1. Schema-based Program Synthesis and the AutoBayes System Part II Johann Schumann SGT, NASA Ames

2. Example Generate a program that finds the maximum value of a function f(x): max f(x) wrt x univariatemultivariate Note: the function might be given as a formula or a vector of data

3. Schemas for univariate optimization schema(max F wrt X, C) :-... as before schema(max F wrt X, C) :- length(X, 1), % F is a vector of data points F(0..n) C = let(sequence([ assign(mymax,0), for(idx(I,0,n), if(select(F,I) > mymax, assign(mymax, select(F,I)), skip)... ]), comment([‘The maximum is found by iterating...’]), mymax). schema(max F wrt X, C) :- length(X, 1), % instantiate numeric solution algorithm % e.g., golden section search C =... schema(max F wrt X, C) :-....

4. Schema for univariate optimization 1.build the derivative: df/dx 2. set it to 0: 0 = df/dx 3.solve that equation for x 4.the solution is the desired maximum schema(max F wrt X, C) :- % INPUT (Problem), OUTPUT (Code fragment) % guards length(X, 1), % calculate the first derivative simplify(deriv(F, X), DF), % solve the equation solve(true, x, 0 = DF, S), % possibly more checks % is that really a maximum? simplify(deriv(DF, X), DDF), (solve(true, x, 0 > DDF, _) -> true ; writeln(‘Proof obligation not solved automatically’) ), XP = [‘The maximum for‘, expr(F), ‘is calculated...’], V = pv_fresh, C = let(assign(V, C, [comment(XP)]), V)..

5. Demo Generation of multiple programs –-maxprog –-maxprog N -fastest (coarse approximation) Control for numeric solvers –pragma schema_control_arbitrary_init_values –pragma schema_control_use_generic_optimize Tracing pragmas The necessity of constraints

6. Multivariate Optimization Task: minimize function F(X) wrt X Algorithm: double* minimze(F){ double* x0 = pick_start(); int converging = 1; while (converging){ double step_length = 0.1; double step_dir = -gradient(F,x0); x1 = x0 + step_length * step_dir; if (fabs(F(x1) - F(x0)) < 0.001) converging = 0; else x0 = x1; } start somewhere go down along the steepest slope when you come to a flat area, return that (local) minimum Many design decisions where to start? how to move? when to stop?

7. Multivariate Optimization schema(max F wrt X, C) :- % IN, OUT % guards: here none length(X,Y), Y > 1, % divide and solve subproblems schema(getStartValue(F,X), C_Start), % recursive schema calls schema(getStepDirection(F,X), C_Dir), schema(getStepSize(F,X), C_Size), % assemble code segment X0=pvar_new(X), % get a new PROGRAM variable C = block([local(X0,double)], series( [ assign(X0, C_start), while_converging(X0, assign(X0, +([X0, *([C_Dir, C_Size]))) ]) ).

8. Multivariate optimization II The schemas generate code in an intermediate language procedural elements local variables, lambda blocks sum(..), while_converging(..) --> loops X0=pvar_new(X), C = block([local(X0,double)], series( [ assign(X0, C_start), while_converging(X0, assign(X0, +([X0, *([C_Dir, C_Size]))) ]) ). double v_0; double E; v_0 = -99; E = 1e10; while (E > 0.001){ y = sin(v_0); v_0 = V_0 - cos(v_0) * 0.01; E = fabs(y - sin(v_0)); } generated code for max sin(v) wrt v Important: variables in specification or program are NOT Prolog variables

9. Why schema-based synthesis? Multiple algorithm variants can be automatically constructed The “best” one is chosen by the user or selected via constraints some possibilities for getStepDir

10. AB Schema Hierarchies Schemas to break down statistical problem –Bayesian independence theorems -- works on Bayesian graphs Schemas to solve complex statistical problems –instantiate (iterative) clustering algorithms –handling of time series problems Schemas to solve atomic problems –instantiate PDF and maximize (symbolically) –instantiate numerical solvers (see last slides) auxiliary schemas –initialization of clustering algorithms –data pre-processing (e.g., [0..1] normalization)

11. AB Schema Hierarchy Static tree structure AB uses two kinds of schemas –schemas for probabilistic problems –schemas for formula

12. Schemas and AB Model The AB schemas have to use all information from the input specification, which is stored in the Prolog data base (AB model) Problem: schemas can modify the model, which must be undone during backtracking –add new statistical variables –remove dependencies for subproblems Solutions: –add model as parameters: schema(Prob, C, M_in, M_out) and everywhere else –keep a model stack (similar to the dynamic calling environments in procedural languages) and use backtrackable asserts/retracts

13. Backtrackable Global Stuff Global data in Prolog are handled using assert/retract or flags. All other data are local to each clause p(X) :- q(X,Z), r(Z). % X, Y, Z local to clause Asserts are not backtrackable p(X) :- assert(keep(X)),..., fail. The “keep(X)” is kept in the data base even after backtracking Work-around: add global variables as parameter to all predicates (impractical) p(X, GL_in, GL_out) :- GL_out = [keep(X)|GL_in],... Backtrackable bassert/bretract requires some low-level additional C-programs (but has clean semantics)

14. Schema Control schema applicability is controlled via guards order of application: order in Prolog file How to enforce/avoid certain schemas –autobayes pragmas, but that’s not really fun –doesn’t work for nested applications: inner loop: symbolic solutions only outer loop: enable numeric loop –generate them all and decide later or pick “fastest” schema control language is a research topic –extend declarative AB language –how to talk about selection of iterative algorithm in a purely declarative language?

15. The AB Infra Structure term utilties rewriting engine symbolic system: –simplifier –abstraction (range, sign, definedness) –solver pretty printer (code, intermediate language) comment generation

16. Term utilities implemented on top of Prolog a lot of functional-programming style predicates for –lists, sets, bags, relations –terms, AC-terms operations –term_substitute, subsumption, differences between term sets...

17. Rewriting Engine A lot of stuff in AB is done using rewriting (but not all) small rewriting engine implemented in Prolog –rewriting rules are Prolog clauses –conditional rewriting, AC-style rewriting –Evaluation: eager: apply first top-down lazy: apply bottom up –continuation: pure bottom-up or dove-tailing –handle for attachment of prover/constraint solver –compilation of rewriting rules for higher efficiency

18. Rewriting Rules Can combine pure rewriting with Prolog programming in the body of the rewrite rule % NAME, STRATEGY, PROVER, ASSUMPTIONS, IN, OUT trig_simplify('sin-of-0', [eval=lazy|_],_,_, sin(0), 0) :- !. trig_simplify('sin-of-pi-over-6',[eval=lazy|_],_,_,sin(*([1/6, pi])), 1/2) :- !. trig_simplify('cos^2+sin^2',[eval=eager|_],_,_, +(Args), +([1|Args3])) :- select(cos(X)**2, Args, Args2), select(sin(X)**2, Args2, Args3), !.

19. Compilation and Rewriting Group and compile rewrite rules (statically) ?- rwr_compile(my_simplifications, [trig_simplify, remove_const_rules ] ). Call the rewriting engine rwr_cond(my_simplifications, true, S, T). Calling with time-out

20. Symbolic System Symbolic system implemented on top of the rewriting engine + Prolog code for solvers, etc assumption-based rewriting –X/Y -- (not(Y = 0)) --> X simplification (lots of rules) calculation of derivatives (deriv(F,X) as operator) Taylor-series expansion,... equation solver –polynomial solver –Gauss-elimination for sets of linear equations –sequentialization of equation systems

21. The AB Intermediate language strict separation between synthesis and code generation small procedural intermediate language with some extensions –sum(..), prod(..), simul_assign(..), while_converging(...) –Annotations for comments, and pre/post/inv formulas code generator for different languages/targets –C++/Octave –C/Matlab, C/standalone –ADA/SparkADA, Java (both “unsupported/in work/bad shape”) Pretty-printer to ASCII, HTML, LaTeX

22. Extending AutoBayes some extensions are straight-forward: add text-book formulas additional symbolic simplification rules might be required adding schemas requires substantial work –“hard-coded” schema as first step –applicability constraints and control –functional mechanisms to handle scalar/vector/matrix cases are available –support for documentation generation –no schema language, Prolog syntax used

23. Non-Gaussian PDF Data characteristics are modeled using probability density functions (PDFs) Example: Gaussians, exponential,... AB contains a number of built-in PDFs, which can be extended (hands-on demo) Having multiple PDFs adds a lot of power over libraries

24. Example For clustering, often Gaussian distribution of data is used. How about angles: 0 == 360 you get 5 clusters A different distribution (vonMises-Fisher) automatically solves this problem In AutoBayes: just replace the “gauss” by “vonmises1” -- no programming required multiple PDFs in one spec

25. Sample Generation We have used: –MODEL ---> P ---(data)--> parameters The model can be read the other way round: generate me random data, which are consistent with the model –MODEL ---> P ---(parameters)--> data Very useful for –model debugging/development –debugging and assessment of synthesized algorithms

26. AutoBayes and Correctness practical synthesis: forget about correct-by- construction, but detailed math derivations, which can be checked externally (e.g., by Mathematica) literature references in documentation/comments generation of test harness and sample data checking of safety properties (“AutoCert”) [Cade2002 slide set]

27. AutoBayes as a Prolog Program AutoBayes is a pretty large program –~180 prolog files, 100,000LoC (with AutoFilter) Heavy use of –meta-programming (call, etc.) –rewriting (using an engine implemented in Prolog) –functional programming elements for all sorts of list/vector/array handling –backtracking and backtrackable global data structures –procedural (non-logical) elements, e.g., file I/O, flags, etc. no use of modules but naming conventions everything SWI Prolog + few C extensions to handle backtrackable global counters and flags

28. AutoBayes Weak Points The input parser is very inflexible (uses Prolog operators) Very bad error messages–often just “no” no “schema language”: AutoBayes extension only by union of Prolog/domain specialist Only primitive control of schema selection: need for a schema- selection mechanism not all schemas are fully documented large code-base, which needs to be maintained

29. Summary AutoBayes suitable for a wide range of data analysis tasks AutoBayes generated customized algorithms AutoBayes schema-based program synthesis + symbolic logic + functional + procedural elements used AutoBayes extension: easy to very hard AutoBayes debugging: a pain, but explanations and LaTeX output very helpful AutoBayes is NASA OpenSource: bugfixes/extensions always welcome AutoBayes has a 160+ pages Users manual AutoBayes useful for classroom projects to PhD projects