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Regarding the set of all classes allowing nop , we leave synthesis complexity open for the eight remaining cases of ( nop inp , [ used ]), ( nop out , [ free ]), ( nop set res ) and ( nop swap ) extended with at least one of set and res |
Key words and phrases: boolean Petri nets, labeled transition systems, net synthesis, types of nets, NP-completeness 1991 Mathematics Subject Classification: Software system structures [MATH] Petri nets, Theory of computation [MATH] Problems, reductions and completeness |
1. Introduction This paper contributes to the analysis of the computational complexity of boolean Petri net synthesis as a function of the specific net class. While the efficient algorithms developed in this paper attack the synthesis problem itself, the proofs for intractable synthesis cases turn to feasibility , the ... |
In NP-completeness proofs, we entirely detach ourselves from the notion of Petri nets. In particular, we use the well known equality between feasibility and the conjunction of the state separation property (SSP) and the event state separation property (ESSP) |
, which are solely defined on input TSs. ESSP is also known for its connection to language viability , meaning that a TS [MATH] has the ESSP if and only if there is a [MATH] -net having the same transitional behavior but not necessarily the same states as [MATH] The presented polynomial time reductions translate the NP... |
into the ESSP for each of the considered 84 boolean net classes. Hence, deciding language viability is NP-complete in all these cases. As we also make sure that given boolean expressions [MATH] are transformed to TSs [MATH] where the ESSP relative to the considered class implies the SSP, we always show the NP-completen... |
While this paper ignores the 128 practically less relevant nop -free classes, it does turn towards the complexity analysis of 36 of the remaining 44 types of nets allowing nop For the 16 extensions of ( nop swap ) with a subset of ( inp out used free ) we sketch how a generalization of Schmitt’s approach |
leads to a polynomial time synthesis algorithm. For the other 20 classes we provide our own polynomial time synthesis algorithm. |
Although we have to leave synthesis complexity open for eight nop -afflicted classes, we nevertheless discuss some of their properties and the consequent difficulties in the conclusions. |
For the sake of readability, we have moved all technical proofs to separate sections at the end of this paper. 2. Preliminary Notions |
This section provides short formal definitions of all preliminary notions used in the paper. For a detailed introduction into the field of Petri net synthesis, we propose the excellent monograph of Badouel, Bernardinello and Darondeau |
Here, a boolean Petri net [MATH] is given by finite and disjoint sets [MATH] of places and [MATH] of transitions, an initial marking [MATH] , and a flow function [MATH] assigning an interaction [MATH] of |
[MATH] to every pair of place [MATH] and transition [MATH] The interactions [MATH] are binary partial functions [MATH] as defined in the listing of Figure For readability, we group interactions by [MATH] [MATH] [MATH] , and [MATH] |
The meaning of a boolean net is to realize a certain behavior by cascades of firing transitions. In particular, a transition [MATH] can fire at a marking [MATH] if interaction [MATH] is defined on [MATH] for all [MATH] and [MATH] is defined on [MATH] for all [MATH] By firing, [MATH] produces the next marking [MATH] tha... |
Given a boolean net [MATH] , its behavior is captured by a finite automaton [MATH] , called the transition system (TS, for short) of [MATH] The state set of [MATH] consists of all markings that, starting from initial state [MATH] , can be reached by a cascade of firing transitions. For every reachable marking [MATH] an... |
Subsets [MATH] define types of nets , subclasses of boolean nets limited to the respective interactions. Hence, in a [MATH] -net, [MATH] for all contained places [MATH] and transitions [MATH] It is clear for [MATH] that the class of [MATH] -nets is a subset of the [MATH] -nets. Notice that [MATH] contains all possible ... |
Boolean net synthesis for a class [MATH] is going backwards from input TS [MATH] to the computation of a [MATH] -net [MATH] with [MATH] isomorphic to [MATH] , if such a net exists. In contrast to [MATH] , the abstract states [MATH] of [MATH] miss any information about markings they stand for. Accordingly, the events [M... |
To prove net synthesis of [MATH] -nets NP-hard, we show the NP-completeness of the corresponding decision version: [MATH] -feasibility is the problem to decide the existence of a [MATH] -net [MATH] with [MATH] isomorphic to the given TS [MATH] On that account, an input TS [MATH] is considered as a directed labeled grap... |
occurs at a state [MATH] , denoted by [MATH] , if [MATH] is defined. TSs in this paper are deterministic by design as their state transition behavior is given by a function. TSs are also required to make every state reachable from [MATH] by a directed path. Aside from that, a TS [MATH] can be simple , which prohibits [... |
To describe feasibility without referencing the sought net [MATH] , we subsequently introduce the state separation property SSP , for short) and the event state separation property ESSP , for short) for TSs, which in conjunction are equivalent to feasibility. These notions require to follow the interpretation of |
, which sees a type of nets [MATH] as a template TS [MATH] for all synthesizable TSs of that class. Leaving out an initial state, they define [MATH] [MATH] and [MATH] for all [MATH] and all [MATH] Based on this, a [MATH] -region of given [MATH] is a pair [MATH] of the support |
[MATH] and the signature [MATH] where every transition [MATH] of [MATH] leads to a transition [MATH] of [MATH] While a region divides [MATH] into the two sets [MATH] for [MATH] , the events are cumulated by [MATH] for all available interactions [MATH] We also use [MATH] for [MATH] |
For a TS [MATH] and a type of nets [MATH] , a pair of states [MATH] is separable for [MATH] if there is a [MATH] -region [MATH] such that [MATH] Accordingly, [MATH] has the SSP for [MATH] if all pairs of distinct states from [MATH] are separable. Secondly, an event [MATH] is called inhibitable at a state [MATH] if ther... |
that a TS [MATH] is [MATH] -feasible, that is, there exists a [MATH] -net [MATH] with [MATH] isomorphic to [MATH] , if and only if [MATH] has both, the SSP and the ESSP for [MATH] Moreover, |
also states that a TS [MATH] has the ESSP for [MATH] if and only if [MATH] is [MATH] -language viable This means, there a [MATH] -net [MATH] with [MATH] language equivalent to [MATH] where every event sequence in [MATH] traverses a [MATH] -rooted directed path in [MATH] if and only if it does in [MATH] |
SSP and ESSP can also be seen as decision problems. Moreover, an SSP atom for [MATH] is to decide for given [MATH] whether the states [MATH] of [MATH] are [MATH] -separable. Similarly, an ESSP atom for [MATH] is to answer for [MATH] if event [MATH] is [MATH] -inhibitable at state [MATH] of [MATH] |
While being introduced to assist in proofs for hard synthesis, regions, SSP and ESSP are also construction tools for boolean nets. In fact, having a region set [MATH] for [MATH] that solves all its SSP and ESSP atoms with respect to some type of nets [MATH] , one can construct a [MATH] -net [MATH] on place set [MATH] ,... |
[MATH] is isomorphic to the state graph of [MATH] Hence, if we can efficiently compute [MATH] then [MATH] is synthesizable in polynomial time. |
Types of nets [MATH] and [MATH] have an isomorphism [MATH] if [MATH] is a transition in the template TS [MATH] if and only if [MATH] is one in the template TS [MATH] We benefit from isomorphisms mapping nop to nop swap to swap inp to out set to res used to free , and vice versa: |
Lemma 2.1 (Without proof) If [MATH] and [MATH] are isomorphic types of nets then a TS [MATH] has the (E)SSP for [MATH] if and only if [MATH] has the (E)SSP for [MATH] |
3. A Reduction Scheme yields the NP-completeness of Feasibility for 77 Boolean Petri Net Classes This section presents our main result: |
Theorem 3.1 Let [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] and [MATH] Deciding [MATH] -feasibility as well as [MATH] -language viability for modest transition systems is NP-complete if |
(1) [MATH] for [MATH] and [MATH] (2) [MATH] or [MATH] , or (3) [MATH] for [MATH] and non empty [MATH] The remainder of this section is devoted to the proof of the main theorem. That input is restricted to modest TSs shows that the problem is intrinsically difficult and that the hardness is not hidden in special structu... |
In total, Theorem 3.1 covers 77 classes. The first condition hits four classes for every set [MATH] [MATH] , and [MATH] The two cases of the second one describe 32 classes each, but they intersect in the eight supersets of [MATH] Condition three brings nine classes. All three conditions cover different classes. Althoug... |
to show the hardness of feasibility in NP for elementary net systems. On the other hand, it is probably impossible to show hardness in NP for all considered classes [MATH] at the same time. Here, we manage to boil it down to six reductions that are all based on one scheme using the NP-complete cubic monotone one-in-thr... |
Starting from the common construction principle, we can choose one of our six reductions by a turn-switch [MATH] In every switch position [MATH] , the chosen reduction works for multiple interaction sets based on mutually shared interactions and isomorphisms. |
Before we can set out the details of our concept, the following subsection introduces our way of easily generating and combining gadget TSs for our NP-completeness proofs. |
3.1. Unions of Transition Systems If [MATH] are TSs with pairwise disjoint states (but not necessarily disjoint events) we say that [MATH] is their union By [MATH] , we denote the entirety of all states in [MATH] and [MATH] is the aggregation of all events. For a flexible formalism, we allow to build unions recursively... |
We lift the concepts of regions, SSP, and ESSP to unions [MATH] as follows: [MATH] -region [MATH] of [MATH] consists of [MATH] and [MATH] such that, for all [MATH] , the projections [MATH] and [MATH] provide a region [MATH] of [MATH] Then, [MATH] has the SSP for [MATH] if for all different states [MATH] of the same TS ... |
To merge a union [MATH] into a single TS, we define the joining [MATH] , which depends on the type of nets [MATH] For our NP-completeness scheme, we require one basic construction [MATH] and an enhanced construction [MATH] If [MATH] are the initial states of [MATH] ’s TSs then [MATH] and [MATH] are TSs with additional ... |
[EQUATION] Hence, [MATH] puts the connector states into a chain of the events from [MATH] and links the initial states of TSs from [MATH] to this chain using events from [MATH] The enhancement [MATH] is obtained from [MATH] by extending [MATH] with additional reverse transitions. Notice that [MATH] and [MATH] are modes... |
Lemma 3.2 Let [MATH] be a type of nets and [MATH] be a union of TSs [MATH] where, for every event [MATH] in [MATH] , there is at least one state [MATH] in [MATH] with [MATH] Moreover, define the joining |
(1) [MATH] if [MATH] and [MATH] and, otherwise, (2) [MATH] if [MATH] [MATH] and for all [MATH] there is exactly one outgoing and one incoming arc at the initial state [MATH] of the TS [MATH] , both labeled with the same event [MATH] occurring at no further arcs in [MATH] |
If [MATH] is defined then [MATH] has the [MATH] -(E)SSP if and only if [MATH] has the [MATH] -(E)SSP. Proof 3.3 If Projecting a [MATH] -region separating [MATH] and [MATH] , respectively inhibiting [MATH] at [MATH] , in [MATH] to the component TSs yields a [MATH] -region separating [MATH] and [MATH] , respectively inhi... |
Only if In the following, if [MATH] has inp and at least one of [MATH] let [MATH] and enter be any of the available interactions from [MATH] Otherwise, if the second condition holds, define [MATH] and let test be any interaction of [MATH] [MATH] -region [MATH] of [MATH] separating [MATH] and [MATH] , respectively inhib... |
[EQUATION] Notice that a [MATH] -region [MATH] like this, which inherits the property of inhibiting [MATH] at [MATH] from [MATH] , do also inhibit [MATH] at all connector states, since [MATH] This has the following consequence: As every event [MATH] has at least one state [MATH] with [MATH] , the ESSP of [MATH] implies... |
For the (E)SSP of [MATH] it is subsequently sufficient to analyze (event) state separation concerning just the connector states and events. To separate the state [MATH] from all the other states of [MATH] we simply define the [MATH] -region [MATH] where only [MATH] and where the signature of all events is nop except fo... |
Hence, taking [MATH] over all [MATH] solves the remaining SSP atoms. Moreover, for [MATH] , the regions [MATH] also inhibit [MATH] and [MATH] at all states, which solves the rest of the ESSP atoms, too. |
Hence, it remains to inhibit the [MATH] - and [MATH] -events in [MATH] For [MATH] the inhibition of [MATH] at all relevant states of [MATH] can be done as follows: If [MATH] [MATH] ), we define the region [MATH] that includes (excludes) exactly the states [MATH] and defines a test signature for [MATH] , a swap signatur... |
3.2. The General Reduction Scheme Our general scheme can be set up to a specific reduction by the turn switch [MATH] In each of its six positions, [MATH] covers a whole collection of net classes. Therefore, we simply understand the positions [MATH] as the type sets managed by the respective reductions: |
[EQUATION] The input to our scheme is the switch position [MATH] and a cubic monotone boolean [MATH] -CNF [MATH] , a set of negation-free [MATH] -clauses over the variables [MATH] such that every variable is a member of exactly three clauses. According to |
, it is NP-complete to decide if [MATH] has a one-in-three model, that is, a subset [MATH] of variables that hit every clause exactly once, which means [MATH] for all [MATH] The result is a union [MATH] of modest gadget TSs with the following properties: |
(1) The variables [MATH] are a subset of [MATH] , the union events. (2) There is a key state [MATH] and a key event [MATH] with [MATH] |
(3) For every [MATH] , there is a [MATH] -region inhibiting [MATH] at [MATH] if and only if [MATH] has a one-in-three model [MATH] |
(4) For every [MATH] , the [MATH] -inhibitability of [MATH] at [MATH] implies that all ESSP atoms and all SSP atoms of [MATH] are solvable. |
A polynomial time reduction scheme with these properties proves Theorem 3.1 as follows: Condition makes [MATH] -ESSP and [MATH] -feasibility the same problem for [MATH] Thus, feasibility is reduced to language viability and we subsequently concentrate on the NP-completeness proof for this problem. In fact, by a one-in-... |
To present an example of our reduction, Figure shows [MATH] for the [MATH] -CNF [MATH] built of the six clauses [MATH] [MATH] [MATH] [MATH] [MATH] , and [MATH] Ignoring the connector states and transitions, the figure also shows the complete union [MATH] together with a [MATH] -region that inhibits the key event [MATH]... |
In the following, let [MATH] be turned to a position in [MATH] and [MATH] be a type of nets from [MATH] To refer to events and states, the generic description of the [MATH] uses lowercase English letters for states and regular events, uppercase [MATH] for events that represent the variables of [MATH] and lowercase Gree... |
For structure, [MATH] is subdivided into a key union [MATH] , which depends on [MATH] and less on [MATH] (in fact, only on the number [MATH] of clauses), and a translator union |
[MATH] , which depends on [MATH] and less on [MATH] While the key union provides [MATH] and [MATH] together with other helpful events, the translator union represents [MATH] The sense in splitting [MATH] is the following: If we have a [MATH] -region of [MATH] that inhibits [MATH] at [MATH] , it is by definition decompo... |
In the next step, we give an abstract description of [MATH] There, we use the variables [MATH] as events. For each [MATH] , we also add a corresponding helper event [MATH] Then, every clause [MATH] is implemented as a translator |
[MATH] , a subunion of [MATH] The TS [MATH] that builds [MATH] with its three copies for [MATH] is shown in Figure .1 for [MATH] and in Figure .5 for [MATH] We let any choice of [MATH] select one TS [MATH] and also define [MATH] and [MATH] to address the other two TSs in a specific consecutive manner. |
The foundation of translation is to make sure for the three variable events [MATH] of [MATH] that in an indicator region exactly one of them can get a signature different from nop Taken across all translators, this implements the requirements of a one-in-three model [MATH] within the union [MATH] To effect this behavio... |
[EQUATION] on states [MATH] , respectively [MATH] , containing transitions labeled with [MATH] [MATH] [MATH] , respectively [MATH] [MATH] [MATH] , at the given positions. Notice, while cross checking with Figure , that states can be the same if they are linked by dots, like [MATH] and [MATH] both represent [MATH] , or ... |
For an indicator region [MATH] of [MATH] , the basis of our construction is a synchronization of certain states on these paths. Firstly, this concerns for all [MATH] the opposing states [MATH] and [MATH] on the two paths, that is, [MATH] Secondly, across all translators, we synchronize all initial and all terminal stat... |
[EQUATION] of the primal paths and make sure that the support of initial differs from terminal states. As part of the interface, the placeholders [MATH] and [MATH] , called materializers , have a significant role in this second synchronization process. Playing together, all materializers [MATH] of [MATH] also synchroni... |
[EQUATION] across the three TSs of [MATH] Recall that the indicator region [MATH] maps the TSs of [MATH] to the type of nets TS [MATH] This includes [MATH] and [MATH] which become paths in TS [MATH] traversing along the states [MATH] By the previous synchronization of states, the events [MATH] on the primal path, respe... |
Figure sketches the first four cases and teaches us that three state changes on the mapped paths come with [MATH] Hence, if swap is not available, like in [MATH] and [MATH] , an indicator region implements the one-in-three behavior. In the other four switch positions [MATH] , we simply have to prevent the indicator reg... |
[MATH] to hinder these troubling swap assignments. See also Figure to get the idea. The general reduction idea in mind, Subsection 3.3 introduces the details of the translator union [MATH] for every [MATH] Subsection 3.4 does the same for [MATH] |
Before we start with our construction, we need some minor tools: Firstly, we use so called generators [MATH] in [MATH] and in [MATH] A template [MATH] serves as a blueprint for freezer gadget TSs as follows: For [MATH] and symbols [MATH] the template generates TS [MATH] from Figure .2 with states [MATH] and placeholder... |
Lemma 3.4 (Without proof) For [MATH] and symbols [MATH] let [MATH] be the generated TS. For a [MATH] -region [MATH] of [MATH] the following conditions hold: |
(1) If [MATH] then [MATH] and [MATH] (2) If [MATH] then [MATH] and [MATH] (3) If [MATH] then [MATH] (4) If [MATH] then [MATH] As second minor notion, we introduce forward-backward transitions [MATH] which simply express the presence of both, [MATH] and [MATH] Third notion are blanc events. Some events occur only once i... |
3.3. Details of the Translator Union This section defines the translator union [MATH] for all cubic monotone boolean [MATH] -CNF [MATH] with [MATH] clauses and every [MATH] The union [MATH] consists of translator subunion [MATH] for [MATH] to cover all clauses in [MATH] and a freezer [MATH] to prevent unwanted swap The... |
Figure .1 defines the gadget TS [MATH] with initial state [MATH] for [MATH] and Figure .5 for [MATH] where the initial state is [MATH] The latter contains six events [MATH] , thus, 18 events for [MATH] Figure .5 just uses [MATH] for [MATH] to preserve clarity. Moreover, for [MATH] , every placeholder [MATH] becomes [MA... |
Notice that Figure .1 and .5 have several colored areas. They demonstrate the [MATH] -related fractions of the three possible indicator regions, later defined in detail by Lemma 3.6 The red region fraction stands for [MATH] , the green for [MATH] and blue for [MATH] In all three settings, the states taking part in the ... |
Figure .2 defines the generator template [MATH] with initial state [MATH] and shows the support of a region where [MATH] While the freezers [MATH] are empty, this template creates the freezers [MATH] and [MATH] The TS [MATH] with initial state [MATH] in Figure .3 builds the freezer [MATH] and TS [MATH] with initial sta... |
The following lemma provides the condition of the interface between [MATH] and [MATH] that is required in an indicator region of the translator union. Aside from [MATH] , the interface consist of [MATH] [MATH] and, for [MATH] and [MATH] [MATH] |
Lemma 3.5 If [MATH] is a cubic monotone boolean [MATH] -CNF with [MATH] clauses, [MATH] a turn switch position for our reduction scheme, [MATH] a type of nets managed by [MATH] and [MATH] [MATH] -region of [MATH] where one of the conditions |
(1) [MATH] [MATH] [MATH] and [MATH] (2) [MATH] [MATH] [MATH] and [MATH] , or (3) [MATH] [MATH] [MATH] [MATH] and [MATH] holds, then [MATH] is an indicator region, meaning [MATH] is a one-in-three model of [MATH] |
The proof of Lemma 3.5 is rather technical and has therefore been moved to Section Next, we have to be able to go the other way around, that is, we need to construct an indicator region [MATH] for any given one-in-three model [MATH] of [MATH] It is important that, on the interface, [MATH] is compatible with a key regio... |
For given [MATH] with [MATH] -clauses and one-in-three model [MATH] , our approach is as follows: We first define for every clause [MATH] the selector [MATH] by [MATH] Hence, [MATH] is the index of the unique variable in [MATH] that is part of the model. Again, [MATH] and [MATH] are the [MATH] -continuations of [MATH] ... |
[MATH] and [MATH] Based on this, we simply define [MATH] for all [MATH] and all [MATH] For a transition [MATH] of [MATH] ,we can then set [MATH] if and only if [MATH] is not in [MATH] Hence, to extend the support with an appropriate signature we only have to worry about these remaining events. Firstly, the idea is to, ... |
Lemma 3.6 (Without proof) For every cubic monotone boolean [MATH] -CNF [MATH] with one-in-three model [MATH] , every [MATH] , every [MATH] and [MATH] , we get an indicator [MATH] -region [MATH] for [MATH] with [MATH] and |
[EQUATION] The lemma does not need a proof, as we only need to verify for every transition [MATH] of [MATH] that there is mapped transition [MATH] in TS [MATH] for every [MATH] |
3.4. Details of the Key Union This subsection defines the key union [MATH] for all numbers [MATH] of clauses and every [MATH] In particular, [MATH] consists of the head |
[MATH] , the duplicator [MATH] , the generator [MATH] , and the freezer [MATH] How these ingredients are constructed depends on [MATH] Firstly, for [MATH] we leave the generator empty and proceed as follows: |
(1) The head [MATH] is chosen as depicted in Figure .1. It provides key event [MATH] and key state [MATH] as well as the full interface [MATH] |
(2) The duplicator [MATH] is constructed from generator templates. It provides the events [MATH] that, for a key region, receive nop and therefore synchronize the head states [MATH] and [MATH] for all [MATH] |
(3) For a key region, the freezer [MATH] assures that [MATH] is assigned inp or out Moreover, it prevents all events of [MATH] and of [MATH] from receiving swap We firstly let [MATH] using [MATH] from Figure .2 and [MATH] from Figure .3. Then, [MATH] is build of [MATH] from Figure .2, [MATH] from Figure .6, as well as ... |
At this point, one may notice that [MATH] and [MATH] actually transform input [MATH] into the same TS and thus, could be consolidated into one switch position. But since there are differences in the constructed regions as defined in Figure , we keep the two switch positions distinguished to make our argumentation simpl... |
For [MATH] we create different key union ingredients as follows: (1) Here [MATH] consists of multiple TSs [MATH] from Figure .4. The head again introduces [MATH] , but here [MATH] is the key state and only [MATH] is provided to the interface. |
(2) The duplicator [MATH] consists of multiple TSs [MATH] from Figure .8. It provides [MATH] to the interface and prevents these events from swap in key regions. |
(3) The generator [MATH] , made of multiple [MATH] from Figure .9, provides [MATH] for the interface and prevents the respective events from swap in a key region. |
(4) The freezer [MATH] consists of the TSs [MATH] from Figure .5, Figure .6 and Figure .7 and provides the interface events [MATH] In a key region, the freezer makes sure that [MATH] is assigned swap if and only if [MATH] gets swap and, furthermore, enforces nop or swap onto event [MATH] , which synchronizes some state... |
The following lemma establishes the interface compatibility of all key-regions with all inhibitor regions as demanded in Lemma 3.5 Moreover, it shows the existence of a key region that is even compatible with the regions from Lemma 3.6 |
Lemma 3.7 Let [MATH] and [MATH] If [MATH] is a [MATH] -key region of [MATH] , that is, where [MATH] is inhibited at the key state, then |
(1) either [MATH] [MATH] and [MATH] or [MATH] [MATH] and [MATH] in case of [MATH] and (2) [MATH] [MATH] [MATH] and [MATH] if and only if [MATH] for [MATH] |
Furthermore, we can always create a [MATH] -key region [MATH] for [MATH] with (1) [MATH] [MATH] and [MATH] if [MATH] or (2) [MATH] [MATH] and [MATH] , otherwise. |
The proof of Lemma 3.7 is again very technical and, thus, can be found in Section The following lemma connects the functionalities of [MATH] and [MATH] for all [MATH] |
Lemma 3.8 If [MATH] is a cubic monotone boolean [MATH] -CNF with [MATH] clauses, [MATH] and [MATH] then [MATH] is [MATH] -feasible if and only if [MATH] has a one-in-three model. |
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