Download - Optimizations for LTL Synthesis

Graz University of Technology

Professor Horst Cerjak, 19.12.20051

Barbara Jobstmann San Jose, Nov 15 Optimizations for LTL Synthesis

Optimizations for LTL Synthesis

Barbara JobstmannRoderick Bloem

Graz University of Technology, Austria

15 November 2006




Motivation

● Synthesis from specification● Correct by construction - no verification● You say what, it says how● Theory well established

● Long history: Church (early 60’s)● Theory: Rabin, Ramadge/Woham, Pnueli/Rosner

● What has changed since then?




Outline

● Introduction

● Approaches and optimizations for LTL synthesis

● Lily

● Conclusion




LTL Synthesis

● Automatically build design from specification● Input

● Set of LTL formulae, e.g. G(s1→ ¬s2), (s1 U s2),…● Partition of the atomic propositions (input/output signals)

Reactive systems: Some signals controlled by system others not● Output

● Automatically created functionally correct finite-state machine (Moore)

● Proposed for LTL by Pnueli, Rosner (POPL'89)

● Difference between monitoring and synthesis● Monitoring: build passive system (nondeterministic)● Synthesis: build reactive system (deterministic)




Key Observation

● Moore machine● Input signal r, output signal a● r=1,r=0 .... input

alphabet● a=1,a=0 .. output

alphabet

● Tree (regular)● r=1,r=0 .... directions D● a=1,a=0 .. alphabet Σ

(labeling)

Σ-labeled D-tree




Idea

1) Build a tree automaton

● Accepts all trees representing moore machines that fulfill spec φ● Directions are input values (D=2I, input signals I)● Alphabet are output values (Σ=2O, output signals O)● Automaton accepts all Σ-labeled D-trees where all paths satisfy the

given formula φ

2) Compute language emptiness

3) Build FSM from the witness (a Σ-labeled D-tree)




Necessary Theory

● Infinite game theory● Automata theory

● Branching mode (Deterministic, Nondeterministic, Universal, Alternating)

● Acceptance condition (Büchi, Co-Büchi, Weak, ..)● Input element (Word,Tree)● Use of KV's abbreviation (e.g.,NBW,UCT,...)




Alternating Word Automata

● N+U branching (edges we can follow and edges we must follow)

● Notation:● Circles represent states● Boxes represent universal

edges● Edges are labeled with sets

of labels




Tree Automata

Universal edges:Foreach direction, follow onlythe matching edges




Universal and tree branching




Safraful Approach [PR89]

1) Build an NBW for φ

2) Convert to DRW● Safra's determinizations algorithm

3) Split alphabet into I/O DRT

4) Check Language Emptiness● Build transducer (fsm)

φ

NBW

Build NBW

DRW+i/o

Build DRW

DRT

Build DRT

FSM

Lang. Emp.




Issues

● 2EXP worst case complexity● Safra's determinization construction

φ

NBW

Build NBW

DRW+i/o

Build DRW

DRT

Build DRT

FSM

Lang. Emp.

expblow-up

expblow-up




Solutions

● Concentrate on subsets of LTL● Alur, Madhusudan, Nam (BMC'03, STTT'05)● Wallmeier, Hütter, Thomas (CIAA'03)● Harding, Ryan, Schobbens (TACAS'05)● Piterman, Pnueli, Sa'ar (VMCAI'06)

● Full LTL (Safraless approach)● Kupferman, Vardi (FOCS'05)● Kupferman, Piterman, Vardi (CAV'06)




Safraless Approach [KV05]

1) Build a UCT ● Negate φ● Build an NBW for ¬φ● Invert NBW → UCT

2) Convert to AWT

3) Convert to NBT

4) Check Language Emptiness

φ+i/o

UCT

Build UCT

AWT

Build AWT

NBT

Build NBT

FSM

Lang. Emp.

L(UCT) = Ltree(φ)

L(AWT) LT(φ)LT(φ) L(AWT)

L(NBT) LT(φ)LT(φ) L(NBT)




List of Optimizations

1) Game-based Heuristic language emptiness

2) Simulation-basedcf. Alur, Henzinger, Kupferman, Vardi (CONCUR’98)cf. Fritz, Wilke (FSTTCS’02)

3) Simplify KV-constructionsBuild AWT, Build NBTcf. Gurumurthy, Kupferman, Somenzi, Vardi (CHARME’05)

4) Process steps incrementalCombine steps

φ+i/o

UCT

Build UCT

AWT

Build AWT

NBT

Build NBT

FSM

Lang. Emp.




Game-based Optimization

● Heuristic language emptiness● Alternating Tree Automaton● Idea

● Find states with empty language (accept no tree)● Runs with non-accepting path are rejected● Environment can force a non-accepting path● Sufficient (but not necessary) for language emptiness

φ+i/o

UCT

Build UCT

AWT

Build AWT

NBT

Build NBT

FSM

Lang. Emp.




Game-based Optimization

● Game● System picks the label and the nondeterminism● Environment picks direction and universality

● State s is winning for environment → LT(s) empty




Example (1)

● φ=GF timer → G(light → light U timer) ● UCT with co-Büchi state (n3)




Example (2)

● Game:● Systems aims to avoid infinitely many visits to n3● Environment aims to force those visits● Co-Büchi game

weak automaton

φ=GF timer → G(light → light U timer)




List of Optimizations

1) Game-based Heuristic language emptiness

2) Simulation-basedcf. Alur, Henzinger, Kupferman, Vardi (CONCUR’98)cf. Fritz, Wilke (FSTTCS’02)

3) Simplify KV-constructionsBuild AWT, Build NBTcf. Gurumurthy, Kupferman, Somenzi, Vardi (CHARME’05)

4) Process steps incrementalCombine steps

φ+i/o

UCT

Build UCT

AWT

Build AWT

NBT

Build NBT

FSM

Lang. Emp.




Lily - Linear Logic sYnthesizer

● First tool to offer synthesis for full LTL● Based on Fabio Somenzi's Wring ● Implements KV05 and all mentioned

optimizations● http://www.ist.tugraz.at/staff/jobstmann/lily/




LTL Specification: Traffic Light

G(F(timer=1)) -> (G(fl=1 -> (fl=1 U timer=1))G(hl=1 -> (hl=1 U timer=1))G(car=1 -> F(fl=1))G(F(hl=1))G(!(hl=1 * fl=1)))

.inputs timer car

.outputs fl hl

hl

hl

fl

sensor(ec)




Generated System: Traffic Lightmodule traffic(hl,fl,clk,car,timer); input clk,car,timer; output fl,hl; wire clk,fl,hl,car,timer; reg state; assign hl = (state == 0); assign fl = (state == 1); initial state=0; always @(posedge clk) begin case(state) 0: begin if (timer==0) state = 0; if (timer==1 && car==1) state = 1; if (car==0) state=0; end 1: begin if (timer==1) state = 0; if (timer==0) state = 1; end endcase endendmodule //traffic




Conclusion

● First implementation of synthesis for full LTL● Optimizations are enabling factor● Our examples are small but useful for property

debugging (or learning LTL)● Future




Thank you for your attention!

Download - Optimizations for LTL Synthesis

Top Related