Logic, Infinite Computation, and Coinduction

University of Texas at Dallas

Logic, Infinite Computation, and Coinduction

Gopal Gupta

Neda Saeedloei, Brian DeVries, Kyle Marple Feliks Kluzniak,

Luke Simon, Ajay Bansal, Ajay Mallya, Richard Min

Applied Logic, Programming-Languages and Systems (ALPS) Lab

The University of Texas at Dallas

Applied Logic, Programming-Languages and Systems (ALPS) Lab @ UTD Slide- 1


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Circular Phenomena in Comp. Sci.

• Circularity has dogged Mathematics and Computer

Science ever since Set Theory was first developed:

– The well known Russell’s Paradox:

• R = { x | x is a set that does not contain itself}

Is R contained in R? Yes and No

– Liar Paradox: I am a liar

– Hypergame paradox (Zwicker & Smullyan)

• All these paradoxes involve self-reference through some type of negation

• Russell put the blame squarely on circularity and sought to ban it from scientific discourse:

``Whatever involves all of the collection must not be one of the collection” -- Russell 1908



Circularity in Computer Science

• Following Russell’s lead, Tarski proposed to ban selfreferential sentences in a language

• Rather, have a hierarchy of languages

• Kripke’s paper challenged this in a1975 paper: argued that circular phenomenon are far more common and circularity can’t simply be banned.

• Circularity has been banned from automated theorem proving and logic programming through the occurs check rule:

An unbound variable cannot be unified with a term containing that variable (i.e., X = f(X) not allowed)

• What if we allowed such unification to proceed (as LP systems always did for efficiency reasons)?




• If occurs check is removed, we’ll generate circular (infinite) structures:

X = [1,2,3 | X] X = f(X)

• Such structures, of course, arise in computing

(circular linked lists), but banned in logic/LP.

• Subsequent LP systems did allow for such circular structures (rational terms), but they only exist as data-structures, there is no proof theory to go along with it.

– One can hold the data-structure in memory within an LP execution, but one can’t reason about it.



Circularity in Everyday Life

• Circularity arises in every day life

– Most natural phenomenon are cyclical

• Cyclical movement of the earth, moon, etc.

• Our digestive system works in cycles

– Social interactions are cyclical:

• Conversation = (1 st speaker, (2 nd Speaker, Conversation)

• Shared conventions are cyclical concepts

• Numerous other examples can be found elsewhere (Barwise & Moss 1996)




• Circular phenomenon are quite common in

Computer Science:

– Circular linked lists

– Graphs (with cycles)

– Controllers (run forever)

– Bisimilarity

– Interactive systems

– Automata over infinite strings/Kripke structures

– Perpetual processes

• Logic/LP not equipped to model circularity directly



Coinduction

• Circular structures are infinite structures

X = [1, 2 | X] is logically speaking X = [1, 2, 1, 2, ….]

•

• Proofs about their properties are infinite-sized

Coinduction is the technique for proving these properties

– first proposed by Peter Aczel in the 80s

• Systematic presentation of coinduction & its application to computing, math. and set theory:

“Vicious Circles” by Moss and Barwise (1996)

• Our focus: inclusion of coinductive reasoning techniques in LP and theorem proving



Induction vs Coinduction

• Induction is a mathematical technique for finitely reasoning about an infinite (countable) no. of things.

• Examples of inductive structures:

– Naturals: 0, 1, 2, …

– Lists: [ ], [X], [X, X], [X, X, X], …

• 3 components of an inductive definition:

(1) Initiality, (2) iteration, (3) minimality

– for example, the set of lists is specified as follows:

[ ] – an empty list is a list (initiality) ……(i)

[H | T] is a list if T is a list and H is an element (iteration) ..(ii) minimal set that satisfies (i) and (ii) (minimality)



Induction vs Coinduction

• Coinduction is a mathematical technique for

(finitely) reasoning about infinite things.

– Mathematical dual of induction

– If all things were finite, then coinduction would not be needed.

– Perpetual programs, automata over infinite strings

• 2 components of a coinductive definition:

(1) iteration, (2) maximality

– for example, for a list:

[ H | T ] is a list if T is a list and H is an element (iteration).

Maximal set that satisfies the specification of a list.

– This coinductive interpretation specifies all infinite sized lists



Example: Natural Numbers

• 



(S) = { 0 }

• N = 

 { succ(x) | x



– where  is least fixed-point.

 S }

• aka “inductive definition”

– Let N be the smallest set such that

• 0  N

• x  N implies x + 1  N

• Induction corresponds to Least Fix Point

(LFP) interpretation.



Example: Natural Numbers and Infinity

• 



• 



(S) = { 0 }  { succ(x) | x  S } unambiguously defines another set

• N’ = 



= N  {  }

–  = succ( succ( succ( ... ) ) ) = succ(  ) =  + 1

– where 

 is a greatest fixed-point

• Coinduction corresponds to Greatest Fixed

Point (GFP) interpretation.



Mathematical Foundations

• Duality provides a source of new mathematical tools that reflect the sophistication of tried and true techniques.

Definition Proof tech.

Mapping

Least fixed point Induction Recursion

Greatest fixed point Coinduction Corecursion

• Co-recursion: recursive def’n without a base case



Applications of Coinduction

• model checking

• bisimilarity proofs

• lazy evaluation in FP

• reasoning with infinite structures

• perpetual processes

• cyclic structures

• operational semantics of “coinductive logic programming”

• Type inference systems for lazy functional languages



Inductive Logic Programming

• Logic Programming

– is actually inductive logic programming.

– has inductive definition.

– useful for writing programs for reasoning about finite things:

- data structures

- properties



Infinite Objects and Properties

• Traditional logic programming is unable to reason about infinite objects and/or properties.

• (The glass is only half-full)

• Example: perpetual binary streams

– traditional logic programming cannot handle bit(0).

bit(1).

bitstream( [ H | T ] ) :- bit( H ), bitstream( T ).

|?- X = [ 0, 1, 1, 0 | X ], bitstream( X ).

• Goal: Combine traditional LP with coinductive LP



Overview of Coinductive LP

• Coinductive Logic Program is a definite program with maximal co-Herbrand model declarative semantics.

• Declarative Semantics: across the board dual of traditional LP:

– greatest fixed-points

– terms: co-Herbrand universe U co (P)

– atoms: co-Herbrand base B co (P)

– program semantics: maximal co-Herbrand model M co (P).



Operational Semantics: co-SLD Resolution

• nondeterministic state transition system

• states are pairs of

– a finite list of syntactic atoms [resolvent] (as in Prolog)

– a set of syntactic term equations of the form x = f(x) or x = t

• For a program p :- p. => the query |?- p. will succeed.

• p( [ 1 | T ] ) :- p( T ). => |?- p(X) to succeed with X= [ 1 | X ].

• transition rules

?-G – definite clause rule

– “coinductive hypothesis rule”

• if a coinductive goal G is called, and G unifies with a call made earlier then G succeeds.

G

….

coinductive success



Correctness

• Theorem (soundness). If atom A has a successful co-SLD derivation in program P, then E(A) is true in program P, where E is the resulting variable bindings for the derivation.

• Theorem (completeness). If A  M co (P) has a rational proof, then A has a successful co-

SLD derivation in program P.

– Completeness only for rational/regular proofs



Implementation

• Search strategy: hypothesis-first, leftmost, depth-first

• Meta-Interpreter implementation.

query(Goal) :- solve([],Goal).

solve(Hypothesis, (Goal1,Goal2)) :solve( Hypothesis, Goal1), solve(Hypothesis,Goal2).

solve( _ , Atom) :- builtin(Atom), Atom.

solve(Hypothesis,Atom):- member(Atom, Hypothesis).

solve(Hypothesis,Atom):- notbuiltin(Atom), clause(Atom,Atoms), solve([Atom|Hypothesis],Atoms).

• A complete meta-interpreter available

• Implementation on top of YAP, SWI Prolog available

• Implementation within Logtalk + library of examples



Example: Number Stream

:- coinductive stream/1.

stream( [ H | T ] ) :- num( H ), stream( T ).

num( 0 ).

num( s( N ) ) :- num( N ).

|?- stream( [ 0, s( 0 ), s( s ( 0 ) ) | T ] ).

1. MEMO: stream( [ 0, s( 0 ), s( s ( 0 ) ) | T ] )

2. MEMO: stream( [ s( 0 ), s( s ( 0 ) ) | T ] )

3. MEMO: stream( [ s( s ( 0 ) ) | T ] )

4.

stream(T)

Answers:

T = [ 0, s(0), s(s(0)) | T ]

T = [ 0, s(0), s(s(0)), s(0), s(s(0)) | T ]

T = [ 0, s(0), s(s(0)) | T ] . . .

T = [ 0, s(0), s(s(0)) | X ] (where X is any rational list of numbers.)



Example: Append

:- coinductive append/3.

append( [ ], X, X ).

append( [ H | T ], Y, [ H | Z ] ) :- append( T, Y, Z ).

|?- Y = [ 4, 5, 6 | Y ], append( [ 1, 2, 3 ], Y, Z).

Answer: Z = [ 1, 2, 3 | Y ], Y=[ 4, 5, 6 | Y]

|?- X = [ 1, 2, 3 | X ], Y = [ 3, 4 | Y ], append( X, Y, Z).

Answer: Z = [ 1, 2, 3 | Z ].

|?- Z = [ 1, 2 | Z ], append( X, Y, Z ).

Answer: X = [ ], Y = [ 1, 2 | Z ] ; X = [1, 2 | X], Y = _

X = [ 1 ], Y = [ 2 | Z ] ;

X = [ 1, 2 ], Y = Z; …. ad infinitum



Example: Comember

member(H, [ H | T ]).

member(H, [ X | T ]) :- member(H, T).

?- L = [1,2 | L], member(3, L). succeeds. Instead:

:- coinductive comember/2. %drop/3 is inductive comember(X, L) :- drop(X, L, R), comember(X, R).

drop(H, [ H | T ], T).

drop(H, [ X | T ], T1) :- drop(H, T, T1).

?- X=[ 1, 2, 3 | X ], comember(2,X). ?- X = [1,2 | X], comember(3, X).

Answer: yes. Answer: no

?- X=[ 1, 2, 3, 1, 2, 3], comember(2, X).

Answer: no.

?- X=[1, 2, 3 | X], comember(Y, X).

Answer: Y = 1;

Y = 2;

Y = 3;



Co-Logic Programming

• combines both halves of logic programming:

– traditional logic programming

– coinductive logic programming

• syntactically identical to traditional logic programming, except predicates are labeled:

– Inductive, or

– coinductive

• and stratification restriction enforced where:

– inductive and coinductive predicates cannot be mutually recursive. e.g., p :- q.

q :- p.

Program rejected, if p coinductive & q inductive



Application of Co-LP

• Co-LP allows one to compute both LFP & GFP

• Computable functions can be specified more elegantly

– Interepreters for Modal Logics can be elegantly specified:

– Model Checking: LTL interpreter elegantly specified

– Timed  -automata: elegantly modeled and properties verified

– Modeling/Verification of Cyber Physical Systems/Hybrid automata

– Goal-directed execution of Answer Set Programs

– Goal-directed SAT solvers (Davis-Putnam like procedure)

– Planning under real-time constraints

– Operational semantics of the π-calculus (incl. timed π -calculus)

• infinite replication operator modeled with co-induction

Co-LP allows systems to be modeled naturally & elegantly



Application: Model Checking

• automated verification of hardware and software systems

•  -automata

– accept infinite strings

– accepting state must be traversed infinitely often

• requires computation of lfp and gfp

• co-logic programming provides an elegant framework for model checking

• traditional LP works for safety property (that is based on lfp) in an elegant manner, but not for liveness .



Verification of Properties

• Types of properties: safety and liveness

• Search for counter-example



Safety versus Liveness

• Safety

– “nothing bad will happen”

– naturally described inductively

– straightforward encoding in traditional LP

• liveness

– “something good will eventually happen”

– dual of safety

– naturally described coinductively

– straightforward encoding in coinductive LP



Finite Automata

automata([X|T], St):- trans(St, X, NewSt), automata(T, NewSt).

automata([ ], St) :- final(St).

trans(s0, a, s1). trans(s1, b, s2). trans(s2, c, s3). trans(s3, d, s0). trans(s2, 3, s0). final(s2).

?- automata(X,s0).

X=[ a, b];

X=[ a, b, e, a, b];

X=[ a, b, e, a, b, e, a, b];

……

……

……



Infinite Automata

automata([X|T], St):- trans(St, X, NewSt), automata(T, NewSt).

trans(s0,a,s1). trans(s1,b,s2). trans(s2,c,s3). trans(s3,d,s0). trans(s2,3,s0). final(s2).

?- automata(X,s0).

X=[ a, b, c, d | X ];

X=[ a, b, e | X ];



Verifying Liveness Properties

• Verifying safety properties in LP is relatively easy: safety modeled by reachability

• Accomplished via tabled logic programming

• Verifying liveness is much harder: a counterexample to liveness is an infinite trace

• Verifying liveness is transformed into a safety check via use of negations in model checking and tabled LP

– Considerable overhead incurred

• Co-LP solves the problem more elegantly:

– Infinite traces that serve as counter-examples are produced as answers



Verifying Liveness Properties

• Consider Safety:

– Question: Is an unsafe state, S u

, reachable?

– If answer is yes, the path to S u is the counter-ex.

• Consider Liveness, then dually

– Question: Is a state, D, that should be dead, live?

– If answer is yes, the infinite path containing D is the counter example

• Co-LP will produce this infinite path as the answer

• Checking for liveness is just as easy as checking for safety



Nested Finite and Infinite Automata

:- coinductive state/2.

state(s0, [s0,s1 | T]):- enter, work, state(s1,T).

state(s1, [s1 | T]):- exit, state(s2,T).

state(s2, [s2 | T]):- repeat, state(s0,T).

state(s0, [s0 | T]):- error, state(s3,T).

state(s3, [s3 | T]):- repeat, state(s0,T).

work. enter. repeat. exit. error.

work :- work.

|?- state(s0,X), absent(s2,X).

X=[ s0, s3 | X ]



An Interpreter for LTL

%--- nots have been pushed to propositions

:- tabled verify/2.

verify(S, [S], A) :- proposition(A), holds(S,A). % p verify(S, [S], not(A)) :- proposition(A), \+holds(S,A). % not(p) verify(S,P, or(A,B)) :- verify(S, P, A) ; verify(S, P, B). %A or B verify(S,P, and(A,B)) :- verify(S,P1, A), verify(S,P2, B). %A and B

(prefix(P2, P1), P=P1 ; prefix(P2,P1), P=P2) verify(S, [S|P], x(A)) :- trans(S, S1), verify(S1, P, A). % X(A) verify(S, P, f(A)) :- verify(S, P, A); verify(S, P, x(f(A))). % F(A) verify(S, P, g(A)) :- coverify(S, P, g(A)). % G(A) verify(S, P,u(A,B)) :- verify(S, P,B); verify(S, P,and(A, x(u(A,B)))). % A u B verify(S, r(A,B)) :- coverify(S, r(A,B)). % A r B

:- coinductive coverify/2.

coverify(S, g(A)) :- verify(S, P, and(A, x(g(A))).

coverify(S, r(A,B)) :- verify(S, P, and(A,B)).

coverify(S, r(A,B)) :- verify(S, P, and(B, x(r(A,B)))).



Verification of Real-Time Systems

“Train, Controller, Gate”

Timed Automata

•  -automata w/ time constrained transitions & stopwatches

• straightforward encoding into CLP( R ) + Co-LP





:- use_module(library(clpr)).

:- coinductive driver/9.

train(X, up, X, T1,T2,T2). % up=idle train(s0,approach,s1,T1,T2,T3) :- {T3=T1}.

train(s1,in,s2,T1,T2,T3):-{T1-T2>2,T3=T2} train(s2,out,s3,T1,T2,T3).

train(s3,exit,s0,T1,T2,T3):-{T3=T2,T1-T2<5}.

train(X,lower,X,T1,T2,T2).

train(X,down,X,T1,T2,T2).

train(X,raise,X,T1,T2,T2).





contr(s0,approach,s1,T1,T2,T1).

contr(s1,lower,s2,T1,T2,T3):- {T3=T2, T1-T2=1}.

contr(s2,exit,s3,T1,T2,T1).

contr(s3,raise,s0,T1,T2,T2):-{T1-T2<1}.

contr(X,in,X,T1,T2,T2).

contr(X,up,X,T1,T2,T2).

contr(X,out,X,T1,T2,T2).

contr(X,down,X,T1,T2,T2).





gate(s0,lower,s1,T1,T2,T3):- {T3=T1}.

gate(s1,down,s2,T1,T2,T3):- {T3=T2,T1-T2<1}.

gate(s2,raise,s3,T1,T2,T3):- {T3=T1}.

gate(s3,up,s0,T1,T2,T3):- {T3=T2,T1-T2>1,T1-T2<2 }.

gate(X,approach,X,T1,T2,T2).

gate(X,in,X,T1,T2,T2).

gate(X,out,X,T1,T2,T2).

gate(X,exit,X,T1,T2,T2).




:- coinductive driver/9.

driver(S0,S1,S2, T,T0,T1,T2, [ X | Rest ], [ (X,T) | R ]) :train(S0,X,S00,T,T0,T00), contr(S1,X,S10,T,T1,T10), gate(S2,X,S20,T,T2,T20), {TA > T}, driver(S00,S10,S20,TA,T00,T10,T20,Rest,R).

|?- driver(s0,s0,s0,T,Ta,Tb,Tc,X,R).

R=[(approach,A), (lower,B), (down,C), (in,D), (out,E), (exit,F),

(raise,G), (up,H) | R ],

X=[approach, lower, down, in, out, exit, raise, up | X] ;

R=[(approach,A),(lower,B),(down,C),(in,D),(out,E),(exit,F),(raise,G),

(approach,H),(up,I)|R],

X=[approach,lower,down,in,out,exit,raise,approach,up | X] ;

% where A, B, C, ... H, I are the corresponding wall clock time of events generated.



DPP – Safety: Deadlock Free

• One potential solution

– Force one philosopher to pick forks in different order than others

• Checking for deadlock

– Bad state is not reachable

– Implemented using Tabled LP

:- table reach/2.

reach(Si, Sf) :- trans(_,Si,Sf).

reach(Si, Sf) :- trans(_,Si,Sfi), no reach(Sfi,Sf).

?- reach([1,1,1,1,1], [2,2,2,2,2]).



DPP – Liveness: Starvation Free

• Phil. waits forever on a fork

• One potential solution

– phil. waiting longest gets the access

– implemented using CLP(R)

• Checking for starvation

– once in bad state, is it possible to remain there forever?

– implemented using co-LP

?- starved(X).

no



Other Applications

• Advanced  -structures can also be modeled and reasoned about:  -PTA and  -grammars

• Non monotonic reasoning:

– CoLP allows goal-directed execution of Answer Set

Programs (ASP)

– Abductive reasoners can be elegantly implemented

– Answer sets programming can be extended to predicates

– ASP can be elegantly extended with constraints: advanced applications such as planning under real-time constraints become possible

• SAT Solvers can be elegantly written

• Operational semantics of pi-calculus can be given

(infinite replication operator modeled with co-induction)



Goal-directed execution of ASP

• Answer set programming (ASP) is a popular formalism for non monotonic reasoning

• Applications in real-world reasoning, planning, etc.

• Semantics given via lfp of a residual program obtained after “Gelfond-Lifschitz” transform

• Popular implementations: Smodels, DLV, etc.

1. No goal-directed execution strategy available

2. ASP limited to only finitely groundable programs

• Co-logic programming solves both these problems.

• Also provides a goal-directed method to check if a proposition is true in some model of a prop. formula



Why Goal-directed ASP?

• Most of the time, given a theory, we are interested in knowing if a particular goal is true or not.

• Top down goal-directed execution provides operational semantics (important for usability)

• Execution more efficient.

– Tabled LP vs bottom up Deductive Databases

• Why check the consistency of the whole knowledgebase?

– Inconsistency in some unrelated part will scuttle the whole system

• Most practical examples anyway add a constraint to force the answer set to contain a certain goal.

– E.g. Zebra puzzle: :- not satisfied .

• Answer sets of non-finitely groundable programs computable & Constraints incorporated in Prolog style.



Negation in Co-LP: co-SLDNF Resolution

• Given a clause such as p :- q, not p.

?- p. fails coinductively when not p is encountered

• To incorporate negation in coinductive reasoning, need a negative coinductive hypothesis rule:

– In the process of establishing not(p), if not(p) is seen again in the resolvent, then not(p) succeeds [co-SLDNF Resolution]

• Also, not not p reduces to p .

• Answer set programming makes the “glass completely full” by taking into account failing computations:

– p :- q, not p. is consistent if p = false and q = false

• However, this takes away monotonicity: q can be constrained to false, causing q to be withdrawn, if it was established earlier.



ASP

• Consider the following program, A : p :- not q. t. r :- t, s.

q :- not p. s.

A has 2 answer sets: {p, r, t, s} & {q, r, t, s}.

• Now suppose we add the following rule to A : h :- p, not h. (falsify p)

Only one answer set remains: {q, r, t, s}

• Gelfond-Lifschitz Method:

– Given an answer set S, for each p  S, delete all rules whose body contains “not p”;

– delete all goals of the form “not q” in remaining rules

– Compute the least fix point, L, of the residual program

– If S = L, then S is an answer set



Goal-directed ASP

• Consider the following program, A’: p :- not q. t. r :- t, s.

q :- not p, r. s. h :- p, not h.

• Separate into constraint and non-constraint rules: only 1 constraint rule in this case.

• Execute the query under co-LP, candidate answer sets will be generated.

• Keep the ones not rejected by the constraints.

• Suppose the query is ?- q. Execution: q  not p, r

 not not q, r  q, r  r  t, s  s  success.

Ans = {q, r, t, s}

• Next, we need to check that constraint rules will not reject the generated answer set.

– (it doesn’t in this case)



Goal-directed ASP

• In general, for the constraint rules of p as head, p

1

:- B

1

. p

2 chk_p

1

:- B

2

. ... p

:- not(p

:- not(p

1

2 n

:- B

), B

1

), B

2

.

.

n

., generate rule(s) of the form: chk_p

2

...

chk_p n

:- not(p), B n

.

• Generate: nmr_chk :- not(chk_p

• For each pred. definition, generate its negative version: not_p :- not(B

1

), not(B

2

1

), ... , not(chk_p

), ... , not(B n

).

n

).

• If you want to ask query Q, then ask ?- Q, nmr_chk.

• Execution keeps track of atoms in the answer set (PCHS) and atoms not in the answer set (NCHS).



Goal-directed ASP

• Consider the following program, P1:

(i) p :- not q. (ii) q:- not r. (iii) r :- not p. (iv) q :- not p.

P1 has 1 answer set: {q, r}.

• Separate into: 3 constraint rules (i, ii, iii)

2 non-constraint rules (i, iv).

p :- not(q).

q :- not(r).

r :- not(p).

q :- not(p).

chk_p :- not(p), not(q).

chk_q :- not(q), not(r).

chk_r :- not(r), not(p).

nmr_chk :- not(chk_p), not(chk_q), not(chk_r).

not_p :- q . not_q :- r, p.

not_r :- p.

Suppose the query is ?- r.

Expand as in co-LP: r  not p  not not q  q (  not r

 fail, backtrack)  not p  success. Ans={r, q} which satisfies the constraint rules of nmr_chk.



Benchmark Results

Top-Down Avg. (s) Smodels Avg. (s) Smodels / Top-Down

13 Queens

15 Queens

19 Queens

20 Queens

21 Queens

22 Queens

23 Queens

24 Queens

8x7 Pigeons

11x10 Pigeons 21.0700

10x10 Pigeons 0.0025




0.0050

0.0120

0.0235

0.8590

0.0560

7.9100

0.1400

2.0500

0.0260

0.0185

0.0760

1.4820

5.3015

19.7560

79.50

216.6700

101.2400

0.1515

131.0700

0.0055

0.0790

0.5155

2.4340

3.70

6.33

63.06

6.17

352.79

10.05

1547.64

49.39

5.83

6.22

2.20

7.90

16.63

34.77



Cyber-Physical Systems (CPS)

• CPS:

-- Networked/distributed Hybrid Systems

-Discrete digital systems with

– Inputs: continuous physical quantities

• e.g., time, distance, acceleration, temperature, etc.

– Outputs: control physical (analog) devices

• Elegantly modeled via co-LP extended with constraints

• Characteristics of CPS:

-- perform discrete computations (modeled via LP)

-- deal with continuous physical quantities (modeled via constraints)

-- are concurrent (modeled via LP coroutining)

-- run forever (modeled via coinduction)



CPS Example

Reactor Temperature Control System rod

1

.

θ m

θ

10

<= θ

θ = θ m add

θ = θ

1 , c

1

θ = θ

M

:= 0 m remove

1 , c

1

:= 0

θ = θ

M no_rod

.

add

2 , c

2

:= 0

θ

10

θ <= θ

M

θ = θ m

θ = θ

M remove

2 , c

2

:= 0

.

θ m rod

2

θ

10

<= θ shutdown r

1

= W out

1 r

2

= W out

2 r

1

>= W add

1 r

1

:= 0 remove

1 r

2

>= W add

2 r

2

:= 0 remove

2 in in

2

1



Rod1 & Rod2

trans_r1(out1, add1, in1, T, Ti, To, W)

:-

{T – Ti >= W, To = Ti}.

trans_r1(in1, remove1, out1, T, Ti, To,

W) :- {To = T}.

r

1

= W out

1 trans_r2(out2, add2, in2, T, Ti, To, W)

:-

{T – Ti >= W, To = Ti}.

trans_r2(in2, remove2, out2, T, Ti, To,

W) :- {To = T}.

r

2

= W out

2 r

1

>= W add

1 r

1

:= 0 remove

1 r

2

>= W add

2 r

2

:= 0 remove

2 in in

2

1



Controller

trans_c(norod, add1, rod1, Tetai, Tetao, T, Ti1, Ti2, To1, To2, F) :-

(F == 1 -> Ti = Ti1; Ti = Ti2),

{Tetai < 550, Tetao = 550, exp(e, (T - Ti)/10) = 5,

To1 = T, To2 = Ti2}.

trans_c(rod1, remove1, norod Tetai, Tetao, T, Ti1, Ti2, To1, To2, F) :-

{Tetai > 510 Tetao = 510, exp(e, (T - Ti1)/10) = 5,

To1 = T, To2 = Ti2}.

trans_c(norod, add2, rod2, Tetai, Tetao, T, Ti1, Ti2, To1, To2, F) :-

(F == 1 -> Ti = Ti1; Ti = Ti2),

{Tetai < 550, Tetao = 550, exp(e, (T - Ti)/10) = 5,

To1 = Ti1, To2 = T}.

trans_c(rod2, remove2, norod, Tetai, Tetao, T, Ti1, Ti2, To1, To2, F) :-

{Tetai > 510, Tetao = 510, exp(e, (T - Ti2)/10) = 9/5,

To1 = Ti1, To2 = T}.

.

θ m trans_c(norod, _, shutdown, Tetai, Tetao, T, Ti1, Ti2, To1, To2, F) :-

(F == 1 -> Ti = Ti1; Ti = Ti2),

{Tetai < 550 Tetao = 550, exp(e, (T - Ti)/10) = 5,

To1 = Ti1, To2 = Ti2}.

rod

1

θ

10

<= θ

θ = θ m add

θ = θ

1 , c

1

θ = θ

M

:= 0 m remove

1 , c

1

:= 0

θ = θ

M no_rod

.

add

2 , c

2

:= 0

θ

10

θ <= θ

M

θ = θ m

θ = θ

M remove

2 , c

2

:= 0

.

θ m rod

2

θ

10

<= θ shutdown


Controller | Rod1 | Rod2

:- coinductive(contr/7).

contr(X, Si, T, Tetai, Ti1, Ti2, Fi) :-

(H = add1; H = remove1; H = add2; H = remove2; H = shutdown),

{Ta > T}, freeze(X, contr(Xs, So, Ta, Tetao, To1, To2, Fo)), trans_c(Si, H, So, Tetai, Tetao, T, Ti1, Ti2, To1, To2, Fi),

((H=add1; H=remove1) -> Fo = 1; Fo = 2),

((H=add1; H=remove1; H=add2; H=remove2) -> X = [ (H, T) | Xs]; X = [ (H, T) ] ).

:- coinductive(rod1/6).

rod1([ (H, T)| Xs], Si1, Si2, Ti1, Ti2, W) :-

H = add1 -> freeze(Xs,rod1(Xs, So1, Si2, To1, Ti2, W));

H = remove1 -> freeze(Xs,rod1(Xs, So1, Si2, To1, Ti2, W); rod2(Xs, So1, Si2, To1, Ti2, W)), trans_r1(Si1, H, So1, T, Ti1, To1, W);

H = shutdown -> {T - Ti1 < A, T - Ti2 < A}.

:- coinductive(rod2/6).

rod2([ (H, T)| Xs], Si1, Si2, Ti1, Ti2, W) :-

H = add2 -> freeze(Xs,rod2(Xs, Si1, So2, Ti1, To2, W));

H = remove2 -> freeze(Xs,rod1(Xs, Si1, So2, Ti1, To2, W); rod2(Xs, Si1, So2, Ti1, To2, W)), trans_r2(Si2, H, So2, T, Ti2, To2, W);

H = shutdown -> {T - Ti1 < A, T - Ti2 < A}.


Controller || Rod1 || Rod2

main(S, T, W) :- {T - Tr1 = W, T - Tr2 = W}, freeze(S, (rod1(S, s0, s0, Tr1, Tr2, W); rod2(S, s0, s0, Tr1, Tr2, W))), contr(S, s0, T, 510, Tc1, Tc2, 1).

• With this elegant modeling, we were able to improve the bounds on W compared to previous work

• HyTech determines W < 20.44 to prevent shutdown

• Subsequently, using linear hybrid automata with clock translation, HyTech improves to W < 37.8

• Using our LP method, we refine it to W < 38.06



Related Publications

1. L. Simon, A. Mallya, A. Bansal, and G. Gupta. Coinductive logic programming. In ICLP’06 .

2. L. Simon, A. Bansal, A. Mallya, and G. Gupta. Co-Logic programming:

Extending logic programming with coinduction. In ICALP’07.

3. G. Gupta et al. Co-LP and its applications, ICLP’07 (tutorial)

4. G. Gupta et al. Infinite computation, coinduction and computational logic. CALCO’11

5. R. Min, A. Bansal, G. Gupta. Co-LP with negation, LOPSTR 2009

6. R. Min, G. Gupta. Towards Predicate ASP, AIAI’09

7. N. Saeedloei, G. Gupta. Coinductive Constraint Programming

FLOPS12.

8. N. Saeedloei, G. Gupta, Timed π-Calculus

9. N. Saeedloei, G. Gupta. Modeling/verification of CPS with coinductive coroutined CLP(R)

10. K. Marple, A. Bansal, R. Min, G. Gupta. Goal-directed Execution of

ASP. PPDP’12

11. K. Marple, G. Gupta, Galliwasp: A Goal-Directed Answer Set Solver .

LOPSTR 2012



Conclusion

• Circularity is a common concept in everyday life and computer science:

• Logic/LP is unable to cope with circularity

• Solution: introduce coinduction in Logic/LP

– dual of traditional logic programming

– operational semantics for coinduction

– combining both halves of logic programming

• applications to verification, non monotonic reasoning, negation in LP, propositional satisfiability, hybrid systems, cyberphysical systems

• Goal-directed impl. of non-monotonic reasoning avail.

• Metainterpreter available: http://www.utdallas.edu/~gupta/meta.tar.gz



Conclusion (cont’d)

• Computation can be classified into two types:

– Well-founded,

• Based on computing elements in the LFP

• Implemented w/ recursion (start from a call, end in base case)

– Consistency-based

• Based on computing elements in the GFP (but not LFP)

• Implemented via co-recursion (look for consistency)

• Combining the two allows one to compute any computable function elegantly:

– Implementations of modal logics (LTL, etc.)

– Complex reasoning systems (NM reasoners)

• Combining them is challenging



LFP vs GFP

G

COMPUTATION

F

P

L

F

P



LFP vs GFP

G

F

P

COMPUTATION

L

F

P



Conclusions: Future Work

• Design execution strategies that enumerate all rational infinite solutions while avoiding redundant solutions p([a|X]) :- p(X).

p([b|X]) :- p(X).

-- If X = [a|X] is reported, then avoid X = [a, a | X], X = [a,a,a|X], etc.

-- A fair depth first search strategy that will produce

X = [a,b|X]

• Combining induction (tabling) and co-induction:

– Stratified co-LP: equivalent to

(SBTAs) stratified Büchi tree automata

– Non-stratified co-LP: inspired by coinductive

Rabin automata ; 3 class of predicates (i) coinductive, (ii) weakly coinductive and (iii) strongly



QUESTIONS?



Coinductive LP: An Example

• Let P

1 be the following coinductive program.

:- coinductive from/2. from(x) = x cons from(x+1) from( N, [ N | T ] ) :- from( s(N), T ).

|?- from( 0, X ).

• co-Herbrand Universe: U co (P

1

) = N    L where

N=[0, s(0), s(s(0)), ... ],  ={ s(s(s( . . . ) ) ) }, and L is the the set of all finite and infinite lists of elements in N,  and L.

• co-Herbrand Model:

M co (P

1

)={ from(t, [t, s(t), s(s(t)), ... ]) | t  U co (P

1

• from(0, [0, s(0), s(s(0)), ... ])  M co (P

1

) implies the query holds

• Without “coinductive” declaration of from, M co (P

1

) }

’)= 

This corresponds to traditional semantics of LP with infinite trees.


Logic, Infinite Computation, and Coinduction

Logic, Infinite Computation, and Coinduction

UT Dallas Computer Science

Circular Phenomena in Comp. Sci.

Circularity in Computer Science

Circularity in Computer Science

Circularity in Everyday Life

Circularity in Computer Science

Coinduction

Induction vs Coinduction

Induction vs Coinduction

Example: Natural Numbers

Example: Natural Numbers and Infinity

Mathematical Foundations

Applications of Coinduction

Inductive Logic Programming

Infinite Objects and Properties

Overview of Coinductive LP

Correctness

Implementation

Example: Number Stream

Example: Append

Example: Comember

Co-Logic Programming

Application of Co-LP

Application: Model Checking

Verification of Properties

Safety versus Liveness

Finite Automata

Infinite Automata

Verifying Liveness Properties

Verifying Liveness Properties

Nested Finite and Infinite Automata

An Interpreter for LTL

“Train, Controller, Gate”

“Train, Controller, Gate”

“Train, Controller, Gate”

“Train, Controller, Gate”

DPP – Safety: Deadlock Free

DPP – Liveness: Starvation Free

Other Applications

Goal-directed execution of ASP

Why Goal-directed ASP?

ASP

Goal-directed ASP

Goal-directed ASP

Goal-directed ASP

Benchmark Results

Cyber-Physical Systems (CPS)

CPS Example

Rod1 & Rod2

Controller

Controller || Rod1 || Rod2

Related Publications

Conclusion

Conclusion (cont’d)

LFP vs GFP

LFP vs GFP

Conclusions: Future Work

QUESTIONS?

Coinductive LP: An Example

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