INTRODUCTION 1 I ISAM: AN ITERATIVE STATE ASSIGNMENT METHOD
Sequential switching circuits are devices whose outputs depend on both current and previous inputs, and they can be classified as synchronous or asynchronous Synchronous circuits rely on clock pulses to synchronize their operations, while asynchronous circuits operate without such timing signals A key advantage of asynchronous design is the ability to transition without waiting for clock pulses; however, this lack of synchronization raises challenges in ensuring that the circuit performs reliably despite variations in signal transmission delays.
An asynchronous sequential circuit's operation can be represented using a flow table, which is a two-dimensional array displaying next-state entries In this table, columns correspond to input states and rows represent the circuit's internal states While flow tables typically include output states, this article focuses solely on the internal workings of the sequential circuit, omitting the output states for clarity.
The current operation of the circuit is known as its present internal state, or simply the present state For instance, if the present state of the circuit illustrated in Figure 1 is labeled as "a," then
When an input II is applied, the circuit transitions to the next state, referred to as "b." If this next state matches the current internal state, it is considered stable concerning that input, indicated by a circled next-state entry Conversely, uncircled entries signify unstable internal states.
Figure I Flow Table for an Asynchronous Sequential
The total circuit state is defined by the combination of the input state and the current internal state In certain flow tables, some total circuit states are never utilized, leading to unspecified next-state entries known as "don't care" states Flow tables that include these "don't care" states are referred to as incompletely specified flow tables Since "don't care" states are not used in the actual circuit synthesis, any value can be assigned to them to simplify the final design This paper addresses both completely specified and incompletely specified flow tables.
Definition: An a synchronous sequential circuit is said to be oper- ating in fundamental mode if the inputs are never changed unless the circuit is in a stable state
Definition: A transition from an unstable state to a stable state is called a direct transition if all internal state variables that are to under- go a change of state are simultaneously excited
A crucial step in designing an asynchronous sequential circuit is obtaining an internal state assignment, which involves encoding each internal state with a binary n-tuple or a set of n-tuples This encoding utilizes n internal state variables, denoted as y1, y2, , yn, allowing for the representation of up to 2n internal states.
In an asynchronous circuit, it is crucial to assign internal states in a way that ensures each transition consistently results in a specific and stable state, regardless of the varying speeds of the circuit components.
A race in an asynchronous sequential circuit occurs when a transition between two states necessitates the simultaneous change of multiple internal state variables If this race results in incorrect operation of the circuit, it is classified as a critical race; if not, it is referred to as a non-critical race.
To ensure reliable circuit operation, every internal state assignment must avoid critical races This can be achieved through two primary strategies: the first is to completely eliminate all races, thereby preventing any critical races from occurring The second strategy allows for the existence of races, provided that they are non-critical in nature, ensuring that circuit functionality remains intact.
Based on these two approaches, two main types of internal state assign- ment techniques have evolved
In synchronous sequential circuits, single transition time (STT) assignments allow all internal state variables to change simultaneously at the start of a transition, leading directly from an unstable to a stable state When each internal state is associated with a unique code, this is referred to as a uni-code single transition time (USTT) assignment In contrast, a uni-code totally sequential (UTS) assignment also uses unique binary codes for each internal state but transitions occur one variable at a time Notably, USTT assignments may experience races, whereas UTS assignments do not.
Single transition time assignment techniques have received con- siderable attention from researchers over the last decade [1~2 1314 ~5]
As a result, there are well-known established methods for generating the USTT assignments and the corresponding next-state equations
Totally sequential assignment techniques 1 on the other hand, have re- ceived considerably less attention The main contributions in this area are due to Hazeltine [ 6] I Maki [ 7, 8] , and Saucier [ 9]
A transition path for a flow table with a UTS assignment is defined as a sequential series of adjacent internal states that connects an unstable state, Sa, to a stable state, Sb This path encompasses both the initial unstable state and the final stable state.
Definition: The distance d between two internal states S and Sb
- - a is the number of bit positions in which the binary code of S differs from a the binary code of Sb
A minimum length (ML) transition occurs when the distance between two internal states, denoted as d, is traversed by exciting each state variable exactly once during the transition Consequently, the path taken during this transition is referred to as the minimum length transition path.
An eros sover occurs when there are transition paths for two transitions, sa to sb and sc to sd, within the same flow table column, ensuring that sb and sd share at least one internal state.
A k-set in a flow table column is defined as a group comprising k-1 unstable states that lead to a single stable state, along with that stable state itself A UTS assignment is considered valid if all transition paths can be constructed without any crossover between the states of different k-sets within the same flow table column Furthermore, a state assignment is termed universal if its validity relies solely on the number of flow table rows; if it depends on other factors, it is classified as non-universal.
Definition: Let s be the minimum number of interna 1 state variables
0 to uniquely code an r-row flow table Ann variable assignment for this
6 flow table is called a near-minimal assignment if s < n < s +£so }
2 where [x} indicates the smallest integer< x
Hazeltine's method focuses on creating transition paths between stable and unstable states on a per-column basis, involving a trial-and-error approach for assignment and path determination This technique results in a non-universal internal state assignment, with transition paths that are not necessarily of minimum length.
ALIAS: An Algorithm for Initial Assignment
Consider an r-row flow table s
To uniquely code each row of the flow table, a minimum number of internal state variables is required The initial assignment procedure aims to assign a unique binary code to each internal state, using the least number of variables possible States that have transitions between them are assigned codes that are numerically close, while states without transitions receive codes that are maximally distant from each other However, meeting these conflicting requirements can be challenging based on the flow table's structure.
Finding an effective initial assignment can be challenging, as it should enable the completion of nearly all transitions in a specified flow table This article introduces two methods for establishing a solid initial assignment, with the first method being PRIME.
This is an initial assignment technique based on USTT assignments
UTS assignments typically necessitate fewer internal state variables compared to USTT assignments This indicates that selecting an appropriate subset of internal state variables from a valid USTT assignment can yield a valid UTS assignment Given the availability of USTT assignment techniques, this method appears to be a promising solution Tracey has created algorithmic techniques to facilitate this process.
USTT assignments involve a technique where all dichotomies for each input column are enumerated to create a Boolean matrix The algorithm subsequently identifies an intersection or partition that encompasses the maximum or near-maximum number of rows within this matrix.
The process begins by discarding covered rows and repeating the procedure on a subset of the original matrix until all rows are covered, resulting in a valid USTT assignment Notably, the first internal state variable covers more dichotomies than subsequent variables, suggesting that the initial assignment generated by this method is likely to be effective.
Tracey's ( 1] matrix reduction algorithm has been programmed and the test problems run indicated the following limitations of this alga- rithm:
(a) S internal state variables do not always guarantee a distinct
0 code for each row of a flow table Under this condition one has to use an initial assignment with s
For larger flow tables, the Boolean matrix size significantly increases, leading to longer processing times for computer programs that utilize cover-finding techniques.
Example #1: Refer to the flow table of Figure 3
Figure 3 Flow Table for Example #l Step 1: List all dichotomies under each input column
Step 2: Form a Boolean rna trix
0 == 3, use Tracey's Boolean matrix reduction algorithm
( 1 1 three time s The first iteration yields a partition
[ (135), (246)] Delete the rows of the Boolean matrix that are covered by this partition The reduced Boolean matrix is given below
The second iteration on the reduced Boolean matrix yields a partition [ (145) 1 (236)] Delete rows of the Boolean rna trix covered by this partition and obtain the reduced
The third iteration yields a partition ((146), (235)) generating a three variable initial assignment given below y 1 [ (1 3 5) 1 (2 4 6) ] y 2 [ (1 4 5) 1 (2 3 6)] y 3 ( (1 4 6) 1 (2 3 5) 1
The validity of the initial assignment is confirmed by constructing all transition paths without crossover, establishing it as a valid UTS assignment However, after the third iteration of the Boolean matrix reduction algorithm, the presence of a non-empty reduced Boolean matrix indicates that the initial assignment does not qualify as a valid USTT assignment.
The variable initial assignment for an r-row flow table includes s0 distinct two-block partitions within the state set PRIME employs a traditional method by utilizing an established state assignment technique to produce s.
Traditional methods for achieving unique state assignments often struggle to meet diverse constraint classes, as noted in Tracey's fundamental theorem This theorem indicates that the necessary constraints for a Unique State Transition Table (USTT) assignment include all dichotomies and state pairs absent from either side of these dichotomies The Boolean matrix reduction algorithm, while effective, does not guarantee a unique code for each internal state until the entire Boolean matrix is addressed This limitation is why the PRIME algorithm may fail to provide distinct codes for each internal state variable However, the approach presented here overcomes this challenge by employing successive partitioning of the entire state set, ensuring a more reliable assignment.
DIAGRAM employs a two-step iterative process for state set partitioning in a flow table Initially, the partitioning algorithm, PART, divides the total state set into disjoint blocks, continuing until each block holds a single element, ensuring unique codes for each internal state In subsequent iterations, the multiple resulting blocks are merged into a two-block partition, streamlining the overall structure.
JOIN Each time JOIN is used a new internal state variable is generated
The iterative procedure of using PART and JOIN terminates when s
0 inter- nal state variables are genera ted Use of the partitioning approach ensures that with generation of s
0 internal state variables each row of the flow table has a distinct code
The effectiveness of the initial assignment in a flow table relies heavily on the algorithms PART and JOIN While making these algorithms more complex could enhance the initial assignment's effectiveness, it would also lead to longer running times for the computer program Since the transition path generation algorithm assesses the validity of the initial assignment, a lengthy initial assignment algorithm that only marginally improves effectiveness is not ideal Consequently, quick and straightforward techniques were adopted for both PART and JOIN to optimize performance.
An inspection of valid internal state assignments indicates that irrespective of the methods of obtaining such assignments the codes for
States with transitions are closer together, while those without transitions are farther apart, which is a key property utilized by PART Initially, transition diagrams were used to visualize these relationships, but as flow tables grew in size, the complexity of these diagrams increased, making manual searches less effective By employing machine search techniques, even complex transition diagrams can be effectively utilized for secondary assignments PART combines transition diagrams with computer search methods to generate initial assignments The algorithm begins by applying a weighting scheme, which can be based on either the states or the transitions within a flow table Since the validity of an internal state assignment relies on the successful completion of all transitions, it is more appropriate to assign weights to transitions rather than individual elements of a k-set, a conclusion supported by comparisons of different weighting strategies.
The weighting scheme applied in this context focuses on the number of elements within a k-set Transitions between elements in a k-set containing only two elements are assigned the highest weight, as these transitions typically must meet the most rigorous criteria for successful completion.
As the number of elements in a k-set increases, the requirements for transitions between states become more flexible, allowing for multiple ways to complete these transitions Consequently, transitions between states in a k-set with more elements are assigned weights that are inversely proportional to the number of elements in the k-set For instance, a transition between states in a 3-set is given a higher weight compared to a transition in a 4-set Additionally, the weighting scheme addresses "don't care" states within a flow table Further details on the weighting scheme will be provided later, along with essential definitions to aid in understanding the algorithm.
A connectivity matrix for an r-row flow table is defined as an rxr binary matrix In this matrix, the element at position (i, j) equals the element at (j, i) and is set to 1 if there is a transition between rows i and j of the flow table; otherwise, both elements are set to 0.
TRAPAGAL: A Transition Path Generation
ALSPT: An Algorithm to Speed up Transitions
SUMMARY AND DISCUSSION
This project focused on developing a robust method for constructing minimum-length transition paths While existing literature offers a limited number of methods for generating transition paths, many are applicable only to specific flow tables and assignments, or they cater to particular internal state assignments This raises a critical question: how can one systematically produce transition paths without crossover, given a valid Unicode totally sequential assignment, whether universal or nonuniversal?
The research introduces a universal transition path generation algorithm that overcomes previous limitations for any type of UTS assignment This method includes three independent algorithms, each varying in complexity, with computer running time correlating directly to their complexity COUNT is the simplest algorithm, offering a shorter running time, while PATH employs an exhaustive search technique, resulting in a longer running time Notably, these algorithms complement each other; for certain flow table classes where COUNT fails to generate a solution, PATH can often find one in a shorter time, demonstrating their respective strengths in different scenarios.
COUNT employs advanced search techniques that significantly enhance solution speed, demonstrating remarkable efficiency across a diverse range of test problems in generating transition paths However, it is important to note that none of the algorithms utilize a fully exhaustive search, which precludes any claims of achieving 100% success.
Incorporating a comprehensive search technique may seem appealing, but its practical application is limited due to the excessive computer time it typically requires.
The availability of an advanced algorithm suggests that reversing its technique could facilitate the generation of state assignments Transition path generation methods create paths based on individual columns, but generating an internal state assignment requires considering interactions among transitions across all input columns of a flow table simultaneously Therefore, these methods cannot be employed as a straightforward solution for producing internal state assignments.
Maki proposed an iterative approach for generating non-universal UTS assignments, although this method has limitations, as it cannot ensure the creation of a minimum variable assignment Furthermore, achieving a minimum variable internal state assignment does not always lead to the most cost-effective circuit realization Consequently, a near-minimal assignment can often be acceptable and, under specific cost criteria, may be the most desirable solution In this regard, the iterative state assignment approach shows significant promise.
Although Maki proposed a broad outline of such an approach and established an upper bound on the number of internal state variables,
This paper introduces a new iterative state assignment method (ISAM) that overcomes previous limitations by systematically generating a valid UTS assignment The method begins with a straightforward procedure to create a "good" minimum variable initial assignment, followed by a transition path generation algorithm to assess its validity If the assignment proves invalid, a variable is added, and the augmented assignment is re-evaluated This iterative process continues until a valid assignment is achieved, highlighting the effectiveness of the transition path generation algorithm as a central component of ISAM.
An iterative assignment algorithm yields a near-minimal UTS assignment, resulting in more spare states compared to a minimum variable UTS assignment These spare states can be leveraged to introduce non-critical races, which help decrease the number of subtransitions in completely sequential transition paths This objective is achieved through a simple algorithm that, after generating a valid UTS assignment, seeks to enhance transition speeds by minimizing subtransitions in sequential paths by incorporating non-critical races.
The iterative state assignment method serves as a powerful algorithm for logic designers, enabling the generation of multiple valid UTS assignments with varying internal state variables This availability allows for effective comparison under different performance criteria and supports the development of improved algorithms for initial assignments Additionally, it provides valuable insights into successful and unsuccessful transitions, guiding the optimal addition of internal state variables Overall, this method holds significant potential as an effective computer-aided design tool for asynchronous sequential circuits.
IV RELATED AREAS OF FUTURE WCRK
It is the opinion of the author that the research leading to the development of iterative state assignment method can be extended to the following research areas
A Routing and Transition Path Generation
The routing problem involves determining an optimal interconnection path between circuit elements on a board, typically aiming to minimize wire length or the number of crossings Similarly, the transition path generation problem focuses on connecting internal states using the shortest possible paths without crossings When constraints prevent the construction of all transition paths, an additional internal state variable is introduced Likewise, if routing interconnections cannot be achieved within the given constraints, an extra layer is added to the design.
There is a significant similarity between the routing problem and another related issue, highlighting the need for a thorough investigation of their relationship Given the extensive attention the routing problem has garnered over the years, exploring this connection could facilitate the application of established routing techniques to transition path generation Additionally, it may pave the way for the development of a unified approach to effectively address both problems.
USTT assignments must meet stricter constraints compared to UTS assignments, leading to quicker circuit realization for USTT at a higher cost, while UTS results in slower realization at a lower cost Tracey's elucidation of the necessary and sufficient conditions for USTT assignments is well understood, but similar conditions for UTS assignments are currently lacking Consequently, the constraints for UTS operations are derived from modifications of those for USTT operations Establishing clear conditions for UTS assignments could enhance the understanding of the relationship between UTS and USTT constraints, potentially paving the way for a unified theory of state assignments that encompasses both types.
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