A generation apparatus generates gain vectors for calculating cumulative expected gains for a transition model in which transition from a current state to a next state occurs in response to an action. The apparatus includes: an acquisition section that acquires gain vectors for a time point next to a target time point that includes cumulative expected gains for and after the next time point for each state at the next time point; a first determination section that determines a value of a transition parameter used for transitioning from the target time point to the next time point, from a valid range of the transition parameter, based on the cumulative expected gains obtained for the gain vectors for the next time point; and a first generation section that generates gain vectors for the target time point from the gain vectors for the next time point, using the transition parameter.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A generation apparatus for generating gain vectors for a transition model, the apparatus comprising: an acquisition section that acquires gain vectors for a next time point after a target time point, said gain vectors including a cumulative expected gains obtained at and after the next time point for each state at the next time point; a first determination section that determines a value of a transition parameter used for transitioning from the target time point to the next time point, from a valid range of the transition parameter, based on cumulative expected gains obtained from the gain vectors at the next time point; and a first generation section that generates gain vectors for the target time point from the gain vectors for the next time point, using the transition parameter, wherein the gain vectors are used to calculate cumulative expected gains in which transition from a current state to a next state occurs in response to an action. 2. The generation apparatus according to claim 1, wherein the first determination section determines a value of the transition parameter for which the cumulative expected gains obtained from the gain vectors at the next time point become equal to or less than a predetermined reference. 3. The generation apparatus according to claim 1, wherein the first determination section determines a value of the transition parameter that minimizes the cumulative expected gains obtained from the gain vectors for the next time point. 4. The generation apparatus according to claim 1, further comprising an initialization section that initializes gain vectors for a future time point; wherein the generation apparatus generates the gain vectors for the target time point, going back from the future time point. 5. The generation apparatus according to claim 1, wherein the acquisition section acquires a set of gain vectors for the next time point that includes at least one gain vector for the next time point; the first determination section determines a value of the transition parameter for each gain vector included in the set of gain vectors for the next time point; and for each gain vector included in the set of gain vectors for the next time point, the first generation section generates a gain vector for the target time point using the transition parameter and adds the gain vector to a set of the gain vectors for the target time point. 6. The generation apparatus according to claim 1, wherein the first determination section determines a transition probability from each state at the target time point to each state at the next time point, from a valid range of the transition probability. 7. The generation apparatus according to claim 6, wherein the first determination section determines the transition probability by linear programming, from the valid range of the transition probability, the range being expressed by a linear inequality of the transition probability. 8. The generation apparatus according to claim 6, wherein the first determination section determines the valid range of the transition probability as being from a reference value up to a constant multiple of a reference value. 9. The generation apparatus according to claim 5, further comprising an elimination section that eliminates a gain vector that does not maximize a value within a probability distribution range of each state, from the set of the gain vectors for the target time point generated by the first generation section. 10. The generation apparatus according to claim 9, wherein the elimination section eliminates a gain vector that does not maximize a value of the cumulative expected gains in a predetermined probability distribution within the range of probability distribution of each state, from the set of the gain vectors for the target time point generated by the first generation section. 11. The generation apparatus according to claim 1, wherein, in response to each of multiple actions performed at the target time point, the first generation section generates the gain vectors for the target time point based on immediate gains expected from a state transition that occurs in response to the action in each state and cumulative expected gains in a destination state of the gain vectors for the next time point. 12. The generation apparatus according to claim 1, wherein said apparatus is implemented by a program of instructions executable by a computer tangibly embodied in one or more computer readable program storage devices. 13. A selection apparatus that selects an action in a transition model, the apparatus comprising: a set acquisition section that acquires a set of gain vectors for a target time point that include a cumulative expected gains obtained for and after the target time point, for each state at the target time point; a probability acquisition section that acquires an assumed probability of being in each state at the target time point; a selection section that selects a gain vector from the set of gain vectors based on the set of gain vectors and the assumed probability; an output section that selects and outputs an action corresponding to the selected gain vector; a second determination section that determines a value of a transition parameter used to transition from the target time point to a next time point, from a valid range of the transition parameter; and a second generation section that generates an assumed probability of being in each state at the next time point after to the target time point, using the transition parameter, wherein a transition from a current state to a next state occurs in response to an action. 14. The selection apparatus according to claim 13, wherein the second determination section determines a value of the transition parameter for which the cumulative expected gains obtained from the selected gain vector become equal to or less than a predetermined reference. 15. The selection apparatus according to claim 13, wherein the second determination section determines a value of the transition parameter that minimizes the cumulative expected gains obtained from the selected gain vector. 16. The selection apparatus according to claim 13, further comprising a generation apparatus that generates gain vectors for calculating cumulative expected gains for the transition from a current state to a next state, wherein the set acquisition section acquires a set of gain vectors generated by the generation section. 17. The selection apparatus according to claim 13, wherein said apparatus is implemented by a program of instructions executable by a computer tangibly embodied in one or more computer readable program storage devices. 18. A method of generating gain vectors for calculating cumulative expected gains in a transition model, the method comprising: acquiring gain vectors for a time point next to a target time point that include cumulative expected gains for and after the next time point for each state at the next time point; determining a value of a transition parameter used for transitioning from the target time point to the next time point, from a valid range of the transition parameter, based on the cumulative expected gains from the gain vectors for the next time point; and generating the gain vectors for the target time point from the gain vectors for the next time point, using the transition parameter, wherein a transition from a current state to a next state occurs in response to an action. 19. A method of selecting an action in a transition model, the method comprising: acquiring a set of gain vectors for a target time point that include cumulative expected gains for and after the target time point, for each state at the target time point; acquiring an assumed probability of being in each state at the target time point; selecting a gain vector from the set of gain vectors based on the set of gain vectors and the assumed probability; selecting and outputting an action corresponding to the selected gain vector; determining a value of a transition parameter used for transitioning from the target time point to a next time point, from a valid range of the transition parameter; and generating an assumed probability of being in each state at the next time point after the target time point, using the transition parameter, wherein a transition from a current state to a next state occurs in response to an action.