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Analyzing Stack and Heap Bounds for Primitive Recursive Programs in PR-Hume Kevin Hammond, Pedro Vasconcelos, Sun Meng, Álvaro Rebón Portillo, Leonid Timochouk University of St Andrews, Scotland Greg Michaelson, Robert Pointon, Graeme McHale, Chunxiu Liu Heriot-Watt University, Scotland Jocelyn Sérot LASMEA, Clermont-Ferrand, France http://www.hume-lang.org

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Hume Higher-order Uniform Meta-Environment David Hume Scottish Enlightenment Philosopher and Sceptic 1711-1776

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Slide 3Kevin Hammond, University of St Andrews Hume Research Objectives Real-Time, Hard Space Functional Programming –Including Device Drivers, Derek! Virtual Testbed for Space/Time Cost Modelling Generative, Domain-Specific Language Design

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Slide 4Kevin Hammond, University of St Andrews Overview 1.Hume Language Design and Examples 2.Stack and Heap Usage for Primitive Recursive Programs 1.Cost Model 2.Inference Algorithm (Type-and-Effect System) 3.Results of the Analysis 4.Conclusions and Further Work

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Slide 5Kevin Hammond, University of St Andrews Hume Design Domain (1)

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Slide 6Kevin Hammond, University of St Andrews Hume Design Domain (2)

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Slide 7Kevin Hammond, University of St Andrews State of the Art... Embedded Systems Engineering –big trend to high level software design (UML etc.) –80% of all embedded software is now written in C/C++ –75% of embedded software is delivered late –bugs can cost $14,000 each to fix! A Major Problem with C/C++ is Poor Memory Management –explicit allocation, deallocation –pointer following –etc. etc. No Accurate Method for Determining Memory Usage –profiling, guesswork(!!), approximation

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Slide 8Kevin Hammond, University of St Andrews A New Direction?

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Slide 9Kevin Hammond, University of St Andrews Hume Design Objectives Targets embedded/critical applications –Hard real-time target –Formally bounded time and space –I/O managed through low-level “ports”/“streams” »Memory-mapped, timed, interrupts or devices –Asynchronous concurrency model (multicore?) –Simple, easily costed, exception handling mechanisms –Transparent design and implementation: correctness by construction –uses Haskell FFI to allow external calls in C/assembler etc. High level of expressiveness/productivity –Rule-based system: concise & clear using functional notation –Runtime errors reduced by strong polymorphic types –Structured reuse through higher order functions –Thread management simplified by implicit concurrency/parallelism –Elimination of memory errors through automatic memory management Reliability, Expressibility, Controllability, Predictability, Costability

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Slide 10Kevin Hammond, University of St Andrews FSA-derived Notation Based on generalised Mealy machines (see Michaelson et al. 2003) Boxes encapsulate a set of rules each mapping inputs to outputs Multiple inputs/outputs are grouped into tuples Sets of boxes are wired into static process networks (automata) Boxes repeat indefinitely once a result is produced (tail recursion) Boxes are asynchronous (ignored inputs/outputs) Wires are single-buffered box b... match (patt 11,..., patt 1k ) -> (expr 11,..., expr 1m ) |... | (patt n1,..., patt nk ) -> (expr 11,..., expr nm ) ;

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Slide 11Kevin Hammond, University of St Andrews Declaration & Metaprogramming Layer Hume Language Structure Coordination Layer Expression Layer

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Slide 12Kevin Hammond, University of St Andrews Expression Layer Purely functional, strict, higher-order, polymorphic, stateless Matches are total Timeouts/space overflows are managed through exceptions varid expr1 … exprn-- function/constructor application (expr1, …, exprn)-- tuples -- vectors (sized) [ expr1, …, exprn ]-- lists (bounded) let decls in expr-- local value declarations expr within cexpr-- timeout/space restriction if expr then expr else expr-- conditional expression case expr of matches-- case expression expr :: type-- type cast expr as type-- type coercion (cost implication)

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Slide 13Kevin Hammond, University of St Andrews Example: Parity Checker type Bit = word 1; type Parity = boolean; parity true = (“true”,true); parity false = (“false”,false); box even_parity2 in (b::Bit, p::Parity) out (show::string, p'::Parity) match (0,true) -> parity true | (1,true) -> parity false | (0,false) -> parity false | (1,false) -> parity true; wire even_parity2 (comm1, even_parity2.p' initially true) (comm2, even_parity2.p); type Bit = word 1; type Parity = boolean; parity true = (“true”,true); parity false = (“false”,false); box even_parity2 in (b::Bit, p::Parity) out (show::string, p'::Parity) match (0,true) -> parity true | (1,true) -> parity false | (0,false) -> parity false | (1,false) -> parity true; wire even_parity2 (comm1, even_parity2.p' initially true) (comm2, even_parity2.p); comm1 p (true,…) p’ b even_parity2 comm2

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Slide 14Kevin Hammond, University of St Andrews Full Hume PR-Hume HO-Hume FSM-Hume Hume Language Levels HW-Hume Full Hume recursive functions recursive data structures PR-Hume primitive-recursive functions primitive-recursive data structures HO-Hume higher-order non-recursive functions non-recursive data structures FSM-Hume 1st-order non-recursive functions non-recursive data structures HW-Hume no functions non-recursive data structures

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Slide 15Kevin Hammond, University of St Andrews Predicting the Cost ?

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Slide 16Kevin Hammond, University of St Andrews A Type-and-Effect Space Cost Model Relates language structure to usage –operational semantics expressed using sequent style Tuned to prototype Hume abstract machine interpreter –allows accuracy to be measured –can be exploited by compiler Both heap and stack can be cleared after a single box step Derived from theoretical work on cost analysis for parallel programs Stack; Heap

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Slide 17Kevin Hammond, University of St Andrews Sized Types Types are annotated with sizes –magnitude of natural –length of a list Sizes can be weakened to any greater size –defined as a subtyping relation –so 10 :: Nat 11 but not 10 :: Nat 9 – means unknown size, greater than any other size, so 10 :: Nat – means undefined size, less than any other size Will be used to determine recursion bounds 10 :: Nat 10 [6,1,2] :: [Nat 6 ] 3

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Slide 18Kevin Hammond, University of St Andrews Latent Costs Define costs for functions Allow costs to be captured for higher-order functions

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Slide 19Kevin Hammond, University of St Andrews Types and Effects for Stack/Heap Usage Size/Cost Expressions Types and Effects

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Slide 20Kevin Hammond, University of St Andrews Cost Rules: Basic Expressions

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Slide 21Kevin Hammond, University of St Andrews Cost Rules: Conditionals/Cases

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Slide 22Kevin Hammond, University of St Andrews Cost Rules: Function Applications

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Slide 23Kevin Hammond, University of St Andrews Cost Rules: Function Decls

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Slide 24Kevin Hammond, University of St Andrews Cost Inference Restrict cost annotations in types to be variables Separately collect constraints on variables So, standard unification can be used on types Constraints must be solved to determine closed costs

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Slide 25Kevin Hammond, University of St Andrews Cost Inference Restrict cost annotations in types to be variables Separately collect constraints on variables So, standard unification can be used on types Constraints must be solved to determine closed costs

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Slide 26Kevin Hammond, University of St Andrews Solving Recurrences For recursive programs, the effect system generates recurrence relations on constraints These are solved to give closed forms –use e.g. Mathematica, called during cost analysis –use an “oracle” of known recurrences –write a new recurrence solver Constraints are monotonically increasing

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Slide 27Kevin Hammond, University of St Andrews Example: Length For the recursive length function length [] = 0; length (x:xs) = 1 + length xs; The type inferred is: length :: [t7]^x21-{x19,x20}->nat^x27, {x19 >=7+6*x21, x20 >=2+4*x21,x27>=x21} {stack,heap}

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Slide 28Kevin Hammond, University of St Andrews Example: Take For the recursive take function take n [] = []; take n (x : xs) = if n > 0 then (x : take (n-1) xs) else []; The type inferred is: take ::nat^x53-{x51,x52}->[t118]^x56 -{x54,x55}->[t118]^x62, {x51>=0,x52>=0,x54>=10+9*min(x53,x56), x55>=7+13*min(x53,x56),x62>=min(x53,x56)}

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Slide 29Kevin Hammond, University of St Andrews Example: Twice/Map For the Higher-Order twice and map2 functions twice f x = f (f x); map2 f [] = []; map2 f (x:[]) = [f x]; map2 f (x:(y:[])) = [f x,f y]; add1 x = 1+x; h x = map2 (twice add1) x; The types inferred are: twice :: (t21-{x14,x15}->t21)-{x2,x3}->t21-{x4,x5}->t21, {x2>=0,x3>=0,x4>=6+max(1+x14,1),x5>=x15+x15} map1 :: (t54-{x62,x63}->t73)-{x45,x46}->[t54]^x64 -{x47,x48}->[t73]^x65, {x45>=0,x46>=0,x47>=6+max(1+max(1+x62,1),2), x48>=max(8+x63,3),x65>=1} add1 :: int-{x23,x24}->int, {x23>=7,x24>=4} h :: [int]^x112-{x75,x76}->[int]^x113, {x75>=30,x76>=25,x113>=1}

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Slide 30Kevin Hammond, University of St Andrews Results !

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Slide 31Kevin Hammond, University of St Andrews Results functionheap (est)stack(est) heap(GHC -O2) length: 10 181(181) 72(72) 1632 length: 100 1711(1711) 612(612) 2357 length: 1000 17011(17011) 6012(6012) 15630 length: 10000170011(170011) 60012(60012) 141626 reverse: 10 381(381) 88(98) 2080 reverse: 100 26862(26862) 810(813) 35395 reverse: 1000 2518512(2518512) 8008(8013)3051874 twice/map2: 1 25(25) 30(30) 1564 twice/map2: 2 38(38) 30(30) 1592 lift 129(144) 89(89) -- functionheap (est)stack(est) heap(GHC -O2) length: 10 181(181) 72(72) 1632 length: 100 1711(1711) 612(612) 2357 length: 1000 17011(17011) 6012(6012) 15630 length: 10000170011(170011) 60012(60012) 141626 reverse: 10 381(381) 88(98) 2080 reverse: 100 26862(26862) 810(813) 35395 reverse: 1000 2518512(2518512) 8008(8013)3051874 twice/map2: 1 25(25) 30(30) 1564 twice/map2: 2 38(38) 30(30) 1592 lift 129(144) 89(89) --

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Slide 32Kevin Hammond, University of St Andrews Results (Pump) Cost model applied to mine drainage example –implemented in prototype Hume abstract machine compiler –compared with measured dynamic runtime costs boxheap estheap actualstack eststack actual pump47381717 environ49472929 water54542424 logger1191053931 others1151067070 wires9684 - - totals480434179171 boxheap estheap actualstack eststack actual pump47381717 environ49472929 water54542424 logger1191053931 others1151067070 wires9684 - - totals480434179171 2.6KB v. 2.4KB 9%

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Slide 33Kevin Hammond, University of St Andrews The Reality !!

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Slide 34Kevin Hammond, University of St Andrews RTLinux Memory Usage (Pump) text data bss dec hexfilename 30146 52 30048 60246 eb56hume_module.o text data bss dec hexfilename 30146 52 30048 60246 eb56hume_module.o Module Size Used by hume_module 61904 0 (unused) rtl_sched 43200 0 [hume_module] rtl_fifo 10016 0 [hume_module] rtl_posixio 7232 0 [rtl_fifo] rtl_time 10064 0 [hume_module rtl_sched rtl_posixio] rtl 27216 0 [hume_module rtl_sched rtl_fifo rtl_posixio rtl_time] Module Size Used by hume_module 61904 0 (unused) rtl_sched 43200 0 [hume_module] rtl_fifo 10016 0 [hume_module] rtl_posixio 7232 0 [rtl_fifo] rtl_time 10064 0 [hume_module rtl_sched rtl_posixio] rtl 27216 0 [hume_module rtl_sched rtl_fifo rtl_posixio rtl_time]

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Slide 35Kevin Hammond, University of St Andrews Comparison with Java/Ada (Pump) Source –867 (234) for Java –251 (51) for Hume –? (206) for Ada Size of object code (byte code) –7991 (2003) for Hume –18316 (7360) for JVM –? (12045) for Ada Memory Requirement –9MB for Java (JVM, MacOSX) –60KB (RTLinux, bare RTS), 400KB (MacOSX, normal RTS) for Hume Execution Time –Hume is 9-12x faster than the JVM –The KVM is 30%-80% slower than the JVM Only one benchmark shown here Direct Real-time comparison requires rewrite to RTSJ But results have been repeated for smaller cases

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Slide 36Kevin Hammond, University of St Andrews Vehicle Sim. Statistics Thu Aug 21 19:06:06 BST 2003 Box Statistics: control: MAXRT = 53120ns, TOT = 1960041024ns, MAXHP = 57, MAXSP = 36 env: MAXRT = 9101600ns, TOT = 1580087776ns, MAXHP = 49099, MAXSP = 129 vehicle: MAXRT = 2973120ns, TOT = 2269933760ns, MAXHP = 49164, MAXSP = 133 Box heap usage: 98440 (99414 est) Box stack usage: 298 (319 est) Stream/MIDI Statistics: output1: MAXRT = 22688ns, TOT = 3188562720ns, MAXHP = 71, MAXSP = 1... Thu Aug 21 19:06:06 BST 2003 Box Statistics: control: MAXRT = 53120ns, TOT = 1960041024ns, MAXHP = 57, MAXSP = 36 env: MAXRT = 9101600ns, TOT = 1580087776ns, MAXHP = 49099, MAXSP = 129 vehicle: MAXRT = 2973120ns, TOT = 2269933760ns, MAXHP = 49164, MAXSP = 133 Box heap usage: 98440 (99414 est) Box stack usage: 298 (319 est) Stream/MIDI Statistics: output1: MAXRT = 22688ns, TOT = 3188562720ns, MAXHP = 71, MAXSP = 1...

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Slide 37Kevin Hammond, University of St Andrews Vehicle Sim. Statistics (2) Wire Statistics: control.0: MAX DELAY = 24544ns, MAXHP = 47 env.0: MAX DELAY = 67072ns, MAXHP = 11 vehicle.0: MAX DELAY = 33056ns, MAXHP = 47 vehicle.1: MAX DELAY = 32448ns, MAXHP = 2 vehicle.2: MAX DELAY = 9118688ns, MAXHP = 11 vehicle.3: MAX DELAY = 9135968ns, MAXHP = 2 Total heap usage: 197022 (199078 est) Total stack usage: 597 (640 est) Sat Aug 23 06:46:19 BST 2003 Wire Statistics: control.0: MAX DELAY = 24544ns, MAXHP = 47 env.0: MAX DELAY = 67072ns, MAXHP = 11 vehicle.0: MAX DELAY = 33056ns, MAXHP = 47 vehicle.1: MAX DELAY = 32448ns, MAXHP = 2 vehicle.2: MAX DELAY = 9118688ns, MAXHP = 11 vehicle.3: MAX DELAY = 9135968ns, MAXHP = 2 Total heap usage: 197022 (199078 est) Total stack usage: 597 (640 est) Sat Aug 23 06:46:19 BST 2003

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Slide 38Kevin Hammond, University of St Andrews Related Work (Analysis) Regions (Tofte) –explicit labelled memory areas, automatic deallocation Cyclone (Morrissett) –C syntax, region inference Sized Types (Hughes & Pareto) –properties of reactive systems, progress, not inference, not cost Camelot/GRAIL (Sannella, Gilmore, Hofmann et al.) –stack/heap inference from JVM bytecode, parametric costs, tail recursion Worst-Case Execution Time Analysis (Wellings et al) –Java/Ada, probabilistic cache/execution costs

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Slide 39Kevin Hammond, University of St Andrews Conclusions Cost Analysis for Primitive Recursive, Higher-Order, Polymorphic Functions –strict, purely functional notation –generates cost equations plus recurrences »recurrences solved by reference to an oracle or external solver –soundness results under construction Good Practical Results Obtained in a number of cases –no loss of accuracy for non-recursive definitions –exact worst-case solutions obtained for many definitions –size-aliasing can cause problems for composing polymorphic definitions

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Slide 40Kevin Hammond, University of St Andrews Further Work/Work in Progress Modelling –soundness proofs »under construction extends Hughes/Pareto MML to inference, different cost domain many technical problems solved, some remaining –resolve size aliasing problem –extend to general data structures –application to other language paradigms: non-strict, object-oriented, C/C++ Real-Time Models –Predictive real-time models need better hardware (especially cache) models –alternative real-time scheduling algorithms should be tried 1MEuro Framework VI Proposal (FET-OPEN) –with Jocelyn Sérot (LASMEA, France), Martin Hofmann (Ludwigs-Maximilian Univerität, Germany) and AbsInt GmbH (Saarbrücken, Germany)

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Slide 41Kevin Hammond, University of St Andrews Recent Papers Inferring Costs for Recursive, Polymorphic and Higher-Order Functional Programs Pedro Vasconcelos and Kevin Hammond To appear in Proc. 2003 Intl. Workshop on Implementation of Functional Languages (IFL ‘03), Edinburgh, Springer-Verlag LNCS, 2004. Winner of the Peter Landin Prize for best paper Hume: A Domain-Specific Language for Real-Time Embedded Systems Kevin Hammond and Greg Michaelson Proc. 2003 Conf. on Generative Programming and Component Engineering (GPCE 2003), Erfurt, Germany, Springer-Verlag LNCS, Sept. 2003. Proposed for ACM TOSEM Fast Track Submission FSM-Hume: Programming Resource-Limited Systems using Bounded Automata Greg Michaelson, Kevin Hammond and Jocelyn Sérot Proc. 2004 ACM Symp. on Applied Computing (SAC ‘04), Nicosia, Cyprus, March 2004 The Design of Hume Kevin Hammond Invited chapter in Domain-Specific Program Generation, Springer-Verlag LNCS State-of-the-art Survey, C. Lengauer (ed.), 2004 Predictable Space Behaviour in FSM-Hume”, Kevin Hammond and Greg Michaelson, Proc. 2002 Intl. Workshop on Implementation of Functional Languages (IFL ‘02), Madrid, Spain, Sept. 2002, Springer-Verlag LNCS 2670, ISBN 3-540-40190-3,, 2003, pp. 1-16

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Slide 42Kevin Hammond, University of St Andrews http://www.hume-lang.org

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Hume Higher-order Uniform Meta-Environment David Hume Scottish Enlightenment Philosopher and Sceptic 1711-1776

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Slide 44Kevin Hammond, University of St Andrews Results (Far Lifts) Cost model applied to recursive lifts boxheap estheap actualstack eststack actual lift1939610 218 206 lift2939 392 218152 floors35 31 21 21 logger715261 20 20 schedule78 70 35 35 totals27061364 512434 boxheap estheap actualstack eststack actual lift1939610 218 206 lift2939 392 218152 floors35 31 21 21 logger715261 20 20 schedule78 70 35 35 totals27061364 512434 12.8KB v. 7.2KB

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