Applied Logic, Programming-Languages and Systems (ALPS) Lab @ UTD Slide- 1 University of Texas at...

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

University of Texas at Dallas

Logic, Infinite Computation, and Coinduction

Gopal GuptaNeda 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



UT Dallas Computer Science• Located in North Dallas in the middle of Telecom Corridor, home of

more than 600 high tech companies (TI, Alcatel, EDS/HP, ….)• 45 T/T faculty members; 13 full-time instructors• 850 UGs, 700 M.S. students, 130 Ph.D. students• BS/MS/Ph.D. in CS, SE, CE and TE• 6 Major research areas: AI, Systems, Theory, SE, N/W, Security• 10 NSF CAREER award winners; 2 Airforce Young Investigators• $9M research expenditure in 2009-10• $12 Million new funding in 2010-11• 40+ grants in 2011-12 totaling more than $12 Million• UTD ranked 167 now in London Times world ranking• Recent work on Frankenstein Virus got national/international

coverage• Please consider applying to our Ph.D. program & for faculty positions



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 self-referential 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 = (1st speaker, (2nd 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 } { succ(x) | x S }

• N =

– where is least fixed-point.

• 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 Uco(P)– atoms: co-Herbrand base Bco(P)– program semantics: maximal co-Herbrand model

Mco(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– 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

….

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 Mco(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: Comembermember(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 inductivecomember(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 Automataautomata([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, Su, reachable?– If answer is yes, the path to Su 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). % pverify(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 Bverify(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 Bverify(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”

-automata w/ time constrained transitions & stopwatches

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

Timed Automata



Verification of Real-Time Systems “Train, Controller, Gate”

:- 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).



Verification of Real-Time Systems:- 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),

reach(Sfi,Sf).

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

no



DPP – Liveness: Starvation Free

?- starved(X). no

• 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



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 available2. 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,

p1:- B1. p2:- B2. ... pn :- Bn., generate rule(s) of the form:

chk_p1 :- not(p1), B1.chk_p2 :- not(p2), B2....chk_pn :- not(p), Bn.

• Generate: nmr_chk :- not(chk_p1), ... , not(chk_pn).• For each pred. definition, generate its negative version:

not_p :- not(B1), not(B2), ... , not(Bn).• 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 ResultsTop-Down Avg. (s)

Smodels Avg. (s)

Smodels / Top-Down

13 Queens 0.0050 0.0185 3.70

15 Queens 0.0120 0.0760 6.33

19 Queens 0.0235 1.4820 63.06

20 Queens 0.8590 5.3015 6.17

21 Queens 0.0560 19.7560 352.79

22 Queens 7.9100 79.50 10.05

23 Queens 0.1400 216.6700 1547.64

24 Queens 2.0500 101.2400 49.39

8x7 Pigeons 0.0260 0.1515 5.83

11x10 Pigeons

21.0700 131.0700 6.22

10x10 Pigeons

0.0025 0.0055 2.20

20x20 Pigeons

0.0100 0.0790 7.90

30x30 Pigeons

0.0310 0.5155 16.63

40x40 Pigeons

0.0700 2.4340 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

no_rod

θ <= θM

Reactor Temperature Control System

θ = ― - 50 10 θ .

θ = ― - 56 10 θ . θ = ― - 60

10 θ .

rod2rod1

θm <= θ θm <= θ

θ = θM

add1 , c1 := 0

θ = θM

add2 , c2 := 0

θ = θm θ = θm

remove1 , c1 := 0 remove2 , c2 := 0 θ = θM

shutdown

θ = θm

in1

r1 >= W

add1

r1 := 0

remove1

out1

r1 = W

in2

r2 >= W

add2

r2 := 0

remove2

out2

r2 = W



Rod1 & Rod2trans_r1(out1, add1, in1, T, Ti, To,

W) :- {T – Ti >= W, To = Ti}.trans_r1(in1, remove1, out1, T, Ti,

To, W) :- {To = T}.

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}.

in1

r1 >= W

add1

r1 := 0

remove1

out1

r1 = W

in2

r2 >= W

add2

r2 := 0

remove2

out2

r2 = W



Controllertrans_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}.

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}.

no_rod

θ <= θM

θ = ― - 50 10 θ .

θ = ― - 56 10 θ . θ = ― - 60

10 θ .

rod2rod1

θm <= θ θm <= θ

θ = θM

add1 , c1 := 0

θ = θM

add2 , c2 := 0

θ = θm θ = θm

remove1 , c1 := 0 remove2 , c2 := 0 θ = θM

shutdown

θ = θm

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 || Rod2main(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 Publications1. 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’115. R. Min, A. Bansal, G. Gupta. Co-LP with negation, LOPSTR 20096. R. Min, G. Gupta. Towards Predicate ASP, AIAI’097. N. Saeedloei, G. Gupta. Coinductive Constraint Programming

FLOPS12.8. N. Saeedloei, G. Gupta, Timed π-Calculus9. 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’1211. 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

LFP

F

P

G

COMPUTATION



LFP vs GFP

LFP

COMPUTATION

GFP



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 produceX = [a,b|X]

• Combining induction (tabling) and co-induction: – Stratified co-LP: equivalent to stratified Büchi tree automata

(SBTAs)– Non-stratified co-LP: inspired by Rabin automata; 3 class of

predicates (i) coinductive, (ii) weakly coinductive and (iii) strongly coinductive



QUESTIONS?



Coinductive LP: An Example• Let P1 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: Uco(P1) = 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: Mco(P1)={ from(t, [t, s(t), s(s(t)), ... ]) | t Uco(P1) }

• from(0, [0, s(0), s(s(0)), ... ]) Mco(P1) implies the query holds• Without “coinductive” declaration of from, Mco(P1’)=

This corresponds to traditional semantics of LP with infinite trees.

Applied Logic, Programming-Languages and Systems (ALPS) Lab @ UTD Slide- 1 University of Texas at...

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