, Semilinearity Petri nets, Inhibitor arcs The reachability relation for Petri nets with one inhibitor arc Nested Petri Nets as normal form for expressions new Overview: Decision algorithm, Logic, Automata

3 Multisets We write a multiset f N B as a set {b f (b) b B}, as a table [ b1 or as an n-ary vector A B N A N B f(b 1 ) f(b ). f(b n ). f N A g N B (f + g) N A B f (b 1 ), b f (b n ) f (b ),..., b n ] /0 with /0(x) = 0 for all x is neutral element for +. N A N B = N A B

4 sgn(f) := {a f(a) > 0}, sgn(m) := f M sgn(f). Restriction: f A := {b f(b) b A} f A := {b f(b) b A}, thus f = f A +f A. A set M = {m 1,...,m k } N A of multi-sets generate linear combinations: M := {a 1 m a k m k i k a i N} More generally, by M 0 := {/0} and M i+1 := M i + M, we can define M := i M i. Linear set: m c + M. Semilinear set: finite union of linear sets. Semilinear sets: Smallest class of sets of multisets containing all finite sets of multisets and being closed under,+ and. [GS65],[ES69]: The semilinear sets are also closed under.

5 New operators Q and Q on multisets For an unambiguous and injective binary relation Q and two sets of Multisets M and N we define { } N Q M := n π1 (Q) +m π (Q) n N,m M, (a,b) Q n(a) = m(b). For example, or { {, , } 5 {(b1,b )} } { {(b3,b 3 )} {, ,, 7 5 3, } 5 3 = } { 8 6 =, { ( , ) (, )} }

6 For π 1 (Q) and π (Q) disjoint, we define Id Q := {{a 1,b 1} (a,b) Q} which is the neutral element for Q. Obviously, it holds N /0 M = N+M which makes + with the neutral element Id /0 = {/0} a special case of the Q operator. Furthermore, for Q with π 1 (Q) and π (Q) disjoint, we define Q (M) as the closure of M Id Q under Q and the addition /0. In other words, 0 Q (M) := Id Q, i+1 Q (M) := i Q (M) QM + Id Q and Q (M) := i i Q (M). Again, /0(M) = M is a special case.

7 Properties: N Q M = M Q 1N For N,M N A we get N M = N Q L Q M with Q := {(a,a ) a A}, Q := {(a,a) a A} and L := {{a 1,a 1,a 1} a A}. In general, N Q L Q M can only be written without brackets because π 1 (Q ) (sgn(m) \ π (Q )) and π (Q ) (sgn(n) \ π 1 (Q )) are disjoint. If, additionally, π (Q ) and sgn(n) are disjoint and sgn(m) and π 1 (Q )) are disjoint, then N Q L Q M = L Q 1 Q (M + N).

8 Semilinearity Q preserves semilinearity: Assume N and M are semilinear sets over A. N semilinear set over A \ π 1 (Q) π 1 (Q). M semilinear set over A \ π (Q) π (Q). E Q := {{a 1,b 1},{c 1} (a,b) Q,c A} = {f (a,b) Q f(a ) = f(b )} semilinear sets over the set A π 1 (Q) π 1 (Q). Thus, N Q M = ((N + M ) E Q ) π 1 (Q) π 1 (Q) is semilinear. Q does not preserve semilinearity: Let M := {, then {(b3,b )} (M) = a b c } c ba not semilinear.

9 Petri net We describe a Petri net as the triple N = (P,T,W) with the places P, the transitions T and the weight function W N P T T P. A transition t T can fire from a marking m N P to a marking m N P, denoted by m[t m, if m W(.,t) = m W(t,.) N P. A firing sequence w = t 1...t n T can fire from m 0 to m n, denoted by m 0 [w m n, if m 1,...m n 1 exist with m 0 [t 1 m 1 [t...[t n m n. Reachability problem: given net N with start- and end markings m 0,m e N P, decide if there is a w T with m 0 [w m e. [May84][Kos84][Lam9]: decidable.

10 Let P + := {p + p P} and P := {p p P} be copies of the places and ˆP := {(p +, p ) p P}. For m define the corresponding copies m := {p m(p) p P} and m + := {p + m(p) p P}. Reachability relation for a transition t: R(t) := {m + m + } m[t m { = r N P+ P } p P r(p ) W(p,t) = r(p + ) W(t, p) N Reachability relation for a set of transitions T as R(T ) := R(t). Monotonicity: Whenever m[w m, then also (m + n)[w (m + n) for any n N P. t T This corresponds to adding Id P := Id ˆP and R(t) = c t + Id P is a linear set using c t with c t (p ) = W(p,t) and c t (p + ) = W(t, p) for all p P.

11 Concatenation of two firing sequences described by the operator P := ˆP iteration described by P := ˆP. The reachability relation of the petri net N is R(N) := R(T ) := P (R(T )). The reachability problem: (m 0 + m+ e ) R(N). Corollary 1 There is a firing sequence w T with m 0 [w m e in N if and only if m + 0 PR(N) P m e = (m 0 + m+ e ) A R(N) = {/0} for A := {(p, p ),(p +, p + ) p P}. In the other case (m 0 + m+ e ) A R(N) = /0.

12 Inhibitor arcs We describe such a Petri net as the 6-tuple (P,T,W,I,m 0,m e ) with the places P, the transitions T, the weight function W N P T T P, the inhibitor arcs I P T and, the start and end markings m 0,m e N P. We will denote an inhibitor arc in the pictures by. m[t m only if p P (p,t) I m(p) = 0. Lemma 1 Each Petri net (P,T,W,I,m 0,m e ) can be changed in such a way that the condition p P,t T (p,t) I W(t, p) = 0 holds without changing the inhibitor arcs or the reachability problem. p x t

13 x t p p p t t a Lemma Each Petri net (P,T,W,I,m 0,m e ) can be changed in a way such that the condition p P,t T (p,t) I m 0 (p) = m e (p) = 0 holds by changing neither the inhibitor arcs, the condition in Lemma 1 nor the reachability problem.

14 The reachability relation for Petri nets with one inhibitor arc Given a Petri-net (P,T,W,{(p 1, ˆt)},m 0,m e ).w (T \ {ˆt}). R 1 = R((P,T \ {ˆt},W)) = P (R(T \ {ˆt})) R = R 1 {r N P,P + r(p 1 ) = r(p+ 1 ) = 0} R 3 = R R(ˆt) R 4 = P\{p1 } (R 3)

15 Lemma 3 Given a Petri-net (P,T,W,{(p 1, ˆt)},m 0,m e ) with only one inhibitor arc (p 1, ˆt) having the property of lemmata 1 and, then there is a firing sequence w T with m 0 [w m e if and only if m + 0 P\{p 1 } R 4 P\{p1 } m e = (m 0 + m+ e ) A R 4 = {/0} A := {(p, p ),(p +, p + ) p P \ {p 1 }}. In the other case (m 0 + m+ e ) A R 4 = /0

16 t 7 p 3 1 t 8 Example: p ˆt 5 7 p 3 with the start marking {p 4, p 3 } and the end marking {p 4, p 3 3}. We have R(t 7 ) = {p 1, p+ 1 3} + Id P, R(t 8 ) = {p 1, p+ 3 1} + Id P and R(ˆt) = {p 3 7, p+ 5} + Id P\{p 1 }. This yields ({[ = p ] R 1 = R((P,{t 7,t 8 })) P, p , [ p 1, p+ 31 ]}) = {[ ] [ ] [ ] [ ] [ ] [ ]} p, 1 p+ p, 13, 1 p+ p, 31, 1 p+, 11 p+ p, 31, 1 p, 11 p+ p, 3, p+, 1 p+ p, 3, p Id P

17 and R = R 1 {(p 1,x),(p + 1,y)}{/0} = {[ p, p+ 3 3 ]} + Id {p,p 3 }. We can cut the firing sequences in (t 7 + t 8 + ˆt) = ((t 7 + t 8 ) + ˆt) into parts in (t 7 + t 8 ) and ˆt all starting and ending with no token on p 1. This yields R 3 = R R(ˆt) and R 4 = {p,p 3 } (R 3) = {[ p, p+ 33 ], [ p 37, p+ 5 ], [ p, p 34, p+ 5 ], [ p 4, p 31, p+ 5 ], [ ] [ ] p, 37 p+, 3 p+ p, 33, 37 p+, 1 p+,..., 36 [ ] [ ] [ ] [ ]} p, 4 p, 3 p+ p, 38, 5 p, 31 p+, 1 p+ p, 37, 36 p+, 4 p+ p, 33, 4 p, 3 p+, 4 p Id {p,p 3 }.

18 Nested Petri Nets as normal form for expressions For every expression e, there is a carrier set C(e) sgn(r(e))}. R(e) N C(e). R is the evaluation function for an expression defined in a way such that is always commutes with the expression operators P, Q, and +, and the additional operator. Expression for an elementary transition: t = L t is an expression for the linear set L t = R(L t ) = c t + Γ t. Example: Γ t = {{p 1, p + 1} p P} leading to Γ t = Id P. C(t) := P P + sgn({c t } Γ t ).

19 Expression for sets of transitions: T = t 1 t... t l for expressions t i T. Expression for a sub-net: N = PT (T ) for N consisting of P T and T. Let C(N) := C(T ) := t T C(t). Expression for a generalized transition: t = L t QA K t, where L t again expresses a linear set, and K t is a set of sub-nets and interpreted as expression K t = N i where the C(N i ) are pairwise disjoint. Ni K t Using Q A := {(a,a) a A} with A = N i K t C(N i ), we define C(t) := {a (c t + g Γ t g)(a) > 0} \ A. This means that the behavior of t is mainly described by the linear set c t +Γ t but it is additionally controlled by the reachability in the sub-nets N i.

20 Example (continued): We identify t 7 = { ˆp 1, ˆp+ 1 3} + Id { ˆp 1, ˆp, ˆp 3 }, t 8 = { ˆp 1, ˆp+ 3 1} + Id { ˆp 1, ˆp, ˆp 3 } and ˆt = {p 3 7, p+ 5} + Id {p,p 3 }. This yields = the expressions T 1 = t 7 t 8 and N 1 { ˆp1, ˆp, ˆp 3 } (T 1). On the next level, we get the generalized transition t = ( /0 + {[ p 1, ˆp 1 ], [ p 31, ˆp 31 ], [ p + 1, ˆp+ 1 ], [ ]} ) p +, 31 ˆp+ 31 {( ˆp, ˆp ),( ˆp 3, ˆp 3 ),( ˆp+, ˆp+ ),( ˆp+ 3, ˆp+ 3 )}N 1, which we visualize as t = t 7 ˆp 1 ˆp 3 t 8 ˆp 3

21 = T = t ˆt and N {p,p 3 } (T ). On the top level, we get [ ] p T 3 = t 3 =, 4 p, 3 p+, 4 p+ 33 {(p,p ),(p 3,p 3 )}N, which we visualize as 3 7 y 4 4 t p 3 p z 5 ˆt

22 Expression T T.1.b t T {c t } Γ t N N i K t T i T.1.b t T i {c t } Γ t N Carrier set C(T ) = C(t) = P T P+ T {w c t,w g,...} C(N i ) = C(T i ) = C(t ) = P T i P + T i {w ct,w g,...}...

23 new Overview The property T The size of an expression Additional operators working on expressions, Logic with mtc The main algorithm establishing property T The reachability relation for Petri nets with inhibitor arcs Priority-Multicounter-Automata Restricted Priority- Multipushdown- Automata

24 The property T Definition 1 An expression T has the property T if t T, N i = PTi (T i ) K t the following 5 conditions hold: 1. In recursive manner, T i has (a) the property T, and (b) For all t T i it holds g {c t } Γ t w g C(t ) g(w g ) = 1, g t T i {c t } Γ t \ {g} g (w g ) = 0.

25 . g {c t } Γ t, p P Ti g(p ) ind(g)(p ) = g(p + ) ind(g)(p + ), where ind(g) := g(w g )g t T i,g {c t } Γ t 3. w C(N i ) \ (P + T i P T i ) Σ g Γt g(w) > There are multisets m +,m R(N i ) with p P Ti m + P Ti (c t + Γ t ) P Ti (( g Γ t g(p ) = 0) m + (p + ) > m + (p )) m P + Ti (c t + Γ t ) P + Ti (( g Γ t g(p + ) = 0) m (p ) > m (p + )).

26 5. c t C(t) R(t). Theorem 1 For every expression T, we can effectively construct a T with R(T ) = R(T ) such that T has property T. Corollary The reachability problem for a Petri net with one inhibitor arc is decidable.

27 The size of an expression m :< m if there is an e with m(e) < m (e) and m(e ) = m (e ) for all e > e. [DM79]: Noetherian order on e s Noetherian order on m s. S(T ) := t T {S(t) 1}. S(t) := (S(K t ),b,b 5 + Γ t ). Here, b i = 0 if Condition T.i is fulfilled, and b i = 1 otherwise. S(K t ) = {S(N i ) 1}. Ni K t

28 S(N i ) := (s m + { P Ti 1},S(T i ),b 1b, C(N i ) ) with s m := max{s s s ((s,.,.,.)) > 0,S(T i )((s,.,.)) > 0}. Example (continued): S(t 7 ) = S(t 8 ) = (/0,0,3), S(T 1 ) = {(/0,0,3) }, S(N 1 ) = ({3 1},{(/0,0,3) },1,6), S(t ) = ({S(N 1 ) 1},1,4), S(T ) = {S(t ) 1,(/0,0,) 1}, S(N ) = ({3 1, 1},S(T ),1,4). Lemma 4 The ordering on S defined above is Noetherian

29 Additional operators working on expressions Lemma 5 Let t = L t Q K t be an expressions for a transition and L be (an expression for) a semi linear set. Then, we can construct an expression T := t L (with R(T ) = (R(L t ) R(L)) Q R(K t )) where the occurring sizes S(t ) with t T increase relatively to S(t) only in the last position in the triple. Lemma 6 Let T and T be expressions for sets of transitions, and Q be a relation. Then, we can construct an expression T := T Q T (with R(T ) = R(T ) Q R(T )) where the occurring sizes S(t) increase only in the last position in the triple and sum up in the first position. Lemma 7 Let N be an expression for a subnet. Then, we can construct an equivalent expression for a transition t(n) with R(t(N)) = R(N) and t P (N) with R(t P (N)) = {m R(N) p P m(p ) = m(p + ) = 0}.

30 Logic Given a formula φ(x 1,...,x k,x 1,...,x k ), then mtc(φ) denotes the smallest set S N k containing all of the following: (x 1,...,x k,x 1,...,x k ) for (x 1,...,x k ) N k (this stands for the identity), (x 1,...,x k,x 1,...,x k ) for φ(x 1,...,x k,x 1,...,x k ) (x 1,...,x k,x 1,...,x k ) for (x 1,...,x k,x 1,...,x k ),(x 1,...,x k,x 1,...,x k ) S, and (x 1 +x 1,...,x k+x N k. k,x 1 +x 1,...,x k +x k ) for a (x 1,...,x k,x 1,...,x k ) S and (x 1,...,x k )

31 Corollary 3 The emptiness and satisfiability is decidable for formulas with an FO+PLUS-formula inside and,, and mtc operators outside. corresponds to expressible with Q Lemma 6 corresponds to expressible since T is already a union remove the element mtc is done by using Lemma 7.

32 The main algorithm establishing property T function reacheq(t ): begin repeat i:= 1 while i 5 and t T, N K t Condition T.i fulfilled do i:=i+1 od if i=6 then return T else T :=T for T according to treatment of Condition T.i until i=6 end reacheq

33 in each step S(T ) decreases (S(reacheq(T )) < S(T ) if T reacheq(t )); due to Lemma 4 the algorithm terminates. Change of size S(t) during the treatment of Condition T.i: S(t) S(N i ) S(t ) for t T i s m + { P Ti 1} S(K t ) b b 5 + Γ t b 1b C(N i ) b b 5 + Γ t T T T T.4 T

34 Condition 1 Recursion and introducing witnesses Let Condition 1 be not fulfilled by T i ; let T i := reacheq(t i ), which terminates by induction since S(T i ) < S(T ). Construct T from T by adding witnesses inside as for T = T = T ˆt in following continued example: Replace t 7 and t 8 by t 7 = { ˆp 1, ˆp+ 1 3,w c t 7 1} + Id ˆp1, ˆp, ˆp 3 and t 8 = { ˆp 1, ˆp + 3 1,w c t 8 1} + Id ˆp1, ˆp, ˆp 3 T 1 = t 7 t 8 and N = 1 { ˆp 1, ˆp, ˆp 3 }(T 1 ). t = (/0 + {{p 1, ˆp 1},{p 3 1, ˆp 3 1},{p+ 1, ˆp+ 1},{p+ 3 1, ˆp+ 3 1},{w ct 1},{w ct 1}} ), 7 8 {( ˆp, ˆp ),( ˆp 3, ˆp 3 ),( ˆp+, ˆp+ ),( ˆp+ 3, ˆp+ 3 )}N 1 and T = t ˆt.

35 The new sizes are now S(t 7 ) = S(t 8 ) = (/0,0,3) = S(t 7), S(T 1 ) = {(/0,0,3) } = S(T 1), S(N 1 ) = ({3 1},{(/0,0,3) },0,8) < S(N 1), S(t ) = ({S(N 1 ) 1},1,6) < S(t ), S(T ) = {S(t ) 1,(/0,0,) 1} < S(T ).

36 Condition Quantitative consistency Let Condition be not fulfilled by T i. The set L := g NC L p P Ti g(p ) ind(g)(p ) = g(p + ) ind(g)(p + ) N i K t is a Presburger set. Construct T := T \ {t} t L using Lemma 5. In the example L is characterized by the following three equations: g(w ct 8 ) = 3g(w ct 7 ), g( ˆp ) g(w c t 7 ) = g( ˆp + ), g( ˆp 3 ) = g( ˆp+ 3 ) g(w c t 8 ). Their solutions are described by the linear set L t = L t L = {[ p ] [ /0 +, 1 ˆp, 1 p+, 1 ˆp+ p, 1 ] [, 31 ˆp, 31 p+, 31 ˆp+ wct, 7 31, w c t 8 3, p, ˆp, p 3, ˆp ]}

37 t = L t {(a,a) a { ˆp, ˆp+, ˆp 3, ˆp+ 3 }}N 1 with S(t ) = ({S(N 1 ) 1},0,3) < S(t ). Adding the witnesses leads to L t {[ p /0 + 1, ˆp 1, p+ 1, ˆp+ 1, w 1 1 ], = [ p 3 1, ˆp 3 1, p+ 3 1, ˆp+ 3 1, w 1 ], [ wct 7, w ]} c t 8, p 3, ˆp, p 3 3, ˆp 3 3, w 3 1 (we omit the witness for /0.) with S(t ) = S(t ) = ({S(N 1 ) 1},0,3). Defining T = t ˆt with S(T ) = S(T ) and N = { ˆp 1, ˆp, ˆp 3 }(T ) with S(N ) = ({3 1, 1},S(T ),0,8) < S(N ) = ({3 1, 1},S(T ),1,4) we get ([ ] t 3 = p { 4, p 3, p+ 4, p+ [w1 ] [ , w ] [ 1, w3 ] [ ]} ) 1, wcˆt 1 {(a,a) a {p,p +,p 3,p+ 3 }}N.

38 Condition 3 Elimination of witnesses Let Condition 3 be not fulfilled by witness w C(N i ) \ (P + T i P T i ). Replace N i by some expression ˆT with R( ˆT ) = R(N i ) (w,w) c t w since for all m L t, we have m(w) = c t (w). Then, we can replace in T := T \ {t} (L t {w} Q\QC(Ni ) (K t \ {N i })) QC(Ni )\{w} ˆT [ ] Example: Consider t with c t = w, p 4, p+ { 5, g Γ t g(w) = 0, K t = {p} (v t j )}, [ ] and c t j = w1, p 6, p+ 7, q 8, q+ { 9, K t j = {q} (u)}.

39 v [ p 04, p+ 06, q 18, q+ 19, p 17, p+ 16, q 8, q+ 9, p 7, p+ 5 ], further- Then t is defined such that c t = [ ] [ ] p more,, 11 p+ p, 01, 1 p+ Γ 11 t and q 8 9 u 6 7 p 4 5 K t = { {p0 } (v 0), {q1 } (u 1), {p1 } (v 1), {q } (u ), {p } (v )}, where p i,q i,v i and u i are replacements caused by disjointness condition in Lemma 6.

40 p 0 q 1 p 1 q p 4 6+x x 6+y y 5 v 0 u 1 v 1 u v The variables x and y illustrate the effect of the periods in Γ t which originate from the (omitted) periods of t j.

41 Condition 4 Elimination of bounded places Condition 4 is decidable by two covering graph constructions for every i: Every node in the covering graph CG (i,+) (CG (i, ), respectively ) has a marking from (N {ω}) P T i ((N {ω}) P+ T i, respectively ). The root of the covering graph CG (i,+) has the marking c t P Ti +ω {p g Γg(p )>0}. For a node in CG (i,+) marked with m, we construct T i with R(T i ) = {g R(T i) g P m} using Lemma 5 as T Ti i := {t {g Lt g P m} t T i }. Compute T i := Ti reacheq(t i ) recursively. For every t T i, add to the covering graph CG (i,+) m := m c t P Ti +{p (c t (p + ) + ω a new node Σ (p + )) p P Ti } g Γ t

42 If m > m for an m on the path from the root to m, then we set m := m +ω(m m ). If for all i a node marked with ω P T i is in CG (i,+) and, analogously, a node marked with ω P+ T i is in CG (i, ), then the Condition 4 is fulfilled. Otherwise, we calculate k := min σ {+, } max min path CG (i,σ) p P Ti max m path m(pσ ) and construct T := T \ {t} U(p) (restrict place p to k). p P Ti Tj,h l 1 U(p) = (L t {p,p + } Q\Q C(Ni ) (K t \ {N i })) QC(Ni )\{p,p + } T k c t (p ),c t (p + ) describes firing sequences in N i with the following property: They start with a marking m 0 with m 0 (p) = j, end with a marking m 1 with m 1 (p) = h, and meanwhile the number tokens on p is always less than l.

43 Example: Let t = (c + Γ ) {p} P {p} P (N i }) with c(p ) = 1, N i = v w t j and t j = (c j + Γ j ) {q} Q {q} Q (u) with c j (p + ) = 1, c j (q ) = 8 and c j (q + ) = 9 q v 8 9 u p w

44 k = 1 firing sequences are restricted to the regular expression ((wv t j ) + v) wv. T0,0 1 and T 1 1,1 a copy of t j. only consist of a copy of v, T 1 1,0 only of a copy of w and T 1 0,1 only of T 0 0,0 = t( P(T 1 0,0 )) correponds to v, T 0 1,1 = T 1 1,0 PT 0 0,0 PT 1 0,1 T 1 1,1 corresponds to wv t j + v and T 1 1,0 = T 1 1,1 PT 0 1,0 = t( P(T 0 1,1 )) PT 1 1,0 PT 0 0,0 Every t in (c + Γ ) {p,p + } Q C(Ni )\{p,p + } T 1 1,0 looks like

45 T 1 1,0 T 1 0,0 q 8 9 u T 1 1,0 T 1 0,0 T 1 1,1

46 Corollary 4 If the conditions 1-4 hold for t, then it holds f g Γ t g + Γ t k (c t + kf) C(t) R(t) This immediately follows from: Lemma 8 If the conditions 1-4 hold for t, then it holds } f g + Γt e (Γ t Γ t ) k {(c t + kf) C(t),(c t + kf + e) C(t) R(t) g Γ t

47 Condition 5 Making the constant firing If Condition 5 is not fulfilled for t then, according to Corollary 4, for f = g, there g Γ exists a (smallest) k such that (c + kf) C(t) R(t). So we decompose L t such that R(L t ) = R(L t + kf) R(c t + jg + (Γ t \ {g}) ). Set g Γ j k T := T \ {t} {t K t = K t,γ t = Γ t c t = c t + kf)} {t j k,g Γ Γ t = Γ t \ {g}) c t = c t + jg)}. The size S(t ) is smaller than S(t) since b 5 is now zero or Γ \ {g} < Γ.

48 The reachability relation for Petri nets with inhibitor arcs Theorem In a Petri-net (P,T,W,I,m 0,m e ) with g N P + p, p P g(p) g(p ) ( t T (p,t) I (p,t) I), we can construct an expression T g such that there is a firing sequence w T with m 0 [w m e if and only if R(T g ) is (= {/0} and) not empty. With Theorem 1 we derive the following: Corollary 5 The reachability problem for a Petri net (P,T,W,I,m 0,m e ) with is decidable. g N P + p, p P g(p) g(p ) ( t T (p,t) I (p,t) I),

49 Example: Start marking: {p 3 3, p 4 }, end marking {p 4 7} t 6 p 3 t 7 3 p 4 3 p 5 t 9 5 p 1 t 8 We find g with{[ g(p 1 ) = 1, g(p ] [) = and g(p 3 )]} = g(p 4 ) = 3 and construct T 1 = {t 6,t 7 } p with R(T 1 ) =, 43 p+, 1 p+ p, 31, 11 p, 31 p+, p+ + Id 41 P, as innermost net of

50 7 p 4 5 t 9 3 p 3 p 5 t 8 p 3 p 4 t 6 3 p 4 p 3 p 1 t7 p This enables the firing sequence w = t 6 t 7 t 7 from [ p 3 1, p ] [ 4 3 to p 4, p ] 4 on the innermost level as, 31 p, 43 p+, 4 p+ R( 4 PT1 (T 1 )) = R(t ) R(T ). Together [ ] p with

51 [ ] p, 5 p+ R(T 3 ) for t 8, we get the firing sequence w = (w)(w)t 8 (w)t 8 (w)t 8 (w)t 8 from [ [ ] p 3, p ] [ 4 7 to p3 5, p ] 4 p on the next level as 3, p 4 7, p+ 3 5, p+ 4 R( PT (T )) = [ ] p R(t 3 ) R(T 3 ). Together with, 31 p+ R(T 45 3 ) for t 9, this enables the firing sequence w = t 9 (w )t9 5 from [ [ ] p 3 3, p ] [ 4 to p4 ] p 7 on the following level as 3 3, p 4, p+ 4 7 R( PT3 (T 3 )) = R(t 4 ) = R(T 4 ).

52 Priority-Multicounter-Automata A priority-multicounter-automaton is a one-way automaton described by the 6- tuple A = (k,z,σ,δ,z 0,E) with the set of states Z, the input alphabet Σ, the transition relation δ (Z (Σ {λ}) {0...k}) (Z { 1,0,1} k ), initial state z 0, the accepting states E Z, the set of configurations C A = Z Σ N k, the initial configuration σ A (x) = z 0,x,0,...,0 }{{} and configuration transition k relation z,ax,n 1,...,n k A z,x,n 1 + i 1,...,n k + i k

53 if and only if z,z Z,a Σ {λ}, (z,a, j),(z,i 1,...i k ) δ, i j n i = 0. Recognized language: L(A) = {w z e E n 1,...,n k N z 0,w,0,...,0 A z e,λ,n 1,...,n k. Alternatively L(A) = {w z 0,w,0,...,0 A z e,λ,0,...,0 }. Using Theorem, we get: Theorem 3 The emptiness problem for priority-multicounter-automata is decidable. The same holds for the halting problem by constructing an automaton which contains its input in the states.

54 Restricted Priority- Multipushdown- Automata Different treatment of one of the two pushdown symbols {0, 1}: A 0 can be pushed to and popped from every pushdown store independently, but a 1 can only be pushed to or popped from a pushdown store if all pushdown stores with a lower order are empty. Restriction: If a 1 is popped from a pushdown store, then a 1 cannot be pushed anymore to this store until it is empty. Theorem 4 The emptiness problem for restricted priority-multipushdown-automata is decidable.

55 This generalizes the result in [JKLP90] that LIN%D 1 (the class of languages generated by linear grammar and deletion of semi Dyck words) is recursive. Conjecture: Decidability still holds in the unrestricted case. Open problem: Pushdown automaton with additional weak counters (without zero-test).