Methods for performing generalized trajectory evaluation

Methods for formal verification of circuits and other finite-state systems are disclosed. Formal definitions and semantics are disclosed for a model of a finite-state system, an assertion graph to express properties for verification, and satisfiability criteria for specification and automated verification of forward implication properties and backward justification properties. A method is disclosed to perform antecedent strengthening on antecedent labels of an assertion graph.A method is also disclosed to compute a simulation relation sequence ending with a simulation relation fixpoint, which can be compared to a consequence labeling for each edge of an assertion graph to verify implication properties properties according to the formal semantics. An alternative method is disclosed to compute the simulation relation sequence from the strengthened antecedent labels of an assertion graph, thereby permitting automated formal verification of justification properties.Finally methods are disclosed to significantly reduce computation through abstraction of models and assertion graphs and to compute an implicit satisfiability of an assertion graph by a model from the simulation relation computed for the model and assertion graph abstractions. Other methods and techniques are also disclosed herein, which provide for fuller utilization of the claimed subject matter.

FIELD OF THE INVENTION

This invention relates generally to automated design verification, and in particular to formal property verification and formal equivalence verification for very large scale integrated circuit designs and other finite state systems.

BACKGROUND OF THE INVENTION

As hardware and software systems become more complex there is a growing need for automated formal verification methods. These methods are mathematically based techniques and languages that help detect and prevent design errors thereby avoiding losses in design effort and financial investment.

Examples of the type of properties being verified include safety properties (i.e. that the circuit can not enter undesirable states) and equivalence properties (i.e. that a high level model and the circuit being verified have equivalent behaviors). There are two well-established symbolic methods for automatically verifying such properties of circuits and finite state systems that are currently considered to be significant. The two most significant prior art methods are known as classical Symbolic Model Checking (SMC) and Symbolic Trajectory Evaluation (STE).

Classical SMC is more widely know and more widely received in the formal verification community. It involves building a finite model of a system as a set of states and state transitions and checking that a desired property holds in the model. An exhaustive search of all possible states of the model is performed in order to verify desired properties. The high level model can be expressed as temporal logic with the system having finite state transitions or as two automata that are compared according to some definition of equivalence. A representative of classical SMC from Carnegie Mellon University known as SMV (Symbolic Model Verifier) has been used for verifying circuit designs and protocols. Currently these techniques are being applied also to software verification.

One disadvantage associated with classical SMC is a problem known as state explosion. The state explosion problem is a failure characterized by exhaustion of computational resources because the required amount of computational resources expands according to the number of states defining the system. SMV, for example, is limited by the size of both the state space of systems and also the state space of properties being verified. Currently, classical SMC techniques are capable of verifying systems having hundreds of state encoding variables. The budget of state encoding variables must be used to describe both the high level model and the low level circuit or system. This limitation restricts classical SMC to verifying circuits up to functional unit block (FUB) levels. For systems with very much larger state spaces, SMC becomes impractical to use.

The second and less well-known technique, STE, is a lattice based model checking technique. It is more suitable for verifying properties of systems with very large state spaces (specifiable in thousands or tens of thousands of state encoding variables) because the number of variables required depends on the assertion being checked rather than on the system being verified. One significant drawback to STE lies in the specification language, which permits only a finite time period to be specified for a property.

A Generalized STE (GSTE) algorithm was proposed in a Ph.D. thesis by Alok Jain at Carnegie Mellon University in 1997. The GSTE proposed by Jain permits a class of complex safety properties with infinite time intervals to be specified and verified. One limitation to Jain's proposed GSTE is that it can only check for future possibilities based on some past and present state conditions. This capability is referred to as implication. For example, given a set of state conditions at some time, t, implication determines state conditions for time, t+1. Another, and possibly more important limitation is that the semantics of the extended specification language were not supported by rigorous theory. As a consequence few practitioners have understood and mastered the techniques required to use GSTE effectively.

DETAILED DESCRIPTION

These and other embodiments of the present invention may be realized in accordance with the following teachings and it should be evident that various modifications and changes may be made in the following teachings without departing from the broader spirit and scope of the invention. The specification and drawings are, accordingly, to be regarded in an illustrative rather than restrictive sense and the invention measured only in terms of the claims.

Methods for formal verification of circuits and other finite-state systems are disclosed herein. For one embodiment, formal definitions and semantics are disclosed for a model of a finite-state system, an assertion graph to express forward implication properties and backward justification properties for verification, and satisfiability criteria for automated verification of forward implication properties and backward justification properties. For one embodiment, a method is disclosed to perform antecedent strengthening on antecedent labels of an assertion graph.

For one alternative embodiment, a method is disclosed to compute a simulation relation sequence ending with a simulation relation fixpoint, which can be compared to a consequence labeling for the edges of an assertion graph to verify implication properties. For another alternative embodiment, a method is disclosed to compute the simulation relation sequence from the strengthened antecedent labels of an assertion graph, thereby permitting automated formal verification of justification properties.

For another alternative embodiment, methods are disclosed to significantly reduce computation through abstraction of models and assertion graphs and to compute an implicit satisfiability of an assertion graph by a model from the simulation relation computed for the model and assertion graph abstractions.

For another alternative embodiment, a method for representing and verifying assertion graphs symbolically is disclosed that provides an effective alternative for verifying families of properties. For another alternative embodiment, a class of lattice domains based on symbolic indexing functions is defined and a method using assertion graphs on a symbolic lattice domain to represent and verify implication properties and justification properties, provides an efficient symbolic manipulation technique using BDDs. Previously disclosed methods for antecedent strengthening, abstraction, computing simulation relations, verifying satisfiability and implicit satisfiability may also be extended to assertion graphs that are symbolically represented. Other methods and techniques are also disclosed herein, which provide for fuller utilization of the claimed subject matter.

Intuitively, a model of a circuit or other finite state system can be simulated and the behavior of the model can be verified against properties expressed in an assertion graph language. Formal semantics of the assertion graph language explain how to determine if the model satisfies the property or properties expressed by the assertion graph. Two important characteristics of this type of verification system are the expressiveness of the assertion graph language and the computational efficiency of carrying out the verification.

For one embodiment, a finite state system can be formally defined on a nonempty finite set of states, S, as a nonempty transition relation, M, where (s1, s2) is an element of the transition relation, M, if there exists a transition in the finite state system from state s1to state s2and both s1and s2are elements of S. M is called a model of the finite state system.

For another possible embodiment, an alternative definition of the model, M, can be set forth as a pair of induced transformers, Pre and Post, such that Pre({s2}) includes s1and Post({s1}) includes s2if (s1,s2) is an element of M. In other words, the Pre transformer identifies any states, s, in S for which there exists a transition to some state, s′, in S. Pre is called a pre-image transformer. The Post transformer identifies any states, s′, in S for which there exists a transition from some state, s, in S. Post is called a post-image transformer.

For one possible embodiment, a finite sequence of states of length, n, is called a finite trace, t, in the model M if it is true of every state, s, occurring in the ith position in the sequence (i being contained within the closed interval [1,n−1]) that some state, s′, for which Post({s}) includes s′, occurs in the i+1th position in the sequence. An infinite trace is a sequence of states, which satisfies the above conditions for all i greater or equal to 1.

For one embodiment, an assertion graph, G, can be defined on a finite nonempty set of vertices, V, to include an initial vertex, vI; a set of edges, E, having one or more copies of outgoing edges originating from each vertex in V; a label mapping, Ant, which labels an edge, e, with an antecedent Ant(e); and a label mapping, Cons, which labels an edge, e, with a consequence, Cons(e). When an outgoing edge, e, originates from a vertex, v, and terminates at vertex, v′, the original vertex, v, is called the head of e (written v=Head(e)) and the terminal vertex, v′, is called the tail of e (written v′=Tail(e)).

For one embodiment,FIG. 1bdepicts an assertion graph,102. The two types of labels used in the assertion graph have the following purposes: an antecedent represents a set of possible pre-existing states and stimuli to a circuit or finite state system to affect its behavior; a consequence represents a set of possible resulting states or behaviors to be checked through simulation of the circuit or finite state system. Antecedent and consequence labels are written as ai/ci for the edges of assertion graph102. For example, from vertex vI, a corresponding system should transition according to edge121and produce resulting states or behaviors according to consequence c1if stimuli and pre-existing state conditions described by antecedent a1are met. On the other hand, the system should transition according to edge120and produce resulting states or behaviors according to consequence c0if stimuli and pre-existing state conditions described by antecedent a0are met. Similarly for vertex v1, the system should transition according to edge122producing consequences c2if antecedent a2is met or according to edge123producing consequences c3if antecedent a3is met. For vertex v2, consequences are trivially satisfied.

It will be appreciated that using an assertion graph, properties may be conveniently specified at various levels of abstraction according to the complexity of the circuit or finite state system being modeled.

For example, using the assertion graph102ofFIG. 1bproperties can be specified at a convenient level of abstraction for some finite state system as depicted in assertion graph103ofFIG. 1c. From vertex, vI, a corresponding circuit should transition around edge131if busy, or transition along edge130to vertex, v1, if not busy and accepting input B. From vertex v1, either the circuit is stalled in which it continues to transition along loop133, or it produces an output along edge132of F(B) which is a function of the input B.

For one possible embodiment, a finite sequence of edges of length, n, is called a finite path, p, in the assertion graph G if it is true of every edge, e, occurring in the ith position in the sequence (i being contained within the closed interval [1,n−1]) that some edge, e′, for which Tail(e)=Head(e′), occurs in the i+1th position in the sequence. If for the first edge, e1, in the sequence, Head(e1)=vI (the initial vertex), then the sequence is called a finite I-path. An infinite path (or infinite I-path) is a sequence of edges, which satisfies the above conditions for all i greater or equal to 1.

An I-path provides an encoding of correlated properties for a finite state system. Each property may be interpreted such that if a trace satisfies a sequence of antecedents, then it must also satisfy a corresponding sequence of consequences.

For example, the assertion graph201depicted inFIG. 2adescribes a collection of correlated properties. The infinite I-path including edge214, edge216, edge215, edge216, . . . indicates that if the system enters state s1, then it alternates between {s3, s6} and {s4, s5}. The infinite I-path including edge213, edge215, edge216, edge215, . . . indicates that if the system enters state s2, then it alternates between {s4, s5} and {s3, s6}.

A rigorous mathematical basis for both STE and GSTE was devised by Ching-Tsun Chou of Intel Corporation in a paper entitled “The Mathematical Foundation of Symbol Trajectory Evaluation,” (Proceedings of CAV'99, Lecture Notes in Computer Science #1633, Springer-Verlag, 1999, pp. 196–207). In order for practitioners to truly understand and make good use of GSTE, it is necessary to have a language semantics that is based on rigorous mathematical theory.

For one embodiment, a strong semantics for assertion graphs may be defined more formally. To say that a finite trace, t, of length n in a model, M, satisfies a finite path, p, of the same length in an assertion graph, G, under an edge labeling, L (denoted by (M, t)|=L(G, p)), means that for every i in the closed interval [1,n], the ith state in trace, t, is included in the set of states corresponding to the label of the ith edge in path, p. To illustrate examples of a state being included in a label set, s1is included in the antecedent set {s1} of edge214inFIG. 2a, and s3is included in the consequence set {s3, s6} of edge216.

To say that a state, s, satisfies an edge, e, in n steps (denoted by (M, s)|=n(G, e)); means that for every k-length trace prefix, tk, starting from s and every k-length path prefix, pk, starting from e, and for every k less than or equal to n, trace prefix, tk, satisfies path prefix, pk, under the consequence edge labeling, Cons, whenever trace prefix, tk, satisfies path prefix, pk, under the antecedent edge labeling, Ant.

To say that the model M satisfies assertion graph G in n steps (denoted by M|=nG), means that for any edge e beginning at initial vertex vI in G, all states, s, in M satisfy edge e in n steps.

Finally, to say that M strongly satisfies G (denoted by M|=STRONGG); means that M satisfies G in n steps for all n greater or equal to 1.

In prior methods for performing STE and GSTE, semantics were used which required strong assumptions with respect to assertion graphs. In STE for example, only finite path lengths traversing the assertion graphs can be generated and used to verify a corresponding system under analysis. This means that for all transitions for which the antecedents are satisfied, along any path of finite length, the corresponding consequences are checked against the behavior of the circuit or system being analyzed. On the other hand, it shall be demonstrated herein that it is desirable for the semantics to consider all transitions along an infinite path to see if the antecedents are satisfied. If any of the antecedents along an infinite path are violated, then it is not necessary to check the consequences for that path.

Strong satisfiability as defined above formally captures a semantics substantially similar to that used in STE and GSTE as proposed in 1997 by Alok Jain. It requires that a consequence hold based solely on past and present antecedents. Strong satisfiability expresses properties that are effects of causes.

Every prefix of every trace in model101trivially satisfies I-path p1except the tracet3=[s1, s3, s5, s6, s5, . . . ],
because the antecedent {s1} is not satisfied by any trace except t3. The consequence labels for path p1can be writtenCons(p1)=[S, {s3, s6}, {s4, s5}, {s3, s6}, . . . ].
For trace t3, every prefix satisfies the consequences on p1since each state in the trace is included in a corresponding label set for the I-path. Therefore t3also satisfies p1.

Similarly, every prefix of every trace in model101trivially satisfies I-path p2except the tracet4=[s2, s4, s6, s5, s6, . . . ],
because the antecedent {s2} is not satisfied by any trace except t4. The consequence labels for path p2can be writtenCons(p2)=[S, {s4, s5}, {s3, s6}, {s4, s5}, . . . ].
For trace t4, every prefix satisfies the consequences on p2since each state in the trace is included in a corresponding label set for the I-path. Therefore t4also satisfies p2. Accordingly model101strongly satisfies assertion graph201.

The method for performing Generalized Symbolic Trajectory Evaluation (GSTE) proposed by Alok Jain, provides implication capabilities for determining future state conditions from a set of initial state conditions. It is also desirable to ask why a set of state conditions occurred. In other words, what possible initial conditions and transitions could cause the system under analysis to end up in a given state? Such a capability is referred to as justification. Strong satisfiability, however, is inadequate for expressing justification properties, which are causes of effects, rather than effects of causes. As an example of a justification property, one might wish to assert the following: if the system enters state s1, and does not start in state s1, then at the time prior to entering state s1, the system must have been in state s0.

For one embodiment,FIG. 2bdepicts an assertion graph202, which attempts to capture the justification property asserted in the above example. Edge228from vertex v1to vertex v2has an antecedent label {s1} corresponding to the effect portion of the property, and edge227from vertex vI to vertex v1has as a consequence label {s0} corresponding to the cause portion of the property. According to strong satisfiability as defined, the model101does not strongly satisfy the assertion graph202.

All traces t3through t8immediately fail the first consequence label on pI and yet all satisfy the first antecedent label on pI. Therefore traces t3through t8do not satisfy pI. Accordingly model101does not strongly satisfy assertion graph202, and what has been demonstrated is that the method proposed by Alok Jain does not provide for justification. In fact, it is substantially impossible to provide for a justification capability within the semantic constraints used by prior STE and GSTE methods. Yet, intuitively, the justification property asserted in the above example is true for model101. To overcome this discrepancy, a new definition of satisfiability is needed.

For one embodiment, a normal semantics for assertion graphs that provides for justification properties may be formally defined. To say that a trace, t, in a model, M, satisfies a path, p, in an assertion graph, G, under an edge labeling, L (denoted by t|=Lp), means that for every i greater than or equal to 1, the ith state in trace, t, is included in the set of states corresponding to the label of the ith edge in path, p.

To say that a state, s, satisfies an edge, e (denoted by s I=e), means that for every trace, t, starting from s and every path, p, starting from e, trace, t, satisfies path, p, under the consequence edge labeling, Cons, whenever trace, t, satisfies path, p, under the antecedent edge labeling, Ant.

To say that the model M satisfies assertion graph G (denoted by M|=G), means that for any edge e beginning at initial vertex vI in G, all states, s, in M satisfy edge e.

Based on the strong semantics and the normal semantics as defined above, it is true to say that if M strongly satisfies G then M satisfies G (expressed symbolically as M|=STRONGGM|=G) for any assertion graph G and any model M. For example, model101satisfies assertion graph201since model101strongly satisfies assertion graph201.

Returning to examine assertion graph202according to the definition of normal satisfiability, the traces t1and t2satisfy the consequence labels of I-path pI, and therefore satisfy pI. The traces t3through t8all violate the second antecedent label of pI, since none of them enter state s1. Since the antecedent labels are not satisfied, the consequence labels need not be satisfied in order to satisfy the I-path. Therefore t3through t8satisfy pI under the normal satisfiability definition. Accordingly, model101satisfies assertion graph202under the definition of normal satisfiability.

Therefore, for one embodiment, a normal semantics, herein disclosed, provides for assertion graphs, which are capable of expressing justification properties.

It will be appreciated that descriptions of models and assertion graphs, herein disclosed, can be modified in arrangement and detail by those skilled in the art without departing from the principles of the present invention within the scope of the accompanying claims. For example, one popular representation method from automata theory uses automatons, which include automata states, an initial automata state, and a set of state transitions, rather than assertion graphs, which include assertion graph components as described above. A path in an assertion graph is analogous to a run in an automaton, and it can be shown that there is an assertion graph corresponding to the automaton, such that a model satisfies the assertion graph if and only if the model satisfies the automaton.

The assertion graph can be seen as a monitor of the circuit, which can change over time. The circuit is simulated and results of the simulation are verified against consequences in the assertion graph. The antecedent sequence on a path selects which traces to verify against the consequences.

For one embodiment, a simulation relation sequence can be defined for model checking according to the strong satisfiability criteria defined above. For an assertion graph G and a model M=(Pre, Post), define a simulation relation sequence, Simn: E→P(S), mapping edges between vertices in G into state subsets in M as follows:Sim1(e)=Ant(e) if Head(e)=vI, otherwiseSim1(e)={ };Simn(e)=Union (Simn−1(e), (Unionfor all e′ such that Tail(e′)=Head(e)(Intersect (Ant(e), Post(Simn−1(e′))) ))), for all n>1.

In the simulation relation defined above, the nth simulation relation in the sequence is the result of inspecting every state sequence along every I-path of lengths up to n. For any n>1, a state s is in the nth simulation relation of an edge e if it is either in the n−1th simulation relation of e, or one of the states in its pre-image set is in the n-Ith simulation relation of an incoming edge e′, and state s is in the antecedent set of e. It will be appreciated that the Union operation and the Intersect operation may also be interpreted as the Join operation and the Meet operation respectively.

For one embodiment,FIG. 3aillustrates a method for computing the simulation relation for a model and an assertion graph. Box311represents initially assigning an empty set to the simulation relation for all edges e in the assertion graph that do not begin at initial vertex vI, and initially assigning Ant(e) to the simulation relation for all edges e that do begin at initial vertex vI. Box312represents marking all edges in the assertion graph active. Box313represents testing the assertion graph to identify any active edges. If no active edges are identified, then the method is complete. Otherwise, an active edge, e, is selected and marked not active as represented by box314. Box315represents recomputing the simulation relation for edge, e, by adding to the simulation relation for edge e, any states which are in both the antecedent set for edge e and the post-image set for the simulation relation of any incoming edge, e′, to e. Box316represents testing the simulation relation for edge e to determine if it was changed by the recomputation. If it has changed, all outgoing edges from e are marked as active, as represented by Box317. In any case, the method flow returns to the test for active edges represented by Box313.

For example,FIG. 4shows changes over time in the assertion graph201resulting from simulation of the model101. Initially only edge413and edge414have state s2and state s1, respectively, associated with them. In the first subsequent iteration, state s3is added to edge426since s3is in the post-image of {s1} in model101and in the antecedent set of edge426in assertion graph201. Similarly s4is added to edge425. In the next iteration, s6is added to edge436because it is in the post-image of {s4} and in the antecedent set of edge436. State s5is added to edge435because it is in the post-image of {s3} and in the antecedent set of edge435. In the final iteration, no new states are added to any edge. Therefore a fixpoint solution is reached.

Comparing the final simulation relation for each edge, with the consequence set for that edge, indicates whether the model101strongly satisfies the assertion graph201. Since {s1} of edge444is a subset of the consequence set S, edge214is satisfied. Since {s2} of edge443is a subset of the consequence set S, edge213is satisfied. Since {s4, s5} of edge445is a subset of the consequence set {s4, s5}, edge215is satisfied. Finally, since {s3, s6} of edge446is a subset of the consequence set {s3, s6}, edge216is satisfied. Therefore the final simulation relation indicates that model101strongly satisfies assertion graph201.

In order to indicate normal satisfiability, a method is needed to propagate future antecedents backwards. For one embodiment, a method can be defined to strengthen the antecedent set of an edge e by intersecting it with the pre-image sets of antecedents on future edges. Since the strengthening method can have rippling effects on the incoming edges to e, the method should be continued until no remaining antecedents can be propagated backwards.

For one embodiment, an antecedent strengthening sequence can be defined for model checking according to the normal satisfiability criteria defined above. For an assertion graph G and a model M=(Pre, Post), define an antecedent strengthening sequence, Antn: E→P(S), mapping edges between vertices in G into state subsets in M as follows:Ant1(e)=Ant(e), andAntn(e)=Intersect (Antn−1(e′), (Unionfor all e′ such that Head(e′)=Tail(e)Pre(Antn−1(e′)) )), for all n>1.

In the antecedent strengthening sequence defined above, a state s is in the nth antecedent set of an edge e if it is a state in the n−1th antecedent set of e, and one of the states in a pre-image set of the n−1th antecedent set of an outgoing edge e′. Again, it will be appreciated that the Union operation and the Intersect operation may also be interpreted as the Join operation and the Meet operation respectively.

For one embodiment,FIG. 3billustrates a method for computing the strengthened antecedents for an assertion graph. Box321represents marking all edges in the assertion graph active. Box322represents testing the assertion graph to identify any active edges. If no active edges are identified, then the method is complete. Otherwise, an active edge, e, is selected and marked not active as represented by box323. Box324represents recomputing the antecedent label for edge, e, by keeping in the antecedent label for edge e, any states that are already contained by the antecedent label for edge e and also contained by some pre-image set for the antecedent label of any edge, e′, outgoing from e. Box325represents testing the antecedent label for edge e to determine if it was changed by the recomputation. If it has changed, all incoming edges to e are marked as active, as represented by Box326. In any case, the method flow returns to the test for active edges represented by Box322.

For example,FIG. 5ashows iterations of antecedent strengthening of graph202on model101. The antecedent sets are shown for edges517as S and518as {s1}. Therefore the antecedent set for edge527is computed as the antecedent set for edge517, S, intersected with the pre-image set of the antecedent set of outgoing edge518, denoted Pre({s1}), which is {s0}. Thus the antecedent set of edge527is strengthened to {s0} and the antecedent sets for edges528and529are unchanged. In the final iteration, no antecedent sets are changed and so a fixpoint solution502is reached and the iterations are terminated.

FIG. 5bshows the final simulation relation resulting from iterations of the method ofFIG. 3aperformed on the antecedent strengthened assertion graph502and using model101. Comparing the final simulation relation labels for each edge, with the consequence set for that edge (as shown in assertion graph202) indicates whether the model101strongly satisfies the strengthened assertion graph502. Since the simulation relation set {s0} of edge547is a subset of the consequence set {s0} of edge227and accordingly of edge537, edge537is satisfied. Since the simulation relation set {s1} of edge548is a subset of the consequence set S of edge228and accordingly of edge538, edge538is satisfied. Since the simulation relation set {s3, s5, s6} of edge549is a subset of the consequence set S of edge229and accordingly of edge539, edge539is satisfied. Therefore model101strongly satisfies the antecedent strengthened assertion graph502, but more importantly model101satisfies assertion graph202according to normal satisfiability as previously defined.

The fact that transition paths of infinite length are being considered does not mean that the list of possible antecedents will be infinite. Since the assertion graph describes a finite state machine, the number of permutations of those finite states is also finite. Therefore a fixpoint does exist and the monotonic methods ofFIG. 3aandFIG. 3bare guaranteed to converge on their respective fixpoints and terminate, given a large enough set of finite resources.

For one embodiment,FIG. 6ashows a method for computing the normal satisfiability of an assertion graph by a model. In block611, the antecedent sets are strengthened for each edge in the assertion graph. In block612, a fixpoint simulation relation is computed using the antecedent strengthened assertion graph. Finally in block613, the simulation relation sets are compared to the consequence sets to see if, for each edge, the simulation relation set is a subset of the consequence set, which is the necessary condition for satisfiability.

For one embodiment,FIG. 6billustrates, in finer detail, a method of computing normal satisfiability. In block621, the strengthened antecedent set fixpoint for each edge e (denoted Ant*(e)) in assertion graph G is computed. In block622, a fixpoint simulation relation set for each edge e (denoted Sim*(e)) is computed using the strengthened antecedents computed for each edge in block621. In block623, the comparison is performed. First, the edges are marked active in block624. Then a test is performed in block625to determine if any active edges remain to be compared. If not, the method is complete and the assertion graph is satisfied by the model. Otherwise, an active edge, e, is selected in block626and set to not active. In block627, the simulation relation set, Sim*(e), is compared to see if it is a subset of the consequence set, Cons(e). If not, the assertion graph is not satisfied by the model. Otherwise the method flow returns to the test at block625to determine if more edges remain to be compared.

For real-world finite-state systems, the number of states to be verified can be vary large and can contribute to a problem known as state explosion, which can, in turn, cause a failure of an automated verification process. One advantage of STE and GSTE, which perform computations in a lattice domain, is that they are less susceptible to state explosion. One lattice domain of interest is the set of all subsets of S, P(S) along with a subset containment relation,. The subset containment relation defines a partial order between elements of P(S), with the empty set as a lower bound and S as an upper bound. The set P(S) together with the subset containment relation,, are called a partially ordered system.

One important strength of trajectory evaluation based on lattice theory comes from abstraction. An abstraction maps the original problem space into a smaller problem space. For instance, a state trace is simply a record of the sequence of state transitions a system undergoes—during a simulation for example. Semantics for a language to describe all possible state transition sequences as disclosed can be easily understood by practitioners. A trajectory can be viewed as an abstraction of multiple state traces, which combines multiple possible state transition paths into equivalence class abstractions. Therefore an elegant semantics for a language to describe all possible trajectories can be defined by combining the semantics for state transition sequences with an abstraction layer to describe trajectories.

For one embodiment an abstraction of the lattice domain (P(S),) onto a lattice domain (P,A) can be defined by an abstraction function A mapping P(S) onto P such that A maps the upper bound S of P(S) to the upper bound U of P; A maps a lattice point S0to the lower bound Z of P if and only if S0is the lower bound of P(S), the empty set; A is surjective (onto); and A is distributive (e.g. A(Union({s1, s2}, {s0}))=Union(A({s1, s2}), A({s0}))=Union(S12, S0)).

FIG. 7illustrates one embodiment of an abstraction function A. The lattice domain718is an abstraction of the lattice domain711through an abstraction function A, which maps cluster713including the upper bound {s0, s1, s2, s3, s4, s5, s6} of lattice domain711to the upper bound U of lattice domain718; the lower bound of lattice domain711to the lower bound717of lattice domain718; cluster710including lattice point {s0} to lattice point S0; cluster712including lattice points {s1}, {s2} and {s1, s2} to lattice point S12, cluster714including lattice points {s3}, {s4} and {s3, s4} to lattice point S34; and cluster716including lattice points {s5}, {s6} and {s5, s6} to lattice point S56.

A concretization of the lattice domain (P,A) back to the lattice domain (P(S),) can be defined by a concretization function A−mapping P into P(S) such that A−maps a lattice point S1of P to the union of all subsets {si, . . . sj} in P(S) for which A({si, . . . sj})=Si. Therefore the concretization for the abstraction illustrated inFIG. 7, is given by A−(U)=S, A−(Z)={ }, A−(S0)={s0}, A−(S12)={s0,s1}, A−(S34)={s3,s4}, A−(S56)={s5,s6}.

Two important points with respect to abstractions are that the partial ordering among points in the original lattice domain are preserved in the abstract lattice domain, and that abstraction may cause potential information loss while concretization will not. For example inFIG. 7, A−(A({s1}))={s1, s2}{s1}, but A(A−(S12))=A({s1,s2})=S12.

For one embodiment, a definition of a model M can be formally defined on a lattice domain (P,) as a pair of monotonic transformers, Pre and Post, such that SiPre(Post(Si)) and that Post(Si)=Z if and only if Si=Z. The second condition ensures that the lower bound, which usually represents the empty set, is properly transformed. An abstraction of M on a lattice domain (PA,A) can be defined as MA=(PreA, PoStA) such thatA(Pre(Si))APreA(A(Si)) and A(Post(Si))APostA(A(Si)), for all Si in P.

For one embodiment, a finite sequence of lattice points of length, n, is called a finite trajectory, T, in the model M if it does not include the lower bound Z and it is true of every pair of lattice points, Si and Si+1, occurring in the ith and i+1th positions respectively in the sequence (i being contained within the closed interval [1,n−1]) that SiPre(Si+1) and Si+1Post(Si). An infinite trajectory is a sequence of lattice points, which satisfies the above conditions for all i greater or equal to 1. Intuitively a trajectory represents a collection of traces in the model.

An assertion graph G on a lattice domain (P,) is defined as before except that the antecedent labeling and the consequence labeling map edges more generally to lattice points Si instead of state subsets. The abstraction of an assertion graph is straightforward. The abstracted assertion graph GAis an assertion graph on a lattice domain (PA,A) having the same vertices and edges as G and for the abstracted antecedent labeling AntAand the abstracted consequence labeling ConsA, AntA(e)=A(Ant(e)) and ConsA(e)=A(Cons(e)) for all edges e in the assertion graphs GAand G.

If A−(A(Cons(e)))=Cons(e) for all edges e in G, then G is said to be truly abstractable and the unique abstraction GA is said to be a true abstraction If assertion graph G is truly abstractable, then the methods previously disclosed are sufficient for antecedent strengthening, determining strong satisfiability and determining normal satisfiability using model and assertion graph abstractions. For example if methods herein previously disclosed determine that an abstracted model MAstrongly satisfies a true abstraction GA, then the original model M strongly satisfies the original assertion graph G, according to the strong satisfiability criteria. Similarly, if methods herein previously disclosed determine that an abstracted model MAsatisfies a true abstraction GA, then the original model M satisfies the original assertion graph G, according to the normal satisfiability criteria.

In general though, an arbitrary assertion graph G is not guaranteed to be truly abstractable. In such cases, using the previously disclosed methods on an abstracted model and an abstracted assertion graph are not guaranteed to indicate satisfiability of the original assertion graph G by the original model M.

For one embodiment, alternative methods provide true implications of strong satisfiability and of normal satisfiability from computations performed on abstracted models and abstracted assertion graphs, which are not necessarily true abstractions. One key observation is that A−(SimA*(e))Sim*(e). A second key observation is that A−(AntA*(e))Ant*(e). In other words the concretization function generates a conservative approximation of a fixpoint simulation relation from a fixpoint simulation relation abstraction and a conservative approximation of a fixpoint strengthened antecedent set from a fixpoint strengthened antecedent set abstraction.

Therefore a method may be constructed which would permit the possibility of false verification failures but would not permit a false indication of assertion graph satisfiability. A result from such a method may be refered to as implicit satisfiability.

For one embodiment,FIG. 8aillustrates a method for implicit strong satisfiability using an abstracted simulation relation. In block811, an abstraction MAof model M is computed. In block812an abstraction GAof assertion graph G is computed, which is not guaranteed to be a true abstraction of assertion graph G. In block814, a simulation relation sequence is computed using the abstracted antecedents for all edges e in GA. In block815, the concretization function is used to conservatively approximate the original fixpoint simulation relation Sim*. In block816, the conservative approximation (denoted SimC) of Sim* is compared to the original consequence set for each edge e in G. If for every edge e in G, SimC(e)Cons(e) then the original model M strongly satisfies the original assertion graph G.

For one embodiment,FIG. 8billustrates a method for implicit normal satisfiability using an abstracted simulation relation. In block821, an abstraction MAof model M is computed. In block822an abstraction GAof assertion graph G is computed, which is not guaranteed to be a true abstraction of assertion graph G. In block823, the abstracted antecedents of GAare strengthened until a fixpoint is reached. In block824, a simulation relation sequence is computed using the strengthened antecedents for all edges e in GA. In block825, the concretization function is used to conservatively approximate the original fixpoint simulation relation Sim*. In block826, the conservative approximation (denoted SimC) of Sim* is compared to the original consequence set for each edge e in G. If for every edge e in G, SimC(e)Cons(e) then the original model M satisfies the original assertion graph G according to the normal satisfiability criteria.

It will be appreciated that the methods herein disclosed may be modified in arrangement and detail by those skilled in the art without departing from the principles of these methods within the scope of the accompanying claims. For example,FIG. 8cillustrates for one alternative embodiment of a modified method for implicit normal satisfiability using an abstracted simulation relation. In block833, the antecedents of an assertion graph G are strengthened until a fixpoint is reached. In block831, an abstraction MAof model M is computed. In block832an abstraction GAof the antecedent strengthened assertion graph G is computed. In block834, a simulation relation sequence is computed using the abstracted strengthened antecedents for all edges e in GA. In block835, the concretization function is used to conservatively approximate the original fixpoint simulation relation Sim*. In block836, the conservative approximation (denoted SimC) of Sim* is compared to the original consequence set for each edge e in G. If for every edge e in G, SimC(e)Cons(e) then the original model M satisfies the original assertion graph G according to the normal satisfiability criteria.

It will be appreciated that for many circuits or other finite state systems, there exists a family of properties related to a particular functionality. For example, an adder circuit may have scalar input values c1and c2and it may be desirable to verify that the adder output would be c1+c2if a particular adder control sequence is satisfied. It will also be appreciated that the number of scalar input combinations is an exponential function of the number of input bits to the adder and therefore it would be tedious if not impractical to express each scalar property as an assertion graph and to verify them individually.

Previously, merging numerous scalar cases into one assertion graph has been problematic. A merged graph may have a size that is also an exponential function of the number of inputs if the merged graph cannot exploit shared structures. Alternatively a merged graph having a reasonable size may fail to verify a property if critical information is lost in lattice operations.

For one embodiment, a method for representing and verifying assertion graphs symbolically provides an effective alternative for verifying families of properties. Once an assertion graph can be adequately represented symbolically, a symbolic indexing function provides a way of identifying assignments to Boolean variables with particular scalar cases. Formally defining a class of lattice domains based on symbolic indexing functions, provides an efficient symbolic manipulation technique using BDDs. Therefore previously disclosed methods for antecedent strengthening, abstraction, computing simulation relations, verifying satisfiability and implicit satisfiability may be extended to assertion graphs that are symbolically represented.

For one embodiment, an m-ary symbolic extension of a lattice domain (P,) can be set forth as a set of symbolic indexing functions {Bm→P} where Bmis the m-ary Boolean product. A symbolic indexing function I in {Bm→P} encodes a group of points on the lattice such that each point is indexed by a particular m-ary Boolean value as follows:I (x)=ORforbin Bm((x=b) AND (I(b)),
wherexdenotes (x1, x2, . . . , xm),bdenotes (b1, b2, . . . , bm) and (x=b) denotes ((x1=b1) AND (x2=b2) AND . . . AND (xm=bm)).

A symbolic indexing function11is less than or equal to a symbolic indexing function I2, denoted I1(x)SI2(x), if and only if for allbn Bm, I1(b)I2(b).

For one embodiment, a symbolic extension of a model M=(Pre, Post) on a lattice domain (P,) can be set forth as a pair of transformers, PreSand PostS, on the lattice domain ({Bm→P},S) such thatPreS(I(x))=ORforbin Bm((x=b) AND Pre(I(b)), andPostS(I(x))=ORforbin Bm((x=b) AND Post(I(b)),
for every I(x) in the set of symbolic indexing functions {Bm→P}. Such a symbolic extension MS=(PreS, PostS) is called a model on the finite symbolic lattice domain ({Bm→P},S).

As an example of a symbolic lattice domain,FIG. 9depicts part of a unary symbolic lattice domain. The unary symbolic indexing funtionI(x)=x AND S1OR x AND S2
encodes two points S1and S2on the lattice domain901. The symbolic indexing function902indexes S1when x=0 corresponding to lattice point903and indexes S2when x=1 corresponding to lattice point904.

FIG. 10shows a model1001on a lattice domain (P,). The model1001has state subsets corresponding to lattice points S1, S2, S3, S4, and S5. In addition lattice lower bound1007corresponds to the empty set of states, and lattice upper bound1005corresponds to all state subsets containing one or more of S1, S2, S3, S4, and S5. The model1001has non-trivial transitions (S1, S3), (S2, S4), (S3, S5), (S4, S5) and (S5, S5).

For one embodiment, an assertion graph GSon a symbolic lattice domain ({Bm→P},S) can be set forth as a mapping GS(b) of m-ary boolean valuesbin Bmto scalar instances of assertion graph GSon the original lattice domain (P,) such that for the symbolic antecedent labeling AntSand the symbolic consequence labeling ConsS,AntS(b)(e)=AntS(e)(b), andConsS(b)(e)=ConsS(e)(b),
for all edges e in the assertion graph GS.FIG. 11ashows two assertion graphs,1101and1102, on a lattice domain (P,) and an assertion graph1103on the unary symbolic lattice domain901that symbolically encodes assertion graphs1101and1102. For example, edge1137in assertion graph1103encodes edge1117in assertion graph1101for x=0 and edge1127for x=1.

The vertices VSof an assertion graph GSon a symbolic lattice domain ({Bm→P}S) can be set forth as a surjective, one-to-one vertex encoding function VS(b) of m-ary boolean valuesbin Bmto vertices V∪{vundef} in the scalar instance GS(b) on the original lattice domain (P,).

A symbolic indexing funtion for the symbolic antecedent labeling isAntS(v,v′)=ORforb,b′ in Bm((v=b) AND AntS(VS(b), VS(b′))),
where AntS(VS(b), vundef)=Z for anybin Bm. By introducing two vertex encoding variables u1and u2to encode the vertices vI, v1, v2, and the undefined vertex vundefas (u1u2), (u1u2), (u1u2), and (u1u2) respectively, the symbolic antecedent encoding function for assertion graph1103becomes

A symbolic indexing function for the symbolic consequence labeling isConsS(v,v′)=ORforb,b′ in Bm((v=b) AND AntS(VS(b), VS(b′))),
where ConsS(VS(b), vundef)=Z for anybin Bm. According to the two variable vertex encoding described above, the symbolic consequence encoding function for assertion graph1103becomes

Given a model MSon the symbolic lattice domain ({Bm→P},S), and an assertion graph GSon the symbolic lattice domain ({Bm→P},S) having edges (v,v′) and (v′,v) wherev′ denotes the successors ofv, andv−denotes the predecessors ofv, a method to symbolically compute the simulation relation sequence of GScan be formally defined. For one embodiment, a symbolic simulation relation sequence SimS(v,v′) can be defined for model checking according to the strong satisfiability criteria as follows:SimS1(v,v′)=(initE(v,v′) AND U) MeetSAntS(v,v′)
where initE is a Boolean predicate for the set of edges outgoing from vI, andSimSn(v,v′)=JoinS(SimSn−1(v,v′), (JoinS for allbin Bm(MeetsS(Ant(v,v′), PostS(SimSn−1(v−,v))))[b/v−])), for all n>1
where JoinSand MeetSare the join, ∪S, and meet, ∩S, operators for the symbolic lattice domain ({Bm→P},S) and [b/v−] denotes replacing each occurrence ofv−in the previous expression with b.

For one embodiment,FIG. 12aillustrates a method for computing the simulation relation for a model and an assertion graph on the symbolic lattice domain ({Bm→P},S) Box1211represents initially assigningZ=(initE(v,v′)U)∩SAntS(v,v′)
to the simulation relation for all edges (v,v′) in the assertion graph that do not begin at initial vertex vI, and initially assigningAntS(v,v′)=(initE(v,v′)U)∩SAntS(v,v′)
to the simulation relation for all edges (v,v′) that do begin at initial vertex vI. Box1215represents recomputing the simulation relation for edge (v,v′) by adding to the simulation relation for edges (v,v′), any states which are in both the antecedent set for edges (v,v′) and the post-image set for the simulation relation of any incoming edges (v−,v) to (v,v′) produced by substituting anybin Bmforv−. Box1216represents testing the simulation relation labeling for edges (v,v′) to determine if it was changed by the recomputation. If it has changed, the method flow returns to the recomputation of simulation relation for edges (v,v′), represented by Box1215. Otherwise a fixpoint has been reached and the method terminates at box1216.

Using the method disclosed above for computing the simulation relation for a model and an assertion graph on the symbolic lattice domain ({Bm→P},S), the simulation relation SimS(v,v′) can be computed. In the first iteration the simulation relation becomesSimS1(v,v)=(u1u2u1′u2′)(xS1xS2).
In the second iteration the simulation relation becomes

Comparing the simulation relation for each edge, with the consequence for that edge indicates whether the symbolic extension of model1001strongly satisfies assertion graph1103. It will be appreciated that a containment comparison may be interpreted and also performed in a variety of ways, for example: by inspection to see if each element in a set Sj is also in a set Sk, or by testing if Sj intersected with Sk equals Sj, or by a computing a logical operation on Boolean expressions Sj and Sk such asSjSk.

Since the simulation relation labelxS1xS2of edge1147is contained by the consequence label U, edge1137is satisfied. Since the simulation relation labelxS3xS4of edge1148is contained by the consequence label U, edge1138is satisfied. Finally since the simulation relation label S5of edge1149is contained by the consequence label S5, edge1139is satisfied. Therefore the final simulation relation indicates that symbolic extension of model1001strongly satisfies assertion graph1103on the symbolic lattice domain ({Bm→P},S). Intuitively this means that the model1001strongly satisfies both assertion graphs1101and1102on the lattice domain (P,).

Accordingly, by applying previously disclosed methods, for example, ofFIG. 6aor ofFIG. 8b, symbolic model checking can be performed using the normal satisfiability criteria if a strengthened antecedent sequence can be computed symbolically.

For one embodiment, an antecedent strengthening sequence AntS(v−,v) can be defined for model checking according to the normal satisfiability criteria as follows:AntS1(v−,v)=AntS(v−,v), andAntSn(v−,v)=MeetS(AntSn−1(v−,v), (JoinS for allbin BmPreS(SimSn−1(v,v′))[b/v′])), for all n>1.

For one embodiment,FIG. 12billustrates a method for computing the strengthened antecedents for an assertion graph on a symbolic lattice domain. In box1221all edges in the assertion graph have their original antecedent label values. Box1224represents recomputing the symbolic antecedent label for edges (v−,v), by keeping in the antecedent label for edges (v−,v), any states that are already contained by the symbolic antecedent label for edges (v−,v) and also contained by some pre-image set for the antecedent label of edges (v,v′), outgoing from (v−,v) and formed by substituting anybin Bmforv′. Box1225represents testing the symbolic antecedent labeling for edges (v−,v) to determine if it was changed by the recomputation. If it has changed, the method flow returns to the recomputation represented by Box1224. Otherwise a fixpoint has been reached and the method terminates at Box1225.

Accordingly, antecedent strengthening may be applied to symbolic model checking to provide normal satisfiability and therefore satisfiability of justification properties on the symbolic lattice domain ({Bm→P},S). It will be appreciated that the methods disclosed herein may be applied orthogonally in combination, thereby producing an exponential number of embodiments according to the combination of disclosed methods.

An assertion graph can be specified in an assertion graph language manually but with an assertion graph language as disclosed, it can also be derived automatically from a high level description, for example, from a register transfer language (RTL) description. Using such an assertion graph language, an assertion graph can also be derived directly from a circuit description.

Both methods for automatically deriving assertion graphs are potentially useful. For instance, if a particular RTL description and a corresponding circuit are very complex, manually generating an assertion graph may be prone to errors, but two assertion graphs could be automatically generated, one from the RTL description and one from the circuit design and the two assertion graphs can then be checked for equivalency. A more typical scenario, though, would be to automatically generate the assertion graph from an RTL description and then to drive the equivalence verification of the RTL description and the circuit description through circuit simulation as previously described.

It will also be appreciated that the methods herein disclosed or methods substantially similar to those herein disclosed may be implemented in one of many programming languages for performing automated computations including but not limited to simulation relation sequences, antecedent strengthening sequences and assertion graph satisfiability using high-speed computing devices. For example,FIG. 13illustrates a computer system to perform computations, for one such embodiment. Computer system1322is connectable with various storage, transmission and I/O devices to receive data structures and programmed methods. Representative data structures1301may include but are not limited to RTL descriptions1311, assertion graphs1312, and finite state models1313. Representative programmed methods1302may include but are not limited to abstraction programs1314, simulation relation programs1315, antecedent strengthening programs1316, and satisfiability programs1317. Components of either or both of the data structures and programmed methods may be stored or transmitted on devices such as removable storage disks1325, which may be accessed through an access device1326in computer system1322or in a storage serving system1321. Storage serving system1321or computer system1322may also include other removable storage devices or non-removable storage devices suitable for storing or transmitting data structures1301or programmed methods1302. Component data structures and programmed methods may also be stored or transmitted on devices such as network1324for access by computer system1322or entered by users through I/O device1323. It will be appreciated that systems such as the one illustrated are commonly available and widely used in the art of designing finite state hardware and software systems. It will also be appreciated that the complexity, capabilities, and physical forms of such design systems improves and changes rapidly, and therefore understood that the design system illustrated is by way of example and not limitation.

The above description is intended to illustrate preferred embodiments of the present invention. From the discussion above it should also be apparent that the invention can be modified in arrangement and detail by those skilled in the art without departing from the principles of the present invention within the scope of the accompanying claims.