Tabled Prolog

David S. Warren

XSB, Inc.

Stony Brook University

Outline• Introduction

– Symmetric, Transitive Relations

• Basic Tabling Uses– Databases (and Datalog)– Grammars– Automata Theory– Dynamic Programming

• Advanced Tabling– Evaluating Recursive Definitions– Program Processing

• Interpreters• Abstract Interpreters

• Beyond Simple Tabling– Negation– Aggregation– Constraints

• Siblings in my family:

• If my sister is my sibling, then I’m hers:

– (symmetric, a problem in Prolog)

Family Relations

sibling(X,Y) :- sibling(Y,X).

sibling(nancy,david).sibling(david,jane).sibling(jane,rick).sibling(rick,emily).

Symmetry

• Symmetric rule will always cause a loop in Prolog.– Why? (Explore the Prolog program…)

– What can we do to “fix” it?• Prolog hackery: cuts, asserts, extra arguments, …

• Is there a more general/universal “fix”?– What is the general problem to be fixed?

– Logic is OK, but Prolog’s evaluation is problemmatical.

Repeated Computation

• The problem is that Prolog repeats computations again and again.

• So we can use “tables” to store the fact that we’ve done a computation, and its results.– Then if we’re about to do a computation and it

is already in the table, we just use the results from there and don’t redo it.

Observations

• Does it solve sibling/2’s problem? (Explore)

• When will it eliminate loops in Prolog?– Always?– Sometimes?

• When? Can we say something general?

Transitivity

• If I am nancy’s sibling and jane is my sibling, then jane is nancy’s sibling:

• Add this rule to the Prolog program. (explore)

• There is a problem here. Will tabling solve it as well?

sibling(X,Y) :- sibling(X,Z), sibling(Z,Y).

Tabling Intuition• Prolog program executed by a growing and shrinking set of

virtual procedural machines:– When calling a predicate, a machine duplicates itself once

for each matching clause.– When an operation fails, that machine disappears.

• For tabling, when calling a predicate, look in table to see if it’s already been called:– If not, record the call in table, call it, and for each machine

that returns, record its answer with its call.– If so, duplicate self for every answer, and suspend self

waiting for more answers; when one shows up, duplicate self for it.

• Asynchonicity is necessary!

Summary

• Many simple rules cause Prolog to loop.

• Tabling eliminates redundant computation by saving previous computations and their results in a table.

• All programs that don’t use structures (lists, or function symbols) will terminate under tabled evaluation.

Basic Applications - Datalog

• Prolog without data structures is a natural relational language: Databases

• Explore examples• Extends relational databases by including recursion:

– supports transitive closure, span-of-control

• How does a DB Query language differ from a Programming language?– To be a “Database,” evaluation must be guaranteed to terminate.

Prolog evaluation doesn’t; Tabled Prolog does.

• Relational Databases include negation (or set difference.)– With recursion, negation is more complicated.– We’ll look at negation later

Context-free Grammars

• Easy to write grammars in Prolog:– CFG Rule: A B C– Prolog Rule: a(S0,S) :- b(S0,S1),c(S1,S).

Input Str: …………………………………S0 S1 S

AB C

CF Grammars

• Normally represent input Str as a list:– Position is represented

– CF Rule: A B t C

– Prolog rule:

a(S0,S) :- b(S0,S1), connect(S1,t,S2), c(S2,S).

With general “connect” fact (often called ‘C’/3): connect([Term|S],Term,S).

Example CF Grammar

Simple Expression Grammar 1 (explore)

expr --> term, [+], expr.expr --> term.term --> factor, [*], term.term --> factor.factor --> [X], {integer(X)}.factor --> ['('], expr, [')'].

[Prolog has DCG preprocessor to add the “input variables” to --> rules for convenience. (See example.)]

CF Grammar Example 2

expr --> expr, [+], term.expr --> term.term --> term, [*], factor.term --> factor.factor --> [X], {integer(X)}.factor --> ['('], expr, [')'].

What’s the difference from Example 1?Why does it matter?

Simple Expression Grammar 2 (explore)

:- auto_table.

CF Grammar Discussion

• Prolog infinitely loops when given left-recursive rules (parses by “recursive descent.”)

• Tabled Prolog handles all CFG’s (parses by “chart parsing”, variant of “Earley recognition.”)

• Complexity?– Polynomial (whereas rec desc is exponential)– In theory cubic, if grammar is in Chomsky form

• But an issue with input representation as lists.• For tabling, better to represent input with facts of form:

word(Loc,Word,Loc+1).

Grammar Questions

What we’ve seen is recognition: accepting or rejecting input Str.

• Is this a Datalog problem?• How do we parse? I.e., construct parse tree.• Why can’t we do it in the same time as

recognition?• Can we do it in same time as recognition +

linear time for each parse? How?

Grammar Parsing

• Consider grammar:

• DCG for parsing:

• Input: aaaaa…aaaaab

• Complexity, O(n³) but no parse!!

A A AA a

:- auto_table.

a(r1(P1,P2)) --> a(P1), a(P2).

a(a) --> [a].

Parsing (better)

:- auto_table.

a --> a, a.

a --> [a].

a(r1(P1,P2),S0,S) :- a(S0,S1),a(S1,S),

a(P1,S0,S1), a(P2,S1,S).

a(a) :- ‘C’(S0,a,S1).

PTQ: A More Complex Grammar

• “The Proper Treatment of Quantification in Ordinary English,” by Richard Montague– “There are no significant differences between logical

languages and natural languages.”– Proposed a formal grammar for (a fragment of) English, and a

formal (model theoretic) semantics!– Examples:

• “John seeks a unicorn” vs. “John finds a unicorn”• “John seeks a woman” de dicto/de re ambiguity• “The temperature is ninety and rising” should not imply “ninety is

rising.”• “Every man loves a woman” is ambiguous.• … more …

Montague Grammar

• My Thesis, took 2-3 years to develop in Lisp– With XSB, took 2-3 days…

• Syntax:– Not Context-free, Left-recursive, infinitely many parses

(variants), …– Complex parsing

• Semantics:– By translation to Intensional Logic (a type theory)– Required simplifications:

• β-reduction, • “extensionalization” (explore…)

Automata Theory (cursory)

• Represent Finite State Machines by facts:– Transition relation: m(MId,Q1,S,S2).– Initial state: mis(MId,Qi).– Final state: mfs(MId,Qf).

• Strs by facts:– Str contents: Str(SId,Loc0,S,Loc1).– Str length: Strlen(SId,Len).

FSA Accepts a Stringaccept(MId,StrName) :- mis(MId,StateStart), recog(MId,StrName,StateStart,StateFinal,0,StrFinal), mfs(MId,StateFinal), Strlen(StrName,StrFinal).

% regular transitionsrecog(MId,StringName,MState0,MState,SLoc0,SLoc) :- string(StringName,SLoc0,Symbol,SLoc1), m(MId,MState0,Symbol,MState1), recog(MId,StringName,MState1,MState,SLoc1,SLoc).% Epsilon transitionsrecog(MId,StringName,MState0,MState,SLoc0,SLoc) :- m(MId,MState0,'',MState1), recog(MId,StringName,MState1,MState,SLoc0,SLoc).

FSA Discussion• Does recognition need tabling? Why or why not?• Exercises: Write programs that:

– Generate an FSA equivalent to the intersection of two FSAs.– Generate an epsilon-free FSA equivalent to a given FSA.– Generate an deterministic FSA equivalent to a given FSA.– Generate a minimal-state FSA equivalent to a given FSA.

• Can you write a program that determines whether the intersection of languages of a CFG and an FSA is non-empty?– Hint: Note the representation of a string is the same as that of an FSA

that recognizes exactly that string.• What would be the difference in the programs written in Prolog vs. Tabled

Prolog?

Dynamic ProgrammingKnapsack Problem (trad.)

• Given a set of items, find whether a packing of a knapsack with items with total weight w exists.

• Items numbered 1 to n: item(I,K) means item #I weighs K

• O(2 ) queries to ks/2, but only O(n*w) different ones. – explore

% ks(I,K) if a subset of items 1..I sums to Kks(0,0). % empty set sums to 0ks(I,K) :- I>0, I1 is I-1, ks(I1,K). % exclude Ith elementks(I,K) :- I>0, item(I,Ki), K1 is K-Ki, % include Ith element K1 >= 0, I1 is I-1, ks(I1,K1).:- ks(n,w).

:- table ks/2.

n

A General Evaluation Strategy for Recursive

Definitions• Have seen examples of tabling as

an extension of Prolog evaluation. • Now consider tabling in a more

general context:– As a General Evaluation Strategy for

Recursive Definitions• Functions and Evaluation• Tabled Evaluation• Multi-valued Functions and Relations

Mathematical Induction

• High school math class…• Define functions on natural numbers

• We proved that f(n) = n²• But how did we know f was well-defined?• How did we evaluate it?

f(0) = 0

f(n) = f(n-1) + 2*n - 1

Evaluating Inductive Definitions

• Evaluate f(8)– Bottom-up evaluation– Top-down (demand-driven) evaluation

f(n) = if n=0 then 0 else n+f(n-1)

8 7 6 5 4 3 2 1 0 : n

: f(n)0

0

1

1

3

3

6

6

10

10

15

15

21

21

28

28

36

36

Bottom-up:

Top-down: : f(n)

Fibonacci

• Bottom-up good for fib.

5 4 3 2 1 0 : n

fib(n) = if n=0 then 1 else if n=1 then 1 else fib(n-1)+fib(n-2)

Bottom-up

Top-down

1

11111

1

111

2

222

3

33

5

5

8

8

: fib(n)

: fib(n)

log2

• Top-down is good for log2

log2: lg(n) = if n = 1 then 0 else 1 + lg(n div 2)

16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 : n

0112222333333334 : lg(n)bu

: lg(n)01234td

Bottom-up vs. Top-downSummary

• Bottom-up and Top-down evaluation are “incommensurate”: neither one is uniformly better than the other.– Bottom-up is exponentially better than top-

down for fib– Top-down is exponentially better than bottom-

up for log2

• Can we get the best of both strategies?

Tabled Evaluation

• Top-down demand-driven, but:– Save intermediate results in a table, so– Future requests use the table.

• Combines top-down demand-driven with bottom-up non-redundancy.

Tabled Evaluation (fib)

• Tabled evaluation similar to bottom-up on fib.

Top-downw/ tabling

1 12358 : fib(n)

5 4 3 2 1 0 : n

Bottom-up 1 12358 : fib(n)

fib(n) = if n=0 then 1 else if n=1 then 1 else fib(n-1)+fib(n-2)

Tabled Evaluation (log2)

• Tabled evaluation similar to top-down on log2

log2: lg(n) = if n = 1 then 0 else 1 + lg(n div 2)

16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 : n

: lg(n)01234td

td w/tabl : lg(n)01234

A Theoretical Oddity?

• Tabled evaluation was proposed in 60’s by D. Michie, but never pursued.– Is the overhead too high?– If a functional programmer is dumb enough to

write fib as doubly recursive, s/he gets what s/he deserves! Write a better program.

• But …

On to Recursive Definitions

• Inductive definitions are:– Defined on the natural numbers (or other well-

founded set)– Required to be defined…

• Explicitly for minimal argument(s)

• In terms of smaller elements for non-minimal arguments

• Recursive definitions aren’t!

New Problems with Recursive Definitions

• Computationally problematical– No ordering, so where is the “bottom” for

bottom-up evaluation?– Demand evaluation loops, e.g., when f(17) =

……f(17)……

• Semantically problematical for functions– May be no solutions: e.g., f(17) = f(17)+1– May be many: e.g., f(17) = f(17)

One Approach:Multi-valued Functions

• Permit functions to be multi-valued: – Result of a function is a set of values

– Allow multiple “definitions” for a function

– Functions are composed point-wise

– Nondeterminism is a useful construct in a variety of applications

• Naturally resolves semantic problems with self-loops:– Define the least fixed point, on the lattice of sets

– f(17)=f(17) interpreted as f(17) = {}

Self-Loops and Top-Down vs. Tabled Evaluation

• Self-loops, when definitions unfold to:– f(17) = … f(17) …

• TD, not remembering anything, can’t avoid loops in definitions with this form.

• Tabled can terminate those loops, since they don’t contribute to defining a result, and look for other ways to determine f(17).

• So tabled evaluation will terminate for definitions for which TD will infinitely loop.

Recursive Definitions of Relations

• Multi-valued functions?

• or Relations?

• Differences:– Syntax, Modes, Higher Order, …– Prolog uses relations, and that is our interest here.

fib(n) = if n=0 then 1 else if n=1 then 1 else fib(n-1)+fib(n-2)

fib(0,1).fib(1,1).fib(N,F) :- N > 1, N1 is N-1, N2 is N-2,

fib(N1,F1), fib(N2,F2), F is F1+F2

Programming Language Implementations

• Programming Languages use recursive definitions as programs and top-down evaluation as the execution strategy.

• Evaluation of recursive definitions was (initially) hard to implement– Fortran didn’t implement it– PL/1 said not to use it for efficiency reasons

• But it made programming MUCH easier– E.g., C.A.R. Hoare’s experience with quicksort

Invention of Quicksort(Reconstructed anecdote inspired by C.A.R. Hoare)

• Quicksort:– Choose a “random” element from array– Partition array with all elements greater than the chosen one to top

and all less to bottom– Recurse on each partition

• Hoare invented and implemented it in Fortran, explicitly handling all the stacks in arrays. Very complicated and difficult.

• Was amazed at how “trivial” it became when it was expressed as a recursive function in Algol60.

CLAIM:

• Tabled Evaluation can make relational programming MUCH easier.– As recursion made programming (and

algorithm development) much easier.– And it ain’t that easy to implement either….

• (but that’s another talk)

Evidence: Applications

• Grammars and Automata Theory, as I hope we have seen.

• Program Analysis– Abstract Interpretation– (Model Checking)

Homework Assignment 1:Program Interpretation

• Use XSB to write an interpreter for Pascal (a subset).

Hints:– Represent state as a list of environments:

• An environment is a set of [variable,value] pairs

• State contains envs containing variables accessible at the current program point: one env for each (statically) enclosing block.

Example ofProgram State Representation

prog P:

var A, B

proc Q:

var C, D

proc R:

var E,F

beginR …call(Q)…

end

beginQ …call(R)…

end

beginP …call(Q)…

end.

[[E,F],[C,D],[A,B]]

[[C,D],[A,B]]

[[A,B]]

Program state at program point

Abstract Syntax Tree

• You will be given a routine that, given a file with Pascal program text, will return its AST

• Each variable use will be represented in the AST by its name and its “scope”– Scope = 1 if local,

– Scope = 2 if declared in immediately enclosing block

– Scope = 3 if declared in next outer block …

More Hints:

• Write functions like:– getVariableValue:

VarName × Scope × State Value– setVariableValue:

VarName × Scope × Value × State State

– interpExpr: AST × State Value– interpStmt: AST × State State

Homework 1: (cont)• In addition to submitting working code, please

answer the following discussion questions:– How did you handle procedure invocation and return?

Describe in particular how the structure of the state was changed?

– How did you use the nondeterministic aspects of the XSB language? I.e., what would have been different had you used Standard ML?

• Due Date: 2 weeks.

Francesco’s HW1 Submission

• On entry to a procedure:– I created a new state by:

• taking a “tail” of the current state, keeping the envs for the block where the called proc is declared,

• And adding a new env to the front for the proc being entered, with (value) parameters initialized.

• On exit from a procedure:– I replaced the “tail” of the state on entry by the tail of the state

returned from the procedure.

• I didn’t use the nondeterministic aspects at all.– (I’ve got better things to do with my time than learn new irrelevant

languages. Is the Prof getting old and losing it?)

Homework 2:Program Analysis

• Write an abstract interpreter for our Pascal subset that will abstract integers to even/odd. I.e., given any program, it determines for each variable whether it will contain only even integers or only odd or might contain either.– Extra credit: Use the same idea to determine for

each variable whether it might be used before it is assigned a value.

Homework 2:Discussion Questions

• How did you use your program for assignment 1?– What did you keep and what did you change?

– Will your abstract interpreter terminate for every input Pascal object program? Why or why not?

– How did you use the nondeterministic aspects of the tabled Prolog language? I.e., what would have been different had you used Standard ML?

• Due Date: tomorrow morning!

Francesco’s HW2 Submission

• I just abstracted integer constants to even/odd, and changed each operation on integers to be an operation on even/odd. E.g. even+odd=odd, even*odd=even, etc. – The problem was comparisons, but if I couldn’t tell the result, I

just returned both true and false. E.g. odd=even returned false, but even=even returned both true and false, as did even<odd, etc.

– I didn’t change ANYTHING ELSE!

• Extra credit: I got it! Vars start uninitted. Constants are initted; Result is initted if both operands initted. EASY!

Evidence for Claim

Francesco’s HW2 Submission (cont)

• It terminates for every program I gave it. I think it will always terminate, but I’m not exactly sure why.

• No way could I have done this by this morning in Standard ML. No way! – The XSB language and tabled evaluation made it work.

(Maybe the Prof isn’t losing it…. yet.)

HW2 Discussion

• Without tabled evaluation, it would loop in while statements, and in recursive procedures.

• Does it always terminate under tabled evaluation? Why?

Yes, there are only finitely many states (assuming the abstract domain of each variable is finite): Each environment has a fixed number of variables;Number of envs in any state is bounded by program nesting.

What did you produce in Homework 2?

• An abstract interpreter for a block structured procedural language.

• Does full inter-procedural analysis– Close to state-of-the-art

• Is “obviously” correct• What does it not do?

– Use a user-defined lattice of abstract values • It uses the subset lattice on values

• But with some more programming, can do that in XSB

Beyond Simple Tabling

• Negation

• Aggregation

• Constraints

Prolog and Negation

• Prolog has the \+ operator– Semantics is “finite failure”

– Formalized as “Clark’s Completion”

• Tabled Prolog (finitely) terminates more often so more negative goals are true– E.g. all Datalog programs.

– Gives a simpler and more interesting theory of finite failure

– What can its semantics be?

Negation and Tabling

• If no recursion through negation, no problem.– Perfect model semantics – Recall that DB (SQL) includes negation, but does NOT

have recursion– But some (many?) programs are illegal.

• Problem with recursion through negation.– What could

• p :- \+ p. mean?• Prolog loops infinitely. Does that mean it has no meaning?

• Nonground negative calls are always problematic.– Will assume/require that all negative calls are ground.

Negation

• The barber shaves everyone in town who doesn’t shave himself:

• I teach DB if Kifer doesn’t, and he teaches it if I don’t:

• What might these mean?

shaves(barber,Y) :- resident(Y), \+ shaves(Y,Y).resident(barber). resident(joe).resident(dave). …

teaches(warren,database) :- \+ teaches(kifer,database).teaches(kifer,database) :- \+ teaches(warren,database).

Two Semantics?• p :- \+ p.

– 3-valued semantics: true, false, and undetermined– p is undetermined here– Well-founded semantics

• Always exists, skeptical

• p :- \+ q. q :- \+ p.– 2-valued semantics.– Here p: true, q:false; or p:false, q:true– Stable Model semantics– Multiple models, sometimes none.

• For first program, no SM; for second all are undet in WF.

Properties of WF and SM Semantics

• Stable Models– NP-hard to compute (for propositional programs)

• Well-founded Semantics– Polynomial to compute (quadratic, but often linear)

– “Relevant”, I.e., goal-directed computation possible.

– “approximates” stable models• All SM’s agree on props that WF semantics determines

• XSB computes this semantics– Uses SLG resolution

Digression: XSB Tabling Builtins

• get_calls(?Goal,-Handle,?Vars)– Unifies Goal with table calls, returns Handle to access

returns, and variables in Goal.• get_returns(+Handle,?Vars,-AHandle)

– Binds Vars to successful returns for Handle.– AHandle is optional; if given returns AHandle to access

delayed literals.• get_delay_lists(+AHandle,-DelayLists)

– Binds DelayLists to a list of lists of delayed literals.– A literal is delayed if it can’t yet be determined as true

or false.• Now explore wfs.P

XSB and WF Negation

• XSB computes a residual program– Ground original program

– Keep only “relevant” rules

– Eliminate any rule with:• Positive body literal that is false in WFM

• Negative body literal that is true in WFM

– Eliminate body literals in remaining rules that are:• Positive and false in WFM

• Negative and true in WFM

Residual Program

• Fact in residual program is true in WFM; • Rule with nonempty body is undetermined in

WFM.• Stable Model of residual program is a stable

model of the “relevant” portion of the original program.

• XSB has xnmr package which uses Smodels (SM computer) to compute various semantics of logic programs with negation.

Aggregation

• Operations on elements in a table.• E.g., minimum. Consider path:

• Can we get shortest path?– Idea: Before we return a distance, look in table to see if

there is already a smaller one for that path; if so, fail.

• Explore shortest_path.P

dist(X,Y,D) :- edge(X,Y,D).dist(X,Y,D) :- dist(X,Z,D1), edge(Z,Y,D2), D is D1+D2.

Constraints

• In Tabled Prolog, an answer is a set of bindings to variables in the goal.– Bindings are equality constraints.– E.g., call p(a,X), answer p(a,b) can be understood as p(a,X)

where X=b.• We can use more general constraints, e.g.,

– Answer: p(a,X) where X > 17.• So computational state represents a set of current constraints on

the current set of variables.– During computation, new constraints get added, and the

current set is simplified.– If the current set becomes inconsistent, then failure.

CLP

– Constraints must be simplified (same as Prolog)

– Constraints must be associated with goals to be stored in the table with answers

– Must handle constraints at:• Call

• Return

• So this is Constraint Logic Programming• What’s the difference between Prolog and Tabled

Prolog when general constraints are used?

Constraint Simplification

• XSB uses Constraint Handling Rules

– (CHR recently added to XSB by Tom Schrijvers)

– Used for Prolog constraint systems (CLP)

– Allows programmer to write rules that control constraint simplification.

– CHR is a language to specify constraint rewriting and simplification rules

• Are applied to the current set of constraints whenever new ones are added.

Call

• What call to make?– Project constraints on variables of the call:

• Take them all, or

• Weaken them, as eg, turn call of p(a,b,X) into p(a,T,X),T=b.

– How to look up in table:• One with equivalent constraints, or

• One with weaker constraints, as eg, given call of p(a,b,X), could use call of p(a,T,X) if exists.

Return

• When new answer: p(X,Y) where cons(X,Y), is produced:– Add if new– Or check if disjoining it with the answers

already there are equivalent to those already there, and if so, don’t add.

• Generalization of: fail if it is implied by answer already there.

Shortest Path revisited• Consider >= as constraint.

ach_path(X,Y,D) :- edge(X,Y,D1), D >= D1.ach_path(X,Y,D) :-

ach_path(X,Z,D1),edge(Z,Y,D2),D >= D1+D2.

• Answer D>=7 implies D>=5, so need keep only D>=5 in table.

• So computing this with tabling computes the shortest path.

XSB as View-Server• Other applications may want access to tables (views).

They “subscribe”– Views should be invalidated when any base fact changes

that (might) change the view.– An application subscribing to a view should be notified

when it is invalidated.

• Allows XSB to serve as model in model-view-controller structured GUI

• XJ is a wrapper for Java Swing package that allows it to use XSB this way.

• Makes data-centric applications very easy to build.

Conclusion

• You can do a lot of neat things in Tabled Prolog

• XSB is a pretty good implementation of it.

• XSB is freely available from xsb.sf.net• And learn more about tabling from a draft-y

book, available from: http://www.cs.sunysb.edu/~warren/xsbbook/book.html