Computing with Finite Automata (part 2) 290N: The Unknown Component Problem Lecture 10

Outline  Reachability analysis  Product computation (“product”)  Verification by language containment (“check”)  Determinization by subset construction (“determinize”)  Don’t-care minimization (“dcmin”)

Image Computation  Given a mapping of minterms from one Boolean space (input space) into another Boolean space (output space) For a set of minterms in the input space For a set of minterms in the input space The image of this set is the set of corresponding minterms in the output spaceThe image of this set is the set of corresponding minterms in the output space For a set of minterms in the output space For a set of minterms in the output space The pre-image of this set is the set of corresponding minterms in the input spaceThe pre-image of this set is the set of corresponding minterms in the input space Input space Output space Image Pre-image

Implicit Image Computation  Implements formula: Image(Y) =  x [R(X,Y) & C(X)]  Implicit methods by far outperform explicit ones Successfully computing images with more than 2^100 minterms in the input/output spaces Successfully computing images with more than 2^100 minterms in the input/output spaces  Operations & and  are basic Boolean manipulations They are efficiently implemented in the BDD package They are efficiently implemented in the BDD package  To avoid large intermediate results (during and after the product computation), operation AND-EXIST can be used, which performs product and quantification simultaneously (in one pass over the BDDs)

Improvements to Image Computation  Represent the transition relation as a set of partitions Natural when the FSM (automaton) is represented by the circuit Natural when the FSM (automaton) is represented by the circuit  Different approaches to computing the image (Coudert, 1989) Input splitting Input splitting Output splitting Output splitting Quantification scheduling Quantification scheduling  Hybrid methods Use partition clustering in addition to quantification scheduling (Berkeley, IWLS 95) Use partition clustering in addition to quantification scheduling (Berkeley, IWLS 95) Use non-linear quantification scheduling (CMU, ICCAD 01) Use non-linear quantification scheduling (CMU, ICCAD 01) Partitioning (OR-decomposition) of the transition relation Partitioning (OR-decomposition) of the transition relation “To split, or to conjoin” (mix the quantification scheduling and input/output splitting) (Somenzi, DAC 2000) “To split, or to conjoin” (mix the quantification scheduling and input/output splitting) (Somenzi, DAC 2000) “The far side of image computation” (Somenzi, ICCAD 2003) “The far side of image computation” (Somenzi, ICCAD 2003)  Tricks and speed-ups Disjoint decomposition Disjoint decomposition Caching of intermediate results, etc Caching of intermediate results, etc

Reachability Analysis  Many applications explore the reachable state space  Given an FSM (automaton) with the transition relation, find all the states reachable from the initial state  Apply image computation repeatedly to compute the sets of reachable states in the next iteration (“onion rings”) until convergence ReachedStates = InitialState; while ( there are new states ) ReachedStatesNew = Image( TransitionRelation, ReachedStates ); if (ReachedStatesNew = ReachedStates ) stop; ReachedStates = ReachedStatesNew ;  Reachability analysis uses sophisticated methods of image computation  Relies on numerous improvements Simplification using don’t-cares Simplification using don’t-cares Iterative squaring Iterative squaring Approximations, etc Approximations, etc

Outline  Reachability analysis  Product computation (“product”)  Verification by language containment (“check”)  Determinization by subset construction (“determinize”)  Don’t-care minimization (“dcmin”)

Product Computation  Starting from the initial states, run two automata “in parallel” and create a new automaton (product automaton) whose states are labeled by the pairs of states: one state from each automaton A state of the product is accepting iff both component states are accepting A state of the product is accepting iff both component states are accepting  The maximum number of states in the product automaton is equal to the product of the number of states of the argument However, many of these states may not be reachable from the initial state However, many of these states may not be reachable from the initial state

Explicit Algorithm  The two automata The linked lists of states {s} and {t} The linked lists of states {s} and {t} The accepting states are marked The accepting states are marked  Additional data structures Q: The FIFO queue of reached product states Q: The FIFO queue of reached product states H: The hash table hashing two component states into the product state H: The hash table hashing two component states into the product state  Initialization Create the initial state of the product automaton (s0,t0) by putting together the initial states of the argument automata, s0 and t0 Create the initial state of the product automaton (s0,t0) by putting together the initial states of the argument automata, s0 and t0 insert this state into Q and H insert this state into Q and H  Computation while Q is not empty, extract one product state (s,t) from Q for all product states (s’,t’) reachable in one transition from (s,t) for all product states (s’,t’) reachable in one transition from (s,t) if (s’,t’) is not in H (that is, (s’,t’) has not been visited) if (s’,t’) is not in H (that is, (s’,t’) has not been visited) create the new product state (s’,t’) create the new product state (s’,t’) insert (s’,t’) into Q and into H insert (s’,t’) into Q and into H else find (s’,t’) using the hash table H else find (s’,t’) using the hash table H add the transition from (s,t) into (s’,t’) add the transition from (s,t) into (s’,t’)

Example of Product

Verification by Language Containment  Works for deterministic, complete automata  Create the product of two automata, S and T  While visiting product states, keep track of the acceptance attribute of the component states Case 1: If we reach product state (s,t), in which s  S is accepting while t  T is non-accepting, the language of T cannot contain the language of S Case 1: If we reach product state (s,t), in which s  S is accepting while t  T is non-accepting, the language of T cannot contain the language of S Case 2: A symmetric case applies when the language of S cannot contain the language of T Case 2: A symmetric case applies when the language of S cannot contain the language of T If we encounter both Case 1 and Case 2, there is no language containment among S and T If we encounter both Case 1 and Case 2, there is no language containment among S and T Can terminate the product computation immediatelyCan terminate the product computation immediately If we do not encounter either Case 1 or Case 2, the languages accepted by S and T are identical If we do not encounter either Case 1 or Case 2, the languages accepted by S and T are identical Need to complete the product computation to prove thisNeed to complete the product computation to prove this

Example of Language Containment Check

Determinization by Subset Construction  Assume that ND transitions in the ND automaton happen at the same time It means that, at any moment, the ND automaton is in a subset of its states It means that, at any moment, the ND automaton is in a subset of its states The subset may contain more than one state The subset may contain more than one state  The point of determinization is to enumerate through all the subsets of states reachable from the initial state under any possible inputs Each subset of states of the ND automaton becomes a single state of the new deterministic automaton Each subset of states of the ND automaton becomes a single state of the new deterministic automaton The languages accepted by the ND automaton and its determinized version are the same The languages accepted by the ND automaton and its determinized version are the same

Determinization Algorithm  The automaton The linked lists of states {s}, with the accepting states marked The linked lists of states {s}, with the accepting states marked  Additional data structures Q: The FIFO queue of reached subsets of states Sk Q: The FIFO queue of reached subsets of states Sk H: The hash table mapping each reached subsets of states Sk into the corresponding state of the determinized automaton H: The hash table mapping each reached subsets of states Sk into the corresponding state of the determinized automaton  Initialization Create the initial state of the determinized automaton by creating the subset of states {s0} composed of the initial state of the ND automaton Create the initial state of the determinized automaton by creating the subset of states {s0} composed of the initial state of the ND automaton insert {s0} into Q and H insert {s0} into Q and H  Computation while Q is not empty, extract one subset of states Si from Q for all subsets of states Sj reachable in one transition from Si for all subsets of states Sj reachable in one transition from Si if Sj is not in H (that is, Sj has not been visited) if Sj is not in H (that is, Sj has not been visited) create the new state of the determinized automaton create the new state of the determinized automaton make the new state accepting if some state of Sj is accepting make the new state accepting if some state of Sj is accepting insert Sj into Q and into H insert Sj into Q and into H else find the new state corresponding to Sj using the hash table H else find the new state corresponding to Sj using the hash table H add the transition from Si into Sj add the transition from Si into Sj

Example of Determinization Regular expression: 0*100*1(00*1)*

Computing Reachable Subsets  Given a subset of states, what are other subsets of states that can be reached in one transition from the given subset?  Naïve explicit approach (using STG) Enumerate the minterms of the Boolean space of conditions Enumerate the minterms of the Boolean space of conditions For each minterm, find the subset of states reachable from the given subset in one iteration For each minterm, find the subset of states reachable from the given subset in one iteration Collect unique subsets Collect unique subsets  Better explicit approach (using STG) Compute partitioning on the condition space defined by all the states from the given subset Compute partitioning on the condition space defined by all the states from the given subset Start by considering the partition defined by one state and gradually refine it by adding other statesStart by considering the partition defined by one state and gradually refine it by adding other states Each partition corresponds to one subset of next states Each partition corresponds to one subset of next states Collect unique subsets Collect unique subsets This approach does not require enumerating through the minterms This approach does not require enumerating through the minterms

Computing Reachable Subsets (continued)  Implicit approach (using the monolithic relation) Restrict the monolithic transition relation Ri(x,cs,ns) to the given subset of states Si(cs): Ri(x,s) =  cs [Ri(x,cs,ns) & Si(cs)] ns  s Restrict the monolithic transition relation Ri(x,cs,ns) to the given subset of states Si(cs): Ri(x,s) =  cs [Ri(x,cs,ns) & Si(cs)] ns  s Given a state subset Si and its transition relation Ri(x,s), while Ri(x,s) is not empty, enumerate through reachable subsets: Given a state subset Si and its transition relation Ri(x,s), while Ri(x,s) is not empty, enumerate through reachable subsets: Extract one minterm m(x,s) from Ri(x,s)Extract one minterm m(x,s) from Ri(x,s) Restrict m(x,s) to only input variables x (call it m(x))Restrict m(x,s) to only input variables x (call it m(x)) Find Sj reachable from Si under m(x): Sj(s) =  x[Ri(x,s) & m(x)]Find Sj reachable from Si under m(x): Sj(s) =  x[Ri(x,s) & m(x)] Find Cij(x) labeling transition Si  Sj: Cij(x)=  s[Ri(x,s)  Sj(s)]Find Cij(x) labeling transition Si  Sj: Cij(x)=  s[Ri(x,s)  Sj(s)] Subtract this transition from Ri(x,s): Ri(x,s) = Ri(x,s) & NOT(Cij(x))Subtract this transition from Ri(x,s): Ri(x,s) = Ri(x,s) & NOT(Cij(x))  Hybrid approach (using state transition relations) Pre-compute BDDs of the transition relations for each state Pre-compute BDDs of the transition relations for each state Given a state subset Si and its transition relation Ri(x,s), while Ri(x,s) is not empty, proceed as described above… Given a state subset Si and its transition relation Ri(x,s), while Ri(x,s) is not empty, proceed as described above…

Don’t-Care Minimization  State minimization of pseudo-non-deterministic FSMs is a complex problem NP-complete, solved exactly for small FSMs only NP-complete, solved exactly for small FSMs only  A fast heuristic minimization is possible (“dcmin”) Requires an accepting don’t-care state to be present Requires an accepting don’t-care state to be present It is the case for most solutions to language equationsIt is the case for most solutions to language equations  Computation Complete with non-accepting state (if not complete) Complete with non-accepting state (if not complete) Define the don’t-care condition of each state to be the condition of its transition into the accepting DC state Define the don’t-care condition of each state to be the condition of its transition into the accepting DC state Create the incompatibility graph Create the incompatibility graph One vertex for each stateOne vertex for each state An edge exists between the two vertices if the care conditions of the two states do not overlapAn edge exists between the two vertices if the care conditions of the two states do not overlap Color the incompatibility graph Color the incompatibility graph Collapse the states, which have the same color Collapse the states, which have the same color

Example