Strongly Equivalent Temporal Logic Programs.
ABSTRACT This paper analyses the idea of strong equivalence for transition systems represented as logic programs under the Answer Set
Programming (ASP) paradigm. To check strong equivalence, we use a linear temporal extension of Equilibrium Logic (a logical
characterisation of ASP) and its monotonic basis, the intermediate logic of HereandThere (HT). Trivially, equivalence in
this temporal extension of HT provides a sufficient condition for temporal strong equivalence and, as we show in the paper,
it can be transformed into a provability test into the standard Linear Temporal Logic (LTL), something that can be automatically
checked using any of the LTL available provers. The paper shows an example of the potential utility of this method by detecting
some redundant rules in a simple actions reasoning scenario.

Conference Paper: AutomataBased computation of temporal equilibrium models
[Show abstract] [Hide abstract]
ABSTRACT: Temporal Equilibrium Logic (TEL) is a formalism for temporal logic programming that generalizes the paradigm of Answer Set Programming (ASP) introducing modal temporal operators from standard Lineartime Temporal Logic (LTL). In this paper we solve some problems that remained open for TEL like decidability, bounds for computational complexity as well as computation of temporal equilibrium models for arbitrary theories. We propose a method for the latter that consists in building a Büchi automaton that accepts exactly the temporal equilibrium models of a given theory, providing an automatabased decision procedure and illustrating the ωregularity of such sets. We show that TEL satisfiability can be solved in exponential space and it is hard for polynomial space. Finally, given two theories, we provide a decision procedure to check if they have the same temporal equilibrium models.Proceedings of the 21st international conference on LogicBased Program Synthesis and Transformation; 07/2011  SourceAvailable from: José Alferes
Conference Paper: Parametrized Equilibrium Logic.
[Show abstract] [Hide abstract]
ABSTRACT: Equilibrium logic provides a logical foundation for the stable model semantics of logic programs. Recently, parametrized logic programming was introduced with the aim of presenting the syntax and natural semantics for parametrized logic programs, which are very expressive logic programs, in the sense that complex formulas are allowed to appear in the body and head of rules. Stable model semantics was defined for such parametrized logic programs. The aim of this paper is to introduce a parametrized version of equilibrium logic that extends parametrized logic programs to general theories, and to show how these can be used to characterize and to study strong equivalence of temporal logic programs.Logic Programming and Nonmonotonic Reasoning  11th International Conference, LPNMR 2011, Vancouver, Canada, May 1619, 2011. Proceedings; 01/2011  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we present a declarative propositional temporal logic programming language called TeDiLog that is a combination of the temporal and disjunctive paradigms in Logic Programming. TeDiLog is, syntactically, a sublanguage of the wellknown Propositional Lineartime Temporal Logic (PLTL). TeDiLog allows both eventualities and alwaysformulas to occur in clause heads and also in clause bodies. To the best of our knowledge, TeDiLog is the first declarative temporal logic programming language that achieves this high degree of expressiveness. We establish the logical foundations of our proposal by formally defining operational and logical semantics for TeDiLog and by proving their equivalence. The operational semantics of TeDiLog relies on a restriction of the invariantfree temporal resolution procedure for PLTL that was introduced by Gaintzarain et al. in 2013. We define a fixpoint semantics that captures the reverse (bottomup) operational mechanism and prove its equivalence with the logical semantics. We also provide illustrative examples and comparison with other proposals.ACM Transactions on Computational Logic 11/2013; Volume 14(Issue 4):Article No. 28  41 pages  ACM (Association for Computing Machinery) New York, NY, USA. · 0.79 Impact Factor
Page 1
Strongly equivalent temporal logic programs?
Felicidad Aguado, Pedro Cabalar,
Gilberto P´ erez and Concepci´ on Vidal
Dept. Computaci´ on,
Corunna University (Spain)
{aguado,cabalar,gperez,eicovima}@udc.es
Abstract. This paper analyses the idea of strong equivalence for tran
sition systems represented as logic programs under the Answer Set Pro
gramming (ASP) paradigm. To check strong equivalence, we use a linear
temporal extension of Equilibrium Logic (a logical characterisation of
ASP) and its monotonic basis, the intermediate logic of HereandThere
(HT). Trivially, equivalence in this temporal extension of HT provides
a sufficient condition for temporal strong equivalence and, as we show
in the paper, it can be transformed into a provability test into the stan
dard Linear Temporal Logic (LTL), something that can be automatically
checked using any of the LTL available provers. The paper shows an ex
ample of the potential utility of this method by detecting some redundant
rules in a simple actions reasoning scenario.
1Introduction
The paradigm of Answer set programming (ASP) [1,2] (based on the stable
models semantics [3]) constitutes one of the most successful examples of logical
nonmonotonic formalisms applied to Knoweldge Representation [4,5] in Arti
ficial Intelligence. Probably, the reasons for this success are both related to its
powerful representational features and, at the same time, to the availability of an
increasing number of efficient ASP solvers (see [6]) that allow its application to
many real scenarios. Concerning the formalism properties, ASP is characterised
by providing nonmonotonic reasoning with a rich and flexible syntax, initially
born from logic programming, but continuously extended thereafter along the
research history in the area, without overlooking its original semantic simplicity.
An important breakthrough in this sense has been the logical characterisation of
ASP in terms of Equilibrium Logic [7] that has opened, for instance, the study
of strong equivalence [8] (the main topic of this paper) and has recently allowed
the extension of the stable models semantics for arbitrary first order theories [9,
10].
As for the practical applications of ASP, perhaps one of the most outstand
ing and frequent uses has been the representation and automated reasoning for
action domains, solving typical problems like prediction, explanation, planning
?This research is partially supported by Spanish Ministry MEC, research project
TIN15455C0302.
Page 2
or diagnostics. Default negation plays here a crucial role, as it allows represent
ing the rule of inertia (that can be stated as “a fluent remains unchanged by
default”) and avoid in this way the frame problem [11]. ASP can also be natu
rally used for solving other typical representational problems in Reasoning about
Actions and Change, and is in fact the basis for a family of high level action
languages [12]. The use of ASP solvers for action domains, however, has some
limitations, as partly explained by the complexity class that these solvers allow
to capture (ΣP
that, when solving a planning problem (which lies in PSPACEcompleteness) we
must fix an a priori plan length so that a ground logic program can be eventually
generated. The search for a minimal plan consists then in gradually incrementing
this plan length until a stable model is found. A first obvious drawback of this
approach is that it is not possible to establish when a given planning problem has
no solution of any length at all. A second and more elaborated problem is that
it is impossible to establish when two descriptions of the same transition system
are strongly equivalent, i.e., when they will behave in the same way regardless
any additional rules we include and any path length we consider.
As we mentioned above, the idea of strong equivalence was introduced in [8],
where the following question was considered: when can we safely replace a piece of
knowledge representation by an “equivalent” one independently of the context?
Formally, we say that two logic programs Π1 and Π2 are strongly equivalent
when, for any arbitrary logic program Π, both Π1∪ Π and Π2∪ Π have the
same stable models. Note that, for a monotonic logic, this property trivially
collapses to regular equivalence (i.e., coincidende of sets of models) of Π1and
Π2. However, when a nonmonotonic entailment is involved, the addition of a
set of rules Π may make different effects in the sets of stable models of Π1
and Π2, so that strong equivalence is indeed an stronger property than regular
equivalence. In [8] it was shown that two logic programs are strongly equivalent
(under the stable models semantics) if and only if they are equivalent under the
intermediate logic of HereandThere [13], the monotonic basis of Equilibrium
Logic.
In this paper we consider the study of strong equivalence for logic programs
that represent transition systems. To this aim, we revisit a temporal extension
of Equilibrium Logic proposed in [14] which consists in the inclusion of modal
operators as those handled in Linear Temporal Logic (LTL) [15,16]. This exten
sion, called Temporal Equilibrium Logic, immediately provides us with a suffi
cient condition for strong equivalence of temporal logic programs: we can simply
check regular equivalence in its monotonic basis, a logic we called Temporal
HereandThere (THT). The main contribution of the paper is the automation
of this test for strong equivalence so that, using a similar translation to those
presented in [17,18,10], we transform a THT formula into LTL and use an
LTL prover afterwards – in particular, we runned our experiments on the Logics
Workbench [19].
The paper is organised as follows. In the next section, we introduce a simple
motivating example, extracted from the Reasoning about Actions literature, to
2in the most general case). In practice, this means for instance
Page 3
show the kind of problems we are interested in, proposing a pair of strong equiv
alence tests in this domain. In Section 3 we revisit the syntax and semantics of
Temporal HereandThere (THT) and we propose afterwards a models selection
criterion, to define the nonmonotonic Temporal Equilibrium Logic (TEL). Sec
tion 4 presents the translation from THT into LTL, whereas Section 5 applies
this translation to answer the questions proposed in Section 3. Finally, Section 6
contains the conclusions and future work.
2A simple motivating example
Consider the following simple and wellknown scenario [20] from Reasoning about
Actions literature.
Example 1. An electric circuit consists of a battery, two switches and a light
bulb. The switches are serially connected, as shown in Figure 2. The system
state is expressed in terms of three propositional fluents sw1,sw2 and light,
whose negations are represented with a bar on top of each fluent symbol. The
state of each switch swican be alternated by performing a corresponding action
togglei.
? ?
light
sw1
sw2
Fig.1. A simple electric circuit.
For simplicity, we assume that we do not handle concurrent actions. A possi
ble representation of this scenario as an ASP logic program, Π1, is shown below:
swj(I + 1) ← togglej(I),swj(I)
swj(I + 1) ← togglej(I),swj(I)
light(I + 1) ← togglej(I),swj(I)
light(I + 1) ← toggle1(I),sw1(I),sw2(I)
light(I + 1) ← toggle2(I),sw2(I),sw1(I)
⊥ ← toggle1,toggle2
f(I + 1) ← f(I),not f(I + 1)
f(I + 1) ← f(I),not f(I + 1)
⊥ ← f(I),f(I)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
Page 4
where j ∈ {1,2} and f ∈ {sw1,sw2,light}, and we assume that each symbol
like f is actually treated as a new predicate. Variable I = 0...n − 1 represents
an integer index for each situation in the temporal narrative. Rules (1)(5) are
the effect axioms, capturing all the direct effects of actions. Rule (6) just avoids
performing concurrent actions. Rules (7),(8) represent the inertia default, for
each fluent f. Finally, (9) expresses that each fluent f and its explicit negation
f cannot be simultaneously true.
As explained in the introduction, a planning problem would include addi
tional facts (representing the initial state) and rules for generating actions and
expressing a plan goal. But the most important additional step is deciding a
finite limit n for variable I, so that the above program can be grounded.
Assume now that we want to modify Π1after noticing that the truth value
of light is actually determined by the value of the two switches, becoming in
this way an indirect effect or ramification1. In other words, we consider now the
addition of rules:
light ← sw1,sw2
light ← sw1
light ← sw2
(10)
(11)
(12)
If we call Π2 = Π1∪ {(10) − (12)}, we may reasonbly propose the following
questions:
(Q1) Can we safely remove now from Π2the effect axioms for fluent light (3)(5)?
(Q2) Assume we removed (3)(5). Since light is “defined” now in terms of sw1
and sw2, can we safely remove from Π2the inertia rules for light?
In order to answer these questions, the current tools for testing strong equiva
lence for ground programs [21,22] cannot provide successful answers in an au
tomated way, as they need previously fixing a numeric value for the path length
n.
3 Linear Temporal HereandThere (THT)
We proceed now to recall the main definitions of the temporal extension of Equi
librium Logic, beginning with its monotonic basis. The logic of Linear Temporal
HereandThere (THT) is defined as follows. We start from a finite set of atoms
Σ called the propositional signature. A (temporal) formula is defined as any
combination of the atoms in Σ with the classical connectives ∧,∨,→,⊥, the
(unary) temporal operators ? (to be read as “always” or “from now on”), ♦
(“eventually”) , ? (“next”) and the (binary) temporal operators U (“until”),
W (“weak until”) and B (“before”). Negation is defined as ¬ϕ
1In fact, this example were used in [20] to illustrate possible representational problems
for a suitable treatment of action ramifications.
def
= ϕ → ⊥
Page 5
whereas ?
allow the abbreviation ?iϕ
A (temporal) interpretation M is an infinite sequence of pairs ?Hi,Ti? with
i = 0,1,2,... where Hi⊆ Tiare sets of atoms. For simplicity, given a temporal
interpretation, we write H (resp. T) to denote the sequence of pair components
H0,H1,... (resp. T0,T1,...). Using this notation, we will sometimes abbreviate
the interpretation as M = ?H,T?. An interpretation M = ?H,T? is said to be
total when H = T. We say that an interpretation M = ?H,T? satisfies a formula
ϕ at some sequence index i, written M,i = ϕ, when any of the following hold:
def
= ¬⊥. As usual, ϕ ↔ ψ stands for (ϕ → ψ) ∧ (ψ → ϕ). We also
def
= ?(?i−1ϕ) for i > 0 and ?0ϕ
def
= ϕ.
1. M,i = p if p ∈ Hi, for any atom p.
2. M,i = ϕ ∧ ψ if M,i = ϕ and M,i = ψ.
3. M,i = ϕ ∨ ψ if M,i = ϕ or M,i = ψ.
4. ?H,T?,i = ϕ → ψ if ?X,T?,i ?= ϕ or ?X,T? = ψ for all X ∈ {H,T}.
5. M,i = ?ϕ if M,i+1 = ϕ.
6. M,i = ?ϕ if for all j ≥ i, M,j = ϕ.
7. M,i = ♦ϕ if there exists some j ≥ i, M,j = ϕ.
8. M,i = ϕ U ψ if there exists j ≥ i, M,j = ψ and M,k = ϕ for all k such
that i ≤ k < j.
9. M,i = ϕ W ψ if either M,i = ϕ U ψ or, for all j ≥ i, M,j = ϕ.
10. M,i = ϕ B ψ if for all j ≥ i, either M,j = ψ or there exists some k,
i ≤ k < j such that M,k = ϕ.
We assume that a finite sequence ?H0,T0?...?Hn,Tn? with n ≥ 0 is an
abbreviation of the infinite sequence ?H
0,...,n and H?
?,T
?? with H?
i= Hi, T?
i= Ti for i =
i= Hn, T?
i= Tnfor i > n.
The logic of THT is an orthogonal combination of the logic of HT and the
(standard) linear temporal logic (LTL) [16]. When we restrict temporal interpre
tations to finite sequences of length 1, that is ?H0,T0? and disregard temporal
operators, we obtain the logic of HT. On the other hand, if we restrict the seman
tics to total interpretations, ?T,T?,i = ϕ corresponds to satisfaction of formulas
T,i = ϕ in LTL.
A theory is any set of formulas. An interpretation M is a model of a theory
Γ, written M = Γ, if M,0 = α, for each formula α ∈ Γ. A formula ϕ is valid
if M,0 = ϕ for any M. The following are valid formulas in THT (and in LTL
Page 6
too):
♦ϕ ↔ ? U ϕ
?ϕ ↔ ⊥ B ϕ
ϕ W ψ ↔ ϕ U ψ ∨ ?ϕ
ϕ U ψ ↔ ϕ W ψ ∧ ♦ψ
¬(ϕ U ψ) ↔ ¬ϕ B ¬ψ
¬(ϕ B ψ) ↔ ¬ϕ U ¬ψ
¬?ϕ ↔ ♦¬ϕ
¬♦ϕ ↔ ?¬ϕ
?¬ϕ ↔ ¬ ? ϕ
?(ϕ ∧ ψ) ↔ ?ϕ ∧ ?ψ
?(ϕ ∨ ψ) ↔ ?ϕ ∨ ?ψ
?(ϕ → ψ) ↔ (?ϕ → ?ψ)
ϕ U ψ ↔ ψ ∨ (ϕ ∧ ?(ϕ U ψ))
ϕ B ψ ↔ ψ ∧ (ϕ ∨ ?(ϕ B ψ))
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
Theorems (13)(15) allow defining ?,♦ and W in terms of U and B. The formulas
(17) and (18) correspond to the De Morgan axioms between operators U and
B. It is easy to see that, together with (13) and (14) they directly imply the
corresponding De Morgan axioms (19) and (20) for ? and ♦. An important
difference with respect to LTL is that, when using these De Morgan axioms,
some care must be taken if double negation is involved. For instance, by (19),
the formula ¬♦¬ϕ is equivalent to ¬¬?ϕ, but this is not in general equivalent to
?ϕ. A simple counterexample is the interpretation ?H,T? with all Ti= {p} but
some Hj = ∅, as it satisfies ¬¬?p but not ?p. As a result, we cannot further
define B (resp. ?) in terms of U (resp. ♦) or vice versa, as happens in LTL.
We have included other LTL standard properties like (21)(24) to show that the
“shifting” behaviour of ? with respect to classical connectives is the same as
in LTL, or (25) and (26) that represent the inductive propagation of U and B
respectively.
The (Linear) Temporal Equilibrium Logic (TEL) is a nonmonotonic version of
THT where we establish a models selection criterion. Given two interpretations
M = ?H,T? and M?= ?H
M ≤ M?, when T = T
for M ≤ M?but M ?= M?.
Definition 1 (Temporal Equilibrium Model). An interpretation M is a
temporal equilibrium model of a theory Γ if M is a total model of Γ and there
is no other model M?< M of Γ.
?,T
?? we say that M is lower than M?, written
?and for all i ≥ 0, Hi⊆ H?
i. As usual, M < M?stands
?
Note that any temporal equilibrium model is total, that is, it has the form
?T,T? and so can be actually seen as an interpretation T in the standard LTL.
Page 7
Example 2. The temporal equilibrium models of theory {♦p} are captured by
the LTL models of the formula α := ¬p U (p ∧ ??¬p).
Proof. Note first that it is easy to check that LTL models of α correspond to
interpretations following the pattern
interpretation M = ?T,T? with some Ti = {p}. Clearly, M,0 = ♦p, but its
minimality will depend on whether there exists another j ?= i with Tj= {p} in T.
Assume there exists such a j: then it is easy to see that the interpretation {H,T}
with Hj= ∅ and Hk= Tkfor k ?= j is also a model of ♦p (since Hi= Ti= {p})
whereas it is strictly lower, so it would not be a temporal equilibrium model.
Assume, on the contrary, that Tiis the unique state containing p, so Tk= ∅ for
all k ?= i. Then, the only possible smaller interpretation would be {H,T} with
Hk= Tk= ∅ for k ?= i and Hi= ∅, but this is not a THT model of ♦p, as p
would not occur in H.
?
∅
?∗{p}
?
∅
?∗. Now, take any a total
? ?
Example 3. Consider the theory Γ just consisting of the formula ?(¬p → ?p).
Its temporal equilibrium models are captured by the LTL models of α := ¬p ∧
?(¬p ↔ ?p).
Proof. Again, note first that the LTL models of α have the form ∅?{p} ∅?+.
be minimal. Once ¬p is fixed at the initial state, we must get p at T1to satisfy
the formula. Then p true at T2would not be minimal, so it must be false, and
so on.
Now, for an informal sketch, note that any solution with p true at T0will not
? ?
Proposition 1. Let Γ be a combination of nonmodal connectives ∧,∨,¬,→,⊥
with expressions like ?ip, being p an atom, and let n be the maximum value for
i in all ?ip occurring in Γ. Then ?T,T? is a temporal equilibrium model of Γ
iff (1) Ti= ∅ for all i > n ; and (2) ?X,X? with X =?n
That is, once ?,♦,U and W are removed, we can reduce TEL to (nontemporal)
Equilibrium Logic for an extended signature with atoms like ?ip.
In [14] it was shown as an example how TEL can be used to encode in a
straightforward way the action language B [12].
An important observation is that proving the equivalence of two temporal
theories in THT is a trivial sufficient condition for their strong equivalence under
TEL – if they have the same THT models, they will always lead to the same set
of temporal equilibrium models, regardless the context. The next section shows
how this THT equivalence test can be transformed into provability in standard
LTL.
i=0{?ip  p ∈ Ti} is an
equilibrium model of Γ, reading each ‘?ip’ as a new atom in the signature.
? ?
4Translating THT into LTL
Given a propositional signature Σ, let us denote Σ∗= Σ ∪{p? p ∈ Σ} which is
going to be the new propositional signature in LTL. For any temporal formula
ϕ we define its translation ϕ∗as follows:
Page 8
1. ⊥∗def
2. p∗def
3. (?ϕ)∗def
4. (ϕ ? ψ)∗def
5. (ϕ → ψ)∗def
= ⊥
= p?for any p ∈ Σ
= ?ϕ∗, if ? ∈ {?,♦,?}
= ϕ∗? ψ∗, when ? ∈ {∧,∨,U,W,B}
= (ϕ → ψ) ∧ (ϕ∗→ ψ∗)
From the last point and the fact that ¬ϕ = ϕ → ⊥, it follows that (¬ϕ)∗=
(ϕ → ⊥) ∧ (ϕ∗→ ⊥) = ¬ϕ ∧ ¬ϕ∗. Similarly, (ϕ ↔ ψ)∗= (ϕ ↔ ψ) ∧ (ϕ∗↔ ψ∗).
We associate to any THT interpretation M = ?H,T? the LTL interpretation
Mt= I in LTL defined as the sequence of sets of atoms Ii= {p? p ∈ Hi} ∪ Ti,
for any i ≥ 0. Informally speaking, Mtconsiders a new primed atom p?per each
one in the original signature p ∈ Σ. In the LTL interpretation, the primed atom
p?represents the fact that p occurs at some point in the H component, whereas
the original symbol p is used to represent an atom in T. As a THT interpretation
must satisfy Hi⊆ Tiby construction, we may have LTL interpretations that do
not correspond to any THT one. In particular, for an arbitrary I, we will only
be able to form some M such that Mt= I when the set of primed atoms at each
Iiis a subset of the nonprimed ones. In other words, only LTL interpretations
I satisfying the axiom:
?(p?→ p)(27)
will have a corresponding THT interpretation M such that I = Mt.
Example 4. M = ((∅,{p,q}),({p},{p,q}),({q},{q})) is a model of the theory
{?(¬p → q) ∧ ♦q}. In the same way, the corresponding interpretation Mt=
({p,q},{p?,p,q},{q?,q}) is a model of
(?(¬p → q) ∧ ♦q)∗
?((¬p → q) ∧ ((¬p)∗→ q?)) ∧ ♦q?↔ ?((¬p → q) ∧ (¬p → q?)) ∧ ♦q?.
In general:
↔ ?(¬p → q)∗∧ (♦q)∗
↔
Theorem 1. Let M = ?H,T? be any THT interpretation and ϕ any formula.
For any i ≥ 0, it holds that
(a) ?H,T?,i = ϕ if and only if Mt,i = ϕ∗in LTL; and
(b) ?T,T?,i = ϕ if and only if Mt,i = ϕ in LTL.
Proof. We proceed by induction. For the base case, it trivially holds for ⊥
whereas for an atom p, we have these equivalence chains
(a) (?H,T?,i = p) ⇔ (p ∈ Hi) ⇔ (p?∈ Ii) ⇔ (Mt,i = p?).
(b) (?T,T?,i = p) ⇔ (p ∈ Ti) ⇔ (p ∈ Ii) ⇔ (Mt,i = p)
Page 9
For the inductive step, we detail the proof for the classical connective →, the
(unary) temporal operator ? and the (binary) temporal operator U. For the
classical connectives ∧ and ∨ the proof is straightforward; for connective B , it
is completely analogous to that for U; and finally, the rest of connectives can be
defined in terms of the previous ones.
1. To prove (a) for the implication we have the chain of equivalent conditions:
For proving (b) it suffices with considering, in each of the three pairs of
conjunctive conditions above, only the second conjunct.
2. For operator ?, note that
(a) ?H,T?,i = ?ϕ
Mt,i = ?ϕ∗
(b) ?T,T?,i = ?ϕ
Mt,i = ?ϕ
3. Finally, for U it follows that:
(a) ?H,T?,i = ϕ U ψ
∃j ≥ i, (M,j = ψ) and ∀k s.t. i ≤ k < j, (M,k = ϕ)
∃j ≥ i, (Mt,j = ψ∗) and ∀k s.t. i ≤ k < j, (M,k = ϕ∗)
Mt,i = ϕ∗U ψ∗
(b) ?T,T?,i = ϕ U ψ
∃j ≥ i, (M,j = ψ) and ∀k s.t. i ≤ k < j, (M,k = ϕ)
∃j ≥ i, (Mt,j = ψ) and ∀k s.t. i ≤ k < j (M,k = ϕ)
Mt,i = ϕ U ψ
Theorem 2 (Main theorem). Let Γ1and Γ2be a pair of temporal theories,
and?Γ1 and?Γ2 the conjunctions of their respective sets of formulas. Then
the formula (27) → (?Γ1↔?Γ2)∗is valid in LTL.
Proof. Once (27) is fixed as hypothesis, Theorem 1 allows us to establish a one
to one correspondence between models of?Γ1↔?Γ2in THT and models of
?Γ1 and?Γ2 are THTequivalent, which is a sufficient condition for strong
As we will show in the next section, we may also use this result to detect a
redundant formula ϕ in some theory Γ. To this aim, we would have to show that
Γ and Γ?= Γ \{ϕ} are strongly equivalent. From the theorem above, it follows
that:
?H,T?,i = ϕ → ψ
Mt,i ?= ϕ or Mt,i = ψ
Mt,i = (ϕ → ψ)∗
def
⇐⇒
?H,T?,i ?= ϕ or ?H,T?,i = ψ
and
?T,T?,i ?= ϕ or ?T,T?,i = ψ
ind
⇐⇒
Mt,i ?= ϕ∗or Mt,i = ψ∗
and
⇐⇒
Mt,i = (ϕ∗→ ψ∗)
and
Mt,i = (ϕ → ψ)
def
⇐⇒
def
⇐⇒ ?H,T?,i + 1 = ϕ
ind
⇐⇒ Mt,i + 1 = ϕ∗
def
⇐⇒
def
⇐⇒ ?T,T?,i + 1 = ϕ
ind
⇐⇒ Mt,i + 1 = ϕ
def
⇐⇒
def
⇐⇒
ind
⇐⇒
def
⇐⇒
def
⇐⇒ Mt,i = (ϕ U ψ)∗
def
⇐⇒
ind
⇐⇒
def
⇐⇒
? ?
Γ1and Γ2are strongly equivalent with respect to temporal equilibrium models if
(?Γ1↔?Γ2)∗in LTL. Thus, if (?Γ1↔?Γ2)∗is LTL valid, this means that
equivalence in TEL.
? ?
Page 10
Corollary 1. Let Γ be a temporal theory,?Γ the conjunction of its formulas
alent if the formula (27) → (?Γ → ϕ)∗is valid in LTL.
5Back to the example
and ϕ some arbitrary temporal formula. Then Γ and Γ ∪{ϕ} are strongly equiv
? ?
When representing now Example 1 in TEL, the encoding is quite obvious. As
happens in Equilibrium Logic, the logic programming notation is directly re
placed by a logical syntax, so that the rule arrow ‘←’, the comma ‘,’ and the
default negation ‘not ’ respectively represent standard implication, conjunction
and negation. On the other hand, in order to capture the transition rules of pro
gram Π1, only a small subset of the linear temporal syntax is actually required.
All the rules depending on the temporal (universally quantified) variable I will
be scoped now by a ? operator, whereas the atoms with argument I + 1 will
be further preceeded by a ? connective. As a result, we obtain the theory Γ1
consisting of the formulas:
?(togglej∧ swj→ ?swj)
?(togglej∧ swj→ ?swj)
?(togglej∧ swj→ ?light)
?(toggle1∧ sw1∧ sw2→ ?light)
?(toggle2∧ sw2∧ sw1→ ?light)
?(toggle1∧ toggle2→ ⊥)
?(f ∧ ¬ ? f → ?f)
?(f ∧ ¬ ? f → ?f)
?(f ∧ f → ⊥)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
that respectively correspond to the rules (1)(9).
In order to answer questions Q1 and Q2 proposed in Section 2, we have
implemented the ϕ∗translation2and feeded the LWB linear temporal logic module
with the resulting formulas extracted from Corollary 1 to detect redundant rules.
In particular, regarding question Q1, we have been able to prove that, given the
theory Γ2consisting of Γ1plus:
?(sw1∧ sw2→ light)
?(sw1→ light)
?(sw2→ light)
(37)
(38)
(39)
(that captures light as an indirect effect) then rule (30), that specified when
light became off depending on each togglej, becomes redundant and can be
2Available at http://www.dc.fi.udc.es/~cabalar/eqwb.html
Page 11
safely removed from Γ2. In fact, we could prove that (30) just followed from the
formulas (29), (38) and (39).
However, when we consider the effect axioms (31),(32) we used to specify
when the light becomes on, our checker just provides a negative answer. Since
we are just dealing with a sufficient condition for strong equivalence, this does
not provide any concluding information about Q1 for these rules. Fortunately,
in this case it is not difficult to see that, in fact, these rules cannot be removed
in any context without changing the general behaviour of the program. To see
why, just think about adding, as a new effect of toggle1, that switch sw2 is
disconnected:
?(toggle1∧ sw2→ sw2)
The addition of this new rule would clearly make a different effect depending on
(31) is present in the theory or not. In particular, when we perform toggle1in
a situation in which sw1∧ sw2, if we do not have (31) the light will just remain
off, whereas if this formula is included, we will get no equilibrium model.
In order to answer question Q2, we considered theory Γ3= Γ2\{(3)−(5)} and
tried to see whether (34),(35) were redundant, for the case f = light. Although
it could seem that light value is “defined” in terms of sw1and sw2by rules (37)
(39), we actually obtained a negative answer, so no conclusion is obtained. It is
easy to see that, in this case, we can find situations in which removing inertia
may cause different effects depending on the context. For instance, if we had
light and do not provide information for sw1and sw2or their explicit negations
in a given state, the inertia would maintain light in the next situation (toggling
yields no effect). If we remove inertia, however, neither light or light would hold
in the next state.
If we look for a stronger relation between light and sw1,sw2we can move to
consider Γ3, consisting of Γ2and the rules:
?(light → sw1)
?(light → sw2)
?(¬light → light)
we finally obtain that inertia (34),(35) for f = light is redundant. The explana
tion for this is simple – note that these rules, together with (37) allow concluding
(among other things) that :
?(light ↔ sw1∧ sw2)
which is a double implication, actually defining light as the conjunction of sw1
and sw2in any situation.
6Conclusions
In this paper we have studied the problem of strong equivalence of logic pro
grams that represent transition systems. To this aim, we have incorporated to
Page 12
logic programs the set of modal operators typically used in Linear Temporal
Logic (LTL). We provided a general semantics for any arbitrary (propositional)
temporal theory, by proposing a temporal extension of Equilibrium Logic [7] (a
logical characterisation of stable models) and its monotonic basis, the logic of
HereandThere (HT). As a result, we obtained a sufficient condition for checking
strong equivalence of temporal logic programs, and we implemented a method
for reducing this condition to provability in standard LTL. The advantage of this
process is that, to the best of our knowledge, it constitutes the first automated
tool for guaranteeing strong equivalence of logic programs representing transi
tion systems, as the previous existing checkers always require fixing a numerical
path length to obtain a ground program.
Of course, the main current drawback of this method is that, when the im
plemented test provides a negative answer, no conclusion can be drawn, since we
have not proved yet whether it also constitutes a necessary condition for strong
equivalence, something that was obtained [8] for the nonmodal case with HT
equivalence. The desirable situation would be, instead, that a negative answer
came with a counterexample, that is, a piece of (temporal) program that added
to the original ones to be tested yields different effects in each case. This point
is left for the immediate future work.
Acknowledgements We wish to thank David Pearce, Agust´ ın Valverde, Manuel
Ojeda and Alfredo Burrieza for their helpful discussions about this work.
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