We propose a new method, called Subclass-oriented Dimensionality Reduction with Pairwise Constraints (SODRPaC), for dimensionality reduction. In a high dimensional space, it is common that a group of data points with one class may scatter in several different groups. Current linear semi-supervised dimensionality reduction methods would fail to achieve fair performances, as they assume two data points linked by a must-link constraint are close each other, while they are likely to be located in different groups. Inspired by the above observation, we classify the must-link constraint into two categories, which are the inter-subclass must-link constraint and the intra-subclass must-link constraint, respectively. We carefully generate cannot-link constraints by using must-link constraints, and then propose a new discriminant criterion by employing the cannot-link constraints and the compactness of shared nearest neighbors. The manifold regularization is also incorporated in our dimensionality reduction framework. Extensive experiments on both synthetic and practical data sets illustrate the effectiveness of our method.

Petri net is a powerful modeling tool for concurrent systems. Subclasses of Petri net are suggested to model certain realistic applications with less computational cost. Structurally weakly persistent net (SWPN) is one of such subclasses where liveness is verified in deterministic polynomial time. This paper studies the computational complexity to verify whether a give net is SWPN. 3UNSAT problem is reduced to the problem to verify whether a net is not SWPN. This implies co-NP completeness of verification problem of SWPN.

Petri net is a graphical and mathematical modeling tool for discrete event systems. This paper treats analysis problems of time Petri nets. In this model, a minimal and a maximal firing delays are assigned to each transition. If a transition is 'enabled' it can fire after minimal delay has passed and must fire before maximal delay has elapsed. Since time Petri net can simulate register machines, it has equivalent modeling power to that of Turing machine. It means, however, that most of the analysis problems of time Petri nets with general net structures are undecidable. In this paper, net structures are restricted to a subclass called partially ordered condition (POC) nets and dissynchronous choice (DC) nets. Firing delays are also restricted to satisfy 'static fair condition' which assures chance to fire for all transitions enabled simultaneously. First, a sufficient condition of liveness of time POC net with the static fair condition is derived. Then it is shown that liveness of time DC net with static fair condition is equivalent to liveness of the underlying nontime net. This means that liveness problem of this class is decidable. Lastly, liveness problem of extended free choice (EFC) net is shown to be decidable.

Up to now, the only useful and well-known structural or initial-marking-based necessary and sufficient liveness conditions of Petri nets have only been those of an extended free-choice (EFC) net and its subclasses such as a free-choice (FC) net, a forward conflict free (FCF) net, a marked graph (MG), and a state machine (SM). All the above subclasses are activated only by deadlock-trap properties (i.e., real d-t properties in this paper), which mean that every minimal structural deadlock (MSDL ND=(SD, TD, FD, MoD)) in a net contains at least one live minimal structural trap (MSTR NT=(ST, TT, FT, MoT)) which is initially marked. However, the necessary and sufficient liveness conditions for EFCF, EBCF, EMGEFCFEBCF, AC (EFCFC), and the net with kindling traps NKT have recently been determined, in which each MSDL without real d-t properties was also activated by a new type of trap of trap, i.e., behavioral traps (BTRs), which are defined by introducing a virtual MSTR, a virtual maximal structural trap (virtual STR), a virtual MSDL, and a virtual maximal structural deadlock (virtual SDL) into a target MSDL. In this paper, a structural or initial-marking-based necessary and sufficient condition for local liveness (i.e., virtual deadlock-trap properties) of each MSDL ND s.t. SDST, SDST, SDST (but ND s.t. SDST is dead owing to real deadlock-trap properties) in a general Petri net N is presented by extending that in NKT. Specifically, live minimal behavioral traps (MBTRs) as well as live maximal behavioral traps (BTRs), i.e., virtual deadlock-trap properties, in a general Petri net N are characterized using the real d-t properties of each MSDL ND s.t. SDST for a general Petri net N, which were also obtained by extending the concept of return paths in NKT in connection with an MSDL which contains at least one MSTR and by using the concepts of T-cornucopias and absolute T-cornucopias in a subclass Ñ of N. In other words, BTRs are defined by introducing a virtual MSTR, a virtual STR, a virtual MSDL, and a virtual SDL into a target MSDL without real d-t properties. Additionally, a structural or initial-marking-based necessary and sufficient condition for liveness of a new subclass Nn of a general Petri net N (i.e., a general Petri net without time) is derived, and the usefulness of the obtained results is also discussed.