Petri net theory has been studied for well over half a century Peterson . As a model for concurrency and causality, Petri net model finds a wide range of applications in system specification and verification. Two equivalent formulations of Petri nets, Vector Addition System (VAS) and Vector Addition System with States (VASS), have been investigated in theoretical setting. In VASS system configurations are formulated by vectors on non-negative integers. Computation rules of systems are captured by vectors on integers. A computation is a sequence of legal transitions of configurations. Many system properties can be formalized and checked algorithmically in Petri net model Esparza , among which reachability has proved to be the most challenging one. A variety of problems in language, logic, concurrency can be reduced to VASS reachability problem Schmitz . Some famous problems are related to reachability problem of extended VASS, say branching VASS Rambow ; Schmitz .
Decidability result of the VASS reachability problem is among the most significant theoretical breakthroughs in computer science. The 1970’s saw decidability proofs for low dimensional VASS reachability Hopcroft and Pansiot ; vanLeeuwen  and an incomplete proof of the full VASS reachability problem by Sacerdote and Tenney Sacerdote and Tenney . In the decade from early 1980’s to early 1990’s, a complete proof of the decidability using decomposition technique was discovered by Mayr Mayr , and refined by Kosaraju Kosaraju  and Lambert Lambert . Starting from 2010 Leroux has been studying the problem in a more logic setting Leroux [2010, 2011, 2012, 2013].
The complexity of the reachability problem for VASS is a long standing open problem. For many years Lipton’s EXPSPACE hardness result Lipton 
, announced in 1976, is the only bound we knew. In the last ten years continuing efforts have been made to attack the problem. The achievements can be classified into three categories. Firstly completeness results have been established for low dimensional VASS. Haase, Kreutzer, Ouaknine and Worrell showed that reachability of-dimensional VASS is NP-complete Haase, Kreutzer, Ouaknine and Worrell . In the -dimensional case, Boldin, Finkel, Göller, Hasse and McKenzie proved that the problem is PSPACE complete Blondin et al  and Englert, Lazić and Totzke pointed out that the problem is NL-complete Englert  if unary encoding is used. Secondly new upper bounds of the general problem have been discovered. Leroux and Schmidt have provided an upper bound in Leroux and Schmidt . An improved upper bound is given in Schmitz . In the latest announcement Leroux and Schmidt , Leroux and Schmidt have proved that for the -dimensional VASS problem is in and consequently that the general VASS problem is in . Lastly but not the least, a non-elementary lower bound has been established by Czerwińsky, Lasota, Lazić, Leroux and Mazowiecki Czerwinsky et al . We now know that the general VASS reachability problem is tower hard. This is a gigantic jump from the EXPSPACE hardness.
The reason that KLMST decomposition algorithm is difficult to analyze is that every decomposition step, which converts a sequence of graphs to a longer sequence of graphs, increases the size of thing exponentially in such a way that may create exponentially more decomposition steps. It was pointed out by Müller Muller  that Mahr’s algorithm is not even primitive recursive due to its reference to Karp and Miller’s coverability trees Karp and Miller . The best upper bound that has been obtained so far is that VASS reachability is in Leroux and Schmidt . The crucial observation that has led to the discovery of the primitive recursive upper bound is that a refinement of Lipton’s approach Lipton , which has an exponential space complexity, can be exploited to reduce the dimensionality of VASS instances. Dimension reduction is part of the decomposition algorithm, although its role has not been paid enough attention it deserves. So far all the arguments for the upper bounds are combinatorial.
Building on Leroux and Schmidt’s observation, a purely algebraic analysis of the decomposition technique is presented in this paper. The algebraic approach reveals a fundamental property of the decomposition procedure and the reduction procedure, they are hugely parallel. This parallelism is sufficient for us to derive an elementary bound on the height of the tower function that bounds the complexity of the decomposition algorithm. It follows immediately from this observation and the tower hardness Blondin et al  that VASS reachability is Tower complete.
The rest of the paper is organized as follows. Section 2 states the preliminaries. Section 3 discusses how linear equation systems help characterize paths in vector addition system. Section 4 does the same for local circular paths. Section 5 describes dimension reduction. Section 6 and Section 7 define reduction algorithm and decomposition algorithm respectively. Section 8 gives an account of the well known reachability checking algorithm in our setting. Section 9 establishes the tower completeness. This section is the main contribution of the paper. Section 10 makes a few comments.
Let be the set of natural numbers (nonnegative integers) and the set of integers. Let denote the set of variables for nonnegative integers. We write for elements in , for elements in , and for elements in . For let be the set and be . For a finite set let denote the number of element of .
A multi-set is a function . For each , the positive value denotes the number of occurrence of in . We will apply the standard set theoretical notation to the multi-sets. We write for instance and , pretending that are sets. We write for the underlying set .
We write for -dimensional vectors in , for vectors in , and for vectors in . All vectors are column vectors. For we write for example for the -th entry of . Let and , where is the transposition operator. We write for a finite sequence of vectors and for the length of . For we write for the -th element appearing in .
Recall that the -norm of is . The -norm of is . The -norm of an integer matrix is , and the -norm is .
2.1 Non-Elementary Complexity Class
VASS reachability is not elementary Czerwinsky et al . To characterize the problem complexity theoretically, one needs complexity classes beyond the elementary class. Schmidt introduced an ordinal indexed class of complexity classes and showed that many problems arising in theoretical computer science are complete problems in this hierarchy Schmitz . In the above sequence and . The class is closed under elementary reduction and is closed under primitive recursive reduction. For the purpose of this paper it suffices to say that contains all the problems whose space complexity is bounded by tower functions of the form
where is an elementary function.
We shall call for example a polynomial time problem a time problem, and an exponential space problem a space problem. Alternatively we may say that a problem can be solved in time or space etc.. For simplicity we shall always omit constant factors when making statements about complexity.
2.2 Integer Programming
We shall need a result in integer linear programmingSchrijver . Let be an integer matrix and . The homogeneous equation system of is given by the linear equation system specified by
A nontrivial solution to (1) is some such that . The set of solutions form a monoid . Since the pointwise ordering is a well quasi order on , must be generated by a finite set of nontrivial minimal solutions. This finite set is called the Hilbert base of , denoted by . The following important result is proved by Pottier Pottier , in which is the rank of .
Lemma 1 (Pottier).
for every .
Let . Nonnegative integer solutions to equation system
can be derived from the Hilbert base of the homogeneous equation system . Let be the finite set of the minimal solutions to with , and be the finite set of the minimal solutions to with . A solution to (2) is of the form
where , and is a natural number for each . The following is an immediate consequence of Lemma 1.
for all .
In polynomial space a nondeterministic algorithm can guess a solution and check if it is minimal. Hence the following.
Both and can be produced in space.
2.3 VASS as Labeled Graph
By a digraph we mean a finite directed graph in which multi-edges and self loops are admitted. A -dimensional vector addition system with states, or -VASS, is a labeled digraph where is the set of vertices and is the set of edges. Edges are labeled by elements of , and the labels are called displacements. A state is identified to a vertex and a transition is identified to a labeled edge. We write for states, and its decorated versions for edges. A transition from to labeled is denoted by . If denotes , we write for the transition , and we write for . We write for the -VASS . In the rest of the paper we refer to either as a graph or as a VASS. The input size of a VASS is to be understood as the length of its binary code.
A -path from to with labels is a sequence of edges for some states , often abbreviated to . A special instance of a -path is a -edge , which is nothing but an edge in . Let denote the -th element in the sequence . For simplicity the G-path is sometimes abbreviated to . We shall write for a G-path when the labels are immaterial. A -cycle is a -path such that . A -loop is a -cycle such that are pairwise distinct. The underlying graph of a -path , denoted by , is the subgraph of defined by the edges appearing in .
A Parikh image for is a vector in . We will write for Parikh images. The displacement is defined by . We define the Parikh image by . A Parikh image often specifies, not necessarily uniquely, a -path. For a -path we write for the Parikh image defined by letting be the number of occurrence of in .
A loop class of a VASS is a set of -loops. The loop class of a subgraph of is the set of -loops appearing in . The loop class of a -cycle is .
A loop class can be identified to a set of disjoint strongly connected component (SCC) of , the underlying graph of the loop class. A trip made from a loop class is a trip in one of the SCCs.
The length of a loop is bounded by . There is some polynomial such that the number of -loops is bounded by
Consequently the number of loop classes is bounded by .
Consider the -VASS defined by the following labeled graph.
The -loops are
Let be the -cycle . The loop class of is .
A located state is a state at a location . A one-step trip with label from a located state to a located state , is a -edge rendering true the equality . Such a one-step trip is denoted by . A trip from to is a sequence of one-step trips between located states concatenated one after another in the following manner
Let denote the above trip. The -path of , denoted by , is . The displacement of , denoted by , is . The length of is denoted by and its -th element is denoted by . We say that is circular if . A walk from to is a trip
where . Given a VASS and located states , the reachability problem asks if there is a walk from to . If inputs are promised to be -dimensional, the problem is denoted by .
2.5 Constraint Graph Sequence
Let be a -VASS. A constraint graph (CG) of is a triple such that and is a subgraph of . We call the entry state and the exit state of the CG. When is the trivial graph with one state and no transition, the CG is essentially a state. A constraint graph sequence (CGS) for is of the following form
where with is a CG of for all , and for all . A diagrammatic illustration is given in Figure 1. We say that is a CGS from to . CGSs are the prime objects of all operations of our algorithms. An instance of consists of a VASS and a pair can be seen as a CGS for . A trip of the form
where , is of type for . A walk of type for is called a witness of for .
Given a VASS and two states , there is a polynomial time algorithm that computes the class of the SCCs of the graph . The SCCs are connected by the edges of . One can nondeterministically select a CGS from to by linearizing the SCCs of .
3 Algebraic Characterization
We start with an algebraic characterization of CGSs. The characteristic system of for is an under specification of by equation system. For each CG with , we introduce a vector of variable for and a vector of variable for . We also introduce a vector of variable for the edges of . We think of as specifying a location for , a location for , and a trip from to inside . The equations for the initial and final locations are respectively
Those concerning are the following, where .
For each there is also an equation connecting the exit state of to the entry state of :
In (6) the notation for example is a -dimensional indicator vector whose -th entry is and all the other entries are . We will call equality (6) Euler Condition, which is a necessary and sufficient condition for the existence of a trip that enters in and leaves from . The system consists of (4) and (5), the equations for all a la (6) and (7), and all the equations connecting them a la (8). A solution to is an assignment of nonnegative integers to the variables that renders valid (4) through (8). The system is satisfiable if it has a nontrivial solution. The CGS is satisfiable if its characteristic system is satisfiable.
Let’s see an example. Consider the -dimensional CGS for defined in Figure 2. Let and denote respectively the left CG and the right CG.
Obviously can be reached from in three steps. This is a trip of type for . It is constructed from the minimal solution to the characteristic equation system that assigns to and , to , to and , and to all the other variables. Now consider the CGS for . There are many walks from to . Four minimal solutions to the characteristic equation system are defined below, where only non-zero assignments are specified.
assigns to , , and .
assigns to and to . It also assigns to , , and .
assigns to , , and , to , and to .
assigns to and to . It also assigns to , , , , and .
The solutions , and form legitimate trips. However does not give rise to any trip. The example tells us that while all trips of type for are solutions to , a solution may not define any trip. If a solution does define a trip, it is possible that no trips admitted by the solution stay inside the first quadrant. So a minimal solution may not admit any walk even if it admits a trip.
Suppose is a solution to and . We say that supports if , where is the pointwise ordering. The minimal solution supports a trip of for if it supports the latter seen as a solution. The difference is a solution to the homogeneous characteristic system obtained from by replacing the constant terms by . The equations are
A solution to is a nontrivial assignment of nonnegative integers to the variables rendering valid (9) through (13). The Euler Condition (11) guarantees that a solution consists of one or more circular trips inside each of the CGs .
In the above example the assignment that maps , , and onto and every other variables onto is a minimal solution to the homogeneous system , in other words . This solution is useless because none of , , and defines any trip for . A useful homogeneous solution is defined by the -cycle whose displacement is and the -cycle whose displacement is . The useless solutions suggest to introduce the following terminology.
A set is connected over if the summation defines a trip of type for .
When is clear from context, we say that is connected. Let denote the class of the connected subsets of over . The largest element of is . Let . We call the principal solution to over . According to Pottier Lemma and its corollaries, every trip of type for supported by is a solution of the form
where and is a positive integer.
We write for the vector of natural number assigned to by a solution to (). The notations are defined analogously. If , defines some circular trip(s) in . If , defines a trip from the entry state to the exit state and possibly also some detached circular trip(s) in . The (circular) trips defined by are not unique, which will not be an issue. Whenever we say a trip defined by , we mean any trip defined by . In fact we will see as a (circular) trip with possibly some additional disjoint circular trips. With these notations we introduce the notion of -fixed component of location.
The location of the entry state , respectively the exit state , is -unfixed at if , respectively ; it is -fixed at if , respectively .
If the location of the entry state is -fixed at , then every trip of type for supported by passes at a location whose -th component must be . If the location of the exit state is -fixed at , then every trip of type for supported by passes at a location whose -th component must be . The following notations will be used.
To define the notion of fixed edges we define Hilbert solution as . The Hilbert solution tells us a lot about the shape of the trips of type for .
An edge in the CG is unbounded if ; it is bounded if .
In the example of Figure 2, all edges in the CGs are unbounded.
A trip of type for supported by some must pass a bounded edge for a fixed number of time specified by . All bounded edges appearing in a CG can be removed by unfolding, producing many smaller CGs, see Section 7 for precise definition. CGs consisting of only unbounded edges of play a key role in KLMST algorithm, hence the following definition.
A CG is unbounded if every edge in is unbounded. A CGS is unbounded if is unbounded for all . An unbounded CGS is also called an -CGS.
4 Local Circular Path
We make a simple yet important observation about partially circular trips in this section. Suppose is a -dimensional VASS and . An -circular path is a -cycle
such that and for all . We intend to understand an -circular path in terms of those of minimal length. For that purpose let’s introduce the homogeneous system defined by the following equations.
where (15) stands for a collection of equations indexed by . The equation (14) is Euler Condition. The system does not specify anything about locations. It only specifies the edges that appear in an -circular path. However it is normally an under-specification of -circular paths. This is because a solution to may consist of two -cycles with disjoint sets of state. For example and such that and and , and moreover for all , may constitute a solution to . But neither -cycle is an -circular path if for some . Notice that by definition a minimal solution cannot contain two disjoint -circular paths.
An -circular path is minimal if it does not contain any proper sub--path that is an -circular path. Evidently an -circular path can be decomposed into a multi-set of minimal -circular paths,
where is the set of the minimal -circular paths and for each . We are interested in representing an -circular path by a set of minimal solutions to . A set of -circular paths are connected if their underlying graphs form a connected subgraph of . A set of minimal solutions to are connected if the -circular paths defined by the solutions are connected. Notice that here connectivity is equivalent to strong connectivity because we are dealing with -cycles.
An -circular path is -equivalent to a set of -circular path if for all . It is easy to see that a minimal -circular path is -equivalent to a connected set of minimal solutions to if the following equality holds for all ,
for all , where is a connected set of minimal solutions to and is a positive integer. The connected set gives rise to a -cycle in which the -cycle defined by each is repeated times. By Pottier Lemma the minimal solutions to are bounded by for some polynomial . There are at most minimal solutions. It follows that for each the difference between the minimal value and the maximum value of the -th component in the -cycle is bounded by
for a polynomial . Notice that is the size of the VASS .
In Section 5 we will make use of circular trips whose underlying -cycles are -circular paths. We shall call the following trip
an -circular trip if is an -circular path. To proceed we need the following definition.
Suppose and . The cartesian product is the -space if for and for . A local space is the -space for some .
An -circular trip in the form of (20) is grounded if it lies in the -space and for each there is some such that . In the following diagram the curly cycles represent -circular trips defined by minimal solutions and the thick curly cycles represent the ones repeated at least once. These curly cycles are connected to form a grounded -circular trip.
Every -circular trip is -equivalent to a grounded -circular trip.
5 Coverability and Pumpability
Pumpability is a fundamental idea in KLMST algorithm. A crucial observation made by Leroux and Schmidt recently is that pumpability is subsumed by coverability Leroux and Schmidt . We give an account of this important connection between coverability and pumpability below. We start by introducing a useful terminology.
Let be a CGS and be a solution to . A trip in is admitted by if it is a trip in the graph .
We say that a trip is admitted in if it is admitted by , where is clear from context.
Fix an -CGS and some . Let . A trip
is a forward -walk in if it is admitted by and for every and every the inequality holds. Let . We say that is forward -coverable in if there is a forward -walk