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Presentation Transcript

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

Family Relations

- Siblings in my family:
- If my sister is my sibling, then I’m hers:
- (symmetric, a problem in Prolog)

sibling(nancy,david).

sibling(david,jane).

sibling(jane,rick).

sibling(rick,emily).

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

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).

A

C

B

S0 S1 S

Input Str: …………………………………

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

[Prolog has DCG preprocessor to add the “input variables”

to --> rules for convenience. (See example.)]

Simple Expression Grammar 1 (explore)

expr --> term, [+], expr.

expr --> term.

term --> factor, [*], term.

term --> factor.

factor --> [X], {integer(X)}.

factor --> ['('], expr, [')'].

CF Grammar Example 2

Simple Expression Grammar 2 (explore)

:- auto_table.

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?

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

A A A

A a

- Consider grammar:
- DCG for parsing:
- Input: aaaaa…aaaaab
- Complexity, O(n³) but no parse!!

:- 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 String

accept(MId,StrName) :-

mis(MId,StateStart),

recog(MId,StrName,StateStart,StateFinal,0,StrFinal),

mfs(MId,StateFinal),

Strlen(StrName,StrFinal).

% regular transitions

recog(MId,StringName,MState0,MState,SLoc0,SLoc) :-

string(StringName,SLoc0,Symbol,SLoc1),

m(MId,MState0,Symbol,MState1),

recog(MId,StringName,MState1,MState,SLoc1,SLoc).

% Epsilon transitions

recog(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

:- table ks/2.

% ks(I,K) if a subset of items 1..I sums to K

ks(0,0). % empty set sums to 0

ks(I,K) :- I>0, I1 is I-1, ks(I1,K). % exclude Ith element

ks(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).

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

- As a General Evaluation Strategy for Recursive Definitions

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

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

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

Top-down:

36

28

21

15

10

6

3

1

0

: f(n)

8 7 6 5 4 3 2 1 0 : n

Bottom-up:

36

28

21

15

10

6

3

1

0

: f(n)

Fibonacci

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

- Bottom-up good for fib.

Top-down

8

5

3

3

2

2

2

1

1

1

1

1

1

1

1

: fib(n)

5 4 3 2 1 0 : n

Bottom-up

8

5

3

2

1

1

: fib(n)

log2

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

- Top-down is good for log2

td

4

3

2

1

0

: lg(n)

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

bu

4

3

3

3

3

3

3

3

3

2

2

2

2

1

1

0

: lg(n)

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)

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

- Tabled evaluation similar to bottom-up on fib.

Top-down

w/ tabling

8

5

3

2

1

1

: fib(n)

5 4 3 2 1 0 : n

Bottom-up

8

5

3

2

1

1

: fib(n)

Tabled Evaluation (log2)

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

- Tabled evaluation similar to top-down on log2

td w/

tabl

4

3

2

1

0

: lg(n)

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

td

4

3

2

1

0

: lg(n)

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.

- Represent state as a list of environments:

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.

Program state at program point

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

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

[[A,B]]

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

- getVariableValue:

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.

- I created a new state by:
- 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

- Use a user-defined lattice of abstract values

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?

- What could
- 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

- So this is Constraint Logic Programming
- What’s the difference between Prolog and Tabled Prolog when general constraints are used?

- 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

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.

- Project constraints on variables of the call:

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

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