We now define desirable properties of transformations of transition systems. In the following, for a function f : A → B, by f−1we denote the inverse function from elements of the image B
of f to subsets of the preimage A of f, defined as f−1(b) ={a ∈ A | f(a) = b}. The inverse
function may also be applied to sets B′ ⊆ B using the definition f−1(B′) = {a ∈ A | f(a) ∈
B′}.
Definition 3.3 (Properties of Transformations). Let τ = ⟨Θ′, σ, λ⟩ be a transformation of a
transition systemΘ = ⟨S, L, c, T, s0, S⋆⟩ into a transition system Θ′ = ⟨S′, L′, c′, T′, s′0, S⋆′⟩.
The following list defines properties thatτ may have, along with a short-hand name for each property. (For example, we say thatτ satisfiesINDSifτ is state-induced, as defined in the first list entry.)
INDS τ isstate-induced if σ is surjective, i.e. if ∀s′ ∈ S′ ∃s ∈ S: s ∈ σ−1(s′).
INDL τ islabel-induced if λ is surjective, i.e. if ∀ℓ′∈ L′∃ℓ ∈ L: ℓ ∈ λ−1(ℓ′).
CONST τ istransition-conservative if ∀s−→ t ∈ T : σ(s)ℓ −−→ σ(t) ∈ Tλ(ℓ) ′. INDT τ istransition-induced if ∀s′ ℓ ′ −→ t′∈ T′∃s−→ t ∈ T : s ∈ σℓ −1(s′)∧t ∈ σ−1(t′)∧ℓ ∈ λ−1(ℓ′). REFT τ istransition-refinable if ∀s′ ℓ ′ −→ t′ ∈ T′∀s ∈ σ−1(s′)∃s ℓ − → t ∈ T : t ∈ σ−1(t′)∧ℓ ∈ λ−1(ℓ′). CONSG τ isgoal-conservative if ∀s ∈ S⋆: σ(s) ∈ S⋆′. INDG τ isgoal-induced if ∀s′∈ S⋆′ ∃s ∈ S⋆: s ∈ σ−1(s′). REFG τ isgoal-refinable if ∀s′ ∈ S⋆′ ∀s ∈ σ−1(s′): s ∈ S⋆. CONSC τ iscost-conservative if ∀ℓ ∈ L: c′(λ(ℓ))≤ c(ℓ). INDC τ iscost-induced if ∀ℓ′ ∈ L′ ∃ℓ ∈ L: ℓ ∈ λ−1(ℓ′)∧ c(ℓ) = c′(ℓ′) REFC τ iscost-refinable if ∀ℓ′ ∈ L′∀ℓ ∈ λ−1(ℓ′): c(ℓ) = c′(ℓ′).
Based on these basic properties, we define the following derived properties, whereA= B+C means thatτ has property A if it has properties B and C:
• conservative: CONS = CONST+CONSG+CONSC
• induced: IND = INDS+INDL+INDT+INDG+INDC
• refinable: REF = REFT+REFG+REFC
Conservative transformations (CONS) are also called abstractions or homomorphisms. Ab- stractions that are also induced (CONS + IND) are called induced abstractions or strict homo- morphisms. Abstractions that are refinable (CONS + REF) are called exact transformations. Anexact induced transformation combines all three properties (CONS + IND + REF).
Finally, we introduce variants ofCONSTandCONSG for a subsetQ⊆ S of the states: CONSQ
T τ istransition-conservative for Q if ∀s−→ t ∈ T with s, t ∈ Q: σ(s)ℓ −−→ σ(t) ∈ Tλ(ℓ) ′.
CONSQ
G τ isgoal-conservative for Q if: ∀s ∈ (S⋆∩ Q): σ(s) ∈ S⋆′.
All derived properties can be analogously applied with respect to a subsetQ⊆ S of the states. For example,τ isconservative for Q (CONSQ) if it satisfiesCONSQ
T+CONSQG+CONSC, and it is astrict homomorphism for Q if it isCONSQ+IND.
Informally speaking, a transformation τ of a transition system Θ into transition system Θ′
is an abstraction (homomorphism) if all behaviors possible in Θ are preserved by τ: every transition of Θ has a corresponding abstract transition in Θ′(CONS
T) of the same or lower cost
(CONSC), and every goal state has a corresponding abstract goal state (CONSG). As we will
show, this is sufficient for deriving admissible and consistent heuristics from τ.
Among these homomorphisms, strict homomorphism are in some sense the most accurate ones: while they include all transitions and goal states that a homomorphism must include, they do not include any additional transitions (INDT) or goal states (INDG) beyond those required by
the homomorphism property. They must not contain any abstract states (INDS) or labels (INDL)
beyond those that the state and label mapping map to. Finally, transformed label costs must correspond to the cost of some original label (INDC), which together with cost-conservativeness
implies that the cost of a transformed label must be the minimum cost of its preimage labels. It is not difficult to show that for every state mapping σ and label mapping λ, there exists a unique transition system Θ′ such that τ = ⟨Θ′, σ, λ⟩ is a strict homomorphism, namely the induced
abstract transition system from Definition 2.11. Hence, strict homomorphisms are uniquely described by their state and label mappings, and we say that Θ′is the transition system induced
by σ and λ. Induced abstractions are practically desirable because they provide the strongest possible heuristics among all abstractions with the same state and label mappings. They are also theoretically desirable because they can be fully understood and analyzed in terms of the state and label mapping.
Exact transformations are conservative “in both directions”: intuitively, refinability means that all behaviors possible in Θ′are also possible in Θ. All transformed transitions s′ ℓ′
−→ t′can be mapped back to original transitions s ℓ
−
→ t for all preimages s of s′ (REF
T), all preimages
of goal states are goal states (REFG), and the label mapping does not affect the label costs
0 1 2 3 x x y z c(x) = 4 c(y) = 3 c(z) = 2 (a) Original transition system.
0 1
2 3
yz
x c0(x) = 4
c0(yz) = 5
(b) Arbitrary transformation (not an abstraction).
03 12
x, yz x, yz
c0(x) = 2 c0(yz) = 2
(c) Abstraction (not a strict homomorphism).
01 2 3 xy xy z c0(xy) = 3 c0(z) = 2
(d) Strict homomorphism (not exact).
02 1 3 x y z c 0(x) = 4 c0(y) = 3 c0(z) = 2 (e) Exact transformation.
Figure 3.1.: Four different transformations of the transition system in part (a). The captions of part (b)–(e) indicate the properties of the transformation.
the same way. To make this formal, we will prove that heuristics based on exact transformations are perfect.
We remark that we define transition-refinability in such a way that given a transition s′ ℓ−→ t′ ′
of Θ′, Θ has a corresponding transition for all preimages s of s′ and some preimage t of t′.
One could alternatively consider a definition where Θ must have a transition for all t and some s. Both definitions give rise to notions of abstract paths being refinable to concrete paths, but the alternative definition does not lead to a perfect heuristic. One could of course also require corresponding transitions to exist for all s and all t, but this is unnecessarily restrictive as our weaker property already leads to a perfect heuristic.
Figure 3.1 illustrates four example transformations with different properties. The original transition system is shown in part (a) of the figure. All other parts of the figures show a transfor- mation of this transition system. We use undecorated numbers and letters to denote states and labels of the original transition system and overlined symbols to denote states and labels of the transformed transition systems. If a state s of the original transition systems is mapped to some state of the transformed transition system, then the name of the state includes s. For example, a transformed state whose preimage consists of states 0 and 3 is denoted by 03. We proceed analogously for labels. All transformations are state-induced and label-induced, so there are no transformed states and labels other than those that the original state and labels map to.
Figure3.1bshows the first transformation, an example of a non-abstraction transformation. Indeed, it satisfies none of the properties of a conservative transformation. It is not goal-
conservative because 3 is not a goal state even though 3 is. It is not transition-conservative because the transformed transition system has no transition corresponding to 0−→ 3. Finally, itx
is not cost-conservative because the cost of yz is higher (5) than the costs of y and z (4 and 3). Figure 3.1cshows an abstraction: all transitions induce abstract transitions, all goal states induce abstract goal states, and all labels are mapped to labels of the same or lower cost. If we removed the transitions from 12 to 03, it would no longer be an abstraction (violatingCONST),
but it would still be an abstraction for the subset of reachable states. The transformation is no strict homomorphism because the transition 03 −→ 03 is not induced by any original transi-yz
tion (violatingINDT) and also because the label cost 2 of x differs from the cost 4 of its only
preimage label x (violatingINDC).
Figure3.1dshows a strict homomorphism: all states, labels, transitions, goal states and label costs are induced by the original transition system. The transformation is not exact for several reasons: for example, it is not transition-refinable because the transition 01 z
−
→ 01 has no match- ing original transition for the preimage 0 of 01, as neither 0 −→ 0 nor 0z −→ 1 are transitions ofz
the original transition system. The transformation is also not cost-refinable because the cost of label xy (3) is lower than the cost of x (4).
Finally, Figure3.1eshows an exact induced transformation: it is conservative, induced and refinable.
Bäckström and Jonsson (2013) describe a similar framework to model abstractions as trans- formations from a labeled digraph G to a labeled digraph G′. As in our case, transformations
must specify how the states and labels of the two digraphs are related. A major difference is that while in our case the states and labels of the two digraphs are related by functions, Bäckström and Jonsson consider more general relations. A transformation in their setting is represented by a set-valued function f that maps states of G to sets of states of G′(with further constraints that
essentially specify that f defines a bijection between equivalence classes of the states of G and G′), and an arbitrary relation R between the labels of G and the labels of G′.
For example, this notion of transformation allows mapping a single state to multiple states, and it is reversible in the sense that for each transformation from G to G′, there exists an inverse
transformation from G′ to G. We restrict ourselves to (functional) state and label mappings be-
cause these are simpler and sufficient for our purposes. A further difference is that the transition graph formalism used by Bäckström and Jonsson does not include notions of initial states, goal states or label costs, although of course these can be associated with transition graphs externally. Bäckström and Jonsson also study properties of transformations, some of which are quite sim- ilar to properties we defined, the main difference being that most of their properties do not con- sider labels but only depend on the relationship between states. In some more detail, Bäckström and Jonsson define several “method properties” for the state mapping f, includingM↑, meaning
that f is functional (rather than set-valued);R↑, meaning in our notation that if s−→ t ∈ G, thenℓ
there is s′ ℓ−→ t′ ′ ∈ G′ such that R(ℓ, ℓ′); andC
↑, meaning in our notation that if R(ℓ, ℓ′)and
s −→ t ∈ G, then there is sℓ ′ ℓ−→ t′ ′ ∈ G′ such that s′ ∈ f(s) and t′ ∈ f(t). They call a trans-
formation a homomorphism if it satisfiesM↑R↑C↑.1 Our propertyCONSTcorresponds to this
notion of homomorphism, while our definition of homomorphism requires additional conditions 1Bäckström and Jonsson use concatenation to denote the combination of properties where we use the “+” symbol.
on goal states and label costs, which are not present in their formalism.
Bäckström and Jonsson further define the converse propertiesR↓ andC↓in the obvious way
and call a transformation satisfying M↑R↕C↕ a strong homomorphism. This corresponds to
addingINDT toCONST. Again, for a transformation to be a strong (strict) homomorphism in
our sense, we require additional conditions on goal states and label costs.
Despite these differences the approach by Bäckström and Jonsson is very similar in spirit and execution to ours. Indeed, it was one of the major inspirations for our definition of transforma- tions between transition systems as well as for the more general idea of studying the merge-and- shrink framework in terms of a family of transformation properties.