Linear cuts in Boolean networks

Boolean networks are popular tools for the exploration of qualitative dynamical properties of biological systems. Several dynamical interpretations have been proposed based on the same logical structure that captures the interactions between Boolean components. They reproduce, in different degrees, the behaviours emerging in more quantitative models. In particular, regulatory conflicts can prevent the standard asynchronous dynamics from reproducing some trajectories that might be expected upon inspection of more detailed models. We introduce and study the class of networks with linear cuts, where linear components—intermediates with a single regulator and a single target—eliminate the aforementioned regulatory conflicts. The interaction graph of a Boolean network admits a linear cut when a linear component occurs in each cycle and in each path from components with multiple targets to components with multiple regulators. Under this structural condition the attractors are in one-to-one correspondence with the minimal trap spaces, and the reachability of attractors can also be easily characterized. Linear cuts provide the base for a new interpretation of the Boolean semantics that captures all behaviours of multi-valued refinements with regulatory thresholds that are uniquely defined for each interaction, and contribute a new approach for the investigation of behaviour of logical models.


Introduction
Boolean networks are a class of non-deterministic discrete event systems used as qualitative dynamical models of biological processes.The study of complex biological processes leads to two types of results: insight about the internal molecular mechanisms, and observation of their state over time and different external stimulations.While the changes of state emerge from the internal mechanisms, they can not be directly compared.The integration of mechanistic knowledge into dynamical models enables to contrast the behaviour emerging from the model with the experimental observations.Such models are valuable tools to identify inconsistencies, evaluate hypothesis and prioritize their experimental validation.Starting with a known initial condition, the model can be used to predict the reachability and stability of a target phenotype, which corresponds to properties of the reachable states of the model.The lack of precise information on the initial conditions and kinetic parameters impedes the construction of comprehensive quantitative models without performing time-consuming exploration of parameters.Boolean and more generally qualitative models have been proposed to cope with this lack of quantitative knowledge (Kauffman, 1969;Thomas, 1973).These models provide a discrete approximation well suited to build large comprehensive models based on incomplete knowledge.They are also amenable to formal analysis, in particular for the identification of attractors (Naldi et al, 2007;Dubrova and Teslenko, 2011;Klarner et al, 2014).Multi-valued networks can be used to account for components for which a higher activity level (denoting for example a higher concentration or a stronger activation) can lead to different effects (new targets, stronger or different effect).Most large networks lack this level of detail and consider only Boolean components (sometimes a few selected multi-valued components).In practice, the coarse-grained predictions obtained with these models are sufficient to reproduce relevant behaviours in a wide range of biological applications (e.g.Sizek et al, 2019;Béal et al, 2021;Bonzanni et al, 2013;Cohen et al, 2015;Collombet et al, 2017).
The analysis of these models often aims initially at the identification of attractors (fixed points or stable oscillations) and reachability properties, which are computationally hard problems in the classical asynchronous semantics.Modelers can attempt to simplify the analysis by first considering trap spaces, stable subspaces that can be efficiently identified using constraint-solving approaches (Klarner et al, 2014).Trap spaces provide a crude approximation of some of the attractors, but may not capture all of them.In addition, they can be used to rule out some reachability properties, since all states outside of the smallest trap space including the initial state are not reachable.On the other hand, reachability analysis inside a given trap space remains hard to solve.These questions are much easier to tackle using the recently proposed most permissive semantics (Paulevé et al, 2020), an over-approximation of the asynchronous semantics which lifts competition between concurrent events by introducing intermediate states representing the inherent uncertainty of Boolean networks.This approach formally accounts for the reachability properties of all possible refinements and where a subset of components are updated.More in detail, for a Boolean network (,  ), given two distinct states ,  (i.e.Δ(, ) ≠ ∅), there exists a transition from  to  • in the synchronous dynamics, if and only if  =  (), • in the asynchronous dynamics, if and only if  = x with  ∈ Δ(,  ()), • in the generalized asynchronous dynamics, if and only if Δ(, ) ⊆ Δ(,  ()).
Note that each state has at most one successor in the synchronous updating, at most  successors in the asynchronous updating and up to 2  − 1 successors in the generalized asynchronous case.
Other updatings have been proposed, in particular the bloc-sequential updating (deterministic, see Robert, 1986) and the use of priority classes (non-deterministic, see Fauré et al, 2006).In addition, one can define stochastic dynamics by adding transition probabilities to non-deterministic updatings.A trajectory from a state  to a state  in any of these updatings implies the existence of a trajectory from  to  in the generalized asynchronous dynamics.By definition, all transitions in the synchronous, asynchronous and priority updatings are also transitions in the generalized asynchronous dynamics.Individual bloc-sequential transitions may not correspond to transitions in the generalized asynchronous dynamics; however, equivalent trajectories always exist.In summary, the reachability properties of the generalized asynchronous dynamics provide an over-approximation of the reachability properties in all other classical updatings.
In addition to the classical updating semantics, the most permissive (MP) semantics has recently been proposed to account for trajectories of multi-valued or continuous refinements which are not captured by the generalized asynchronous dynamics (Paulevé et al, 2020).This semantics introduces intermediate states representing uncertainty during the transitions from regular Boolean states: when a component is in an intermediate state, its target can behave as if it were in either of the classical Boolean state.In this work, we propose an alternative definition of this semantics in Definition 1.This semantics gives an over-approximation of all classical semantics, including the generalized asynchronous and allows to further recover additional relevant dynamical trajectories observed in any multi-valued refinements of the Boolean network.From a computational perspective, while the most permissive semantics increases the cost of explicit simulations due to its large number of trajectories, it also enables efficient analytical methods for the identification of attractors and reachability properties.
We conclude this section with some additional nomenclature.For a path or trajectory  in the asynchronous dynamics given by the sequence of states  0 , . . .,   , we call direction sequence of the path  the sequence  0 , . . .,  −1 of the directions of the edges in the path.In other words, the sequences satisfy     ≠  +1   for  = 0, . . .,  −1.If the direction sequence contains no repetition, we say that  is a geodesic.For convenience, we will call a geodesic in asynchronous dynamics an asynchronous geodesic.
A fixed point (also called stable state or steady state), is a state  such that  () = .Given a fixed point , we have Δ(,  ()) = ∅, and this state has no successor in any updating.
An implicant of a function is a subspace such that the function is true in all states of the subspace.An implicant is prime if it is not contained in any larger implicant (i.e. if it has a minimal set of fixed variables).
A trap space (also called stable motif), is a subspace  such that for each  ∈ ,  () ∈ .One can think of trap spaces as partial fixed points.If a state belongs to a trap space, then all its successors in any updating also belong to this trap space.We call a trap space minimal if it is not a superset of any other trap space.Note that the overlap of two trap spaces is also a trap space and that there is a unique minimal trap space containing a given state .
A trap set is a subset of the state space that is closed for the dynamics.An attractor is an inclusion-minimal trap set.It can consist of an isolated state (it is then a fixed point), or of multiple states; in the latter case it is called a cyclic or complex attractor.Note that trap sets and attractors may depend on the updating semantics, while fixed points and trap spaces are structural properties of the network itself.Each trap space is also a trap set and contains at least one attractor for any updating; the number of minimal trap spaces is thus a lower bound for the number of attractors.

Partial orders in asynchronous and permissive trajectories
Here we investigate structural conditions for existence of permissive and asynchronous geodesics.For this, we define permissive trajectories, which reproduce the most permissive semantics using classical Boolean states and subspaces instead of an extended state space based on the addition of transitory states.We will then use implicants associated to the functions  and their differences with the initial state to identify partial orders enabling permissive geodesics.The partial orders that satisfy additional constraints correspond to geodesics in the classical asynchronous dynamics.
Given the state  and a subspace , the three following sets of components form a partition of : Observe that the state  is in the subspace  if and only if Δ(, ) = ∅.
Definition 1.A permissive trajectory is a succession of states  0 ,  1 , . . .,   such that for any  <  there is a component  such that Δ(  ,  +1 ) = {} and the smallest subspace containing all states ( 0 , . . .,   ) contains at least one state  such that   () ≠    .By extension, a permissive geodesic is a permissive trajectory where each component is used at most once.
Observe that any classical asynchronous trajectory is a permissive trajectory and that any generalized asynchronous trajectory can also be reproduced by a permissive trajectory.We can further define a bijection between permissive trajectories and trajectories starting with a pure Boolean state in the MP semantics.
Proposition 1.Given any permissive trajectory from  to , there exists a permissive trajectory from  to  of length at most 2.This property corresponds to Lemma 1 in the MP supplementary.Note that we get a bound of 2 steps here instead of the 3 bound in MP definition as the transitions from transitory states to regular Boolean states are implicit in the definition of the permissive trajectories.
Proposition 2. Let  be a state and let  be such that [, ] is the minimal trap space containing .Then all maximal permissive geodesics starting in  end in .
We are interested in studying reachability from a given an initial condition .In particular we are interested in determining whether a target state is reachable from  by looking at the implicants defining the network  .To this end, we introduce implicant maps, that is, possible choices of implicants for a given target, and give a characterisation of implicant maps that provide paths to the target as either permissive trajectories or asynchronous trajectories.
Definition 2. Given a state  and a set of components  ⊆ , the map I :  → {0, 1, ★}  is an implicant map of  for the state  if for each component  ∈  and each state  ∈ I () we have   () ≠   .
The set of strong requirements Δ + (, I ()) of the component  combines the set of requirements of  with the set of components blocked by : Intuitively, we want to establish if an implicant map defines a geodesic from  to x .Δ(, I ()) is the set of components that need to change to enable a change in component .On the other hand, some components can only be updated before a change in component , thus creating some potential "conflicts" that forbid some updating orders.The sets Δ + capture these possible conflicts.To talk about absence of conflicts we introduce the notion of consistency.
We need two additional auxiliary constructions.Given an implicant map I :  → {0, 1, ★}  , define the graphs  (I,  An implicant map I is strongly consistent if for each  ∈  we have − → Δ + (I, , ) ⊆  \ {}.The following result establishes that the conditions of consistency and strong consistency exactly characterize the ability of an implicant map to define a permissive geodesic or an asynchronous geodesic.
Proposition 3. Given a state  and a set of components  ⊆ , there is a permissive geodesic from  to x if and only if there if a consistent implicant map of  for .
Furthermore, there is an asynchronous geodesic from  to x if and only if there is a strongly consistent implicant map of  for .
The proof for the permissive geodesic case proceeds similarly, with the sets of full requirements replacing the sets of strong full requirements and  (I, ) replacing  + (I, ).
Let I and I be two different implicant maps for  in .We say that I is a generalization of I if for each  ∈  we have I () ⊆ I ().Observe that if I is (strongly) consistent, then all its generalizations are also (strongly) consistent.We say that I is a prime implicant map if it has no generalization.Observe that if I is a prime implicant map, then for each  ∈ , I () is a prime implicant of the function   or of its negation (depending on the value of   ).In this case, the sets of requirements and blockers, and by extension the (strong) full requirements, associated to each component are minimal.
Lemma 4. If I is a prime implicant map of  for , given  ∈ : (i) for all  ∈ Δ(, I ()), the interaction graph of  has an edge from  to ; (ii) if  ∈ ∇(, I ( )) for some  ∈ , then the interaction graph of  has an edge from  to .In particular, for all  ∈ Δ + (, I ()) \ Δ(, I ()) the interaction graph of  has an edge from  to .
() If  is such that  ∈ ∇(, I ( )), then I  ( ) ≠ ★ and   () ≠   for all  ∈ I ( ).If  is not a regulator of , then   (ȳ  ) ≠   for all  ∈ I ( ) and I admits a generalization as in the previous point.
The following proposition is a corollary of the lemma.Here, given a directed graph , we write G for the undirected graph obtained by ignoring the directions of all edges.
Proposition 5. Consider a Boolean network (,  ) with interaction graph  and I :  → {0, 1, ★}  a prime implicant map for .Then, for all ,  ∈ : then there is a path of length greater than zero from  to  in ; (ii) if  ∈ − → Δ + (I, , ) then there is a path of length greater than zero from  in  in G; if  ∈ − → Δ + (I, , ) \ − → Δ (I, , ) then there is at least one edge (ℎ, ) in the path such that (, ℎ) is an edge in .

𝐿-cuttable Boolean networks
In the previous section, we identified conditions on the implicant maps associated to a given initial state for the existence of a geodesic in permissive trajectories or in classical asynchronous trajectories.In presence of a permissive geodesic, we observed that conflicts captured by the implicant map and the associated auxiliary graph can prevent the existence of the corresponding asynchronous geodesic.Here we will define a topological class of networks in which such conflicts do not exist.In this case, all consistent implicant map are also strongly consistent, and thus all permissive geodesics exist in the asynchronous dynamics.
In the following, we say that a component of a network is linear if it has a single regulator and a single target.In the next definition we introduce the class of linearly-cuttable networks, that is, networks that admit a set of linear components separating all potential regulatory conflicts.We will show that, for asynchronous dynamics associated to linearly cuttable Boolean networks, trap spaces provide good approximation of attractors; in addition, we will prove some general reachability properties.
Definition 3. Given a directed graph  on , a linear cut of  is a set  ⊆  of linear components such that (i) every cycle in  contains at least one component of , (ii) every path of length greater than zero in  from a component with multiple targets to a component with multiple regulators contains at least a component of .
A linear cut  in minimal if there is no linear cut strictly included in .
A Boolean network  = (,  ) is -cuttable if  ⊂  is a linear cut for its interaction graph .
We will also need the notion of canonical states.For an -cuttable network (,  ), Note that if  has a linear cut , then each loop (cycle of length one) is a connected component (since the unique vertex of the loop is necessarily linear).Such a component is called an isolated loop.For all the properties we consider in the following, it is easy to see that if  is obtained from  by adding isolated loops, and  satisfies the given properties, then  also satisfies the same properties.Therefore, in all the following, we assume that  has no loop.
Remark 1.Consider a linear cut  and suppose that there is an edge from  to  ≠  vertices in .Since  is the unique regulator of , all cycles and all paths in  as in Definition 3 (ii) that contain  must also contain .As a consequence,  \ {  } is also a linear cut for .Since we assume that  has no isolated loop, it follows that any minimal linear cut for  is also an independent set of .In addition, there exists at least one -canonical configuration.
We now prove properties of implicant maps for networks with linear cuts.
Remark 2. Consider  -canonical and I :  → {0, 1, ★}  a prime implicant map for  and  ∈  ∩ .Since  has only one regulator , if  ∈  we must have I  () = x  and I  () = ★ for all  ≠ , which gives Δ(, Lemma 6.Given an -canonical initial state  in an -cuttable network, all consistent implicant maps for  have a strongly consistent generalization.
Proof.Consider a consistent implicant map I and take a generalisation I of I that is prime.Suppose that I is a consistent but not strongly consistent implicant map for , i.e., there is at least one component  such that . By Proposition 5 (),  is part of a cycle in G, with at least one edge ( , ) such that  ∈ Δ + (, I ()) \ Δ(, I ()) and (, ) ∈  (at least one edge is associated to a blocker).
If all edges are associated to blockers, the cycle is also a cycle in .By definition of -cuttable network, this cycle contains at least one component of .As  is canonical, the components of  have no blockers (Remark 2) and we have a contradiction.
Thus the cycle contains at least one edge associated to a direct requirement and another edge ( , ) associated to a blocker.Take the maximal sub-path  in the cycle that contains ( , ) and is composed of edges associated to blockers, and call  and  the first and last vertex in the path.Then  contains edges (  ,  ) and ( ,  ) that are not part of , and since the path  is associated to blockers,  contains a path from  to  .That is, the reverse  of the path  is a path in  from a vertex with multiple targets ( ) to a vertex with multiple regulators (  ).By definition of -cuttable network,  contains an element of .Since all edges of  are associated to blockers, this is again in contradiction with Remark 2.
By combining the lemma with Proposition 2 and Proposition 3 we obtain the following.
Corollary 7. Let (,  ) be a Boolean network with interaction graph  and  ⊂  a linear cut.All permissive geodesics starting in an -canonical state  exist in the asynchronous dynamics.In particular: (i) [, ] is the minimal trap space containing  if and only if there exists a maximal geodesic from  to .
(ii) for all subsets of components  ⊆ Δ(,  ()) there exists a path from  to x (all the successors in the generalized asynchronous dynamics are reachable from ).
(iii) The smallest subspace containing the states that are reachable from  is a trap space.
(iv) The smallest subspace containing an attractor is a trap space.
(v) If  belongs to an attractor , there is a geodesic from  to xΔ( ) , and xΔ( ) is -canonical.
(vi) If  is the last vertex of a geodesic starting from  and   () ≠   for some  ∈ [, ] and  ∉ Δ(, ), then there is a geodesic from  to ȳ .
The conclusions of the corollary do not hold for states that are not canonical: for instance, in the asynchronous dynamics of the Boolean network with two variables defined by  ( 1 ,  2 ) = ( 2 ,  1 ) there are no paths from the noncanonical state 01 to 10, while there are transitions to 00 and 11.This example also shows that point () of Definition 3 in cannot be relaxed.

Reachability of trap spaces from canonical states
Lemma 8. Let (,  ) be a Boolean network and  a geodesic from  to  with direction sequence .Let  ∈ Δ(, ) and suppose that  has no edge from  to a vertex that appears after  in .Then there exists a geodesic from  to ȳ whose direction sequence is obtained from  by deleting .
Theorem 9. Let (,  ) be a Boolean network with interaction graph  and  ⊂  a linear cut.Let  be an -canonical configuration and [, ] be the minimal trap space containing .For every trap space  ⊆ [, ] there is a path in the asynchronous dynamics from  to  of length at most 2.
Since  ∩  = ∅, for any  ∈  we have   ≠   and thus  appears in the direction sequence of .Let  0 , . . .,  −1 be an enumeration of  as in the direction sequence of .
We first prove the following property.Let us prove that there is a geodesic  =  0 ,  1 , . . .,   = z from  to z with direction sequence  0 , . . .,  −1 .We have to prove that    (  ) ≠     for 0 ≤  < .Since  is a trap space,   () =   ≠   for all  ∈ , therefore it is sufficient to show that    =   for any regulator  of   .We proceed by induction on .Let  be a regulator of  0 .By (I) we have  ∉ , so  0  =   =   .Let 0 <  <  and let  be a regulator of   .If  ∉ , then    =   =   by definition of .Otherwise, by (I) we have  ∈ { 0 , . . .,  −1 } thus   ≠    , and we deduce that    =   .
Example 1.The theorem does not hold if the initial state is not -canonical.The Boolean network The fixed points of  are 00000, 10110 and 11111.
Consider the state  = 11011, which is not -canonical (  () 1 ≠  1 ).The fixed point 11111 is a direct successor for  in the asynchronous dynamics of  .In addition, 00000 is reachable from  in the asynchronous dynamics of  via the path 11011 → 01011 → 01001 → 00001 → 00000.As a consequence, the minimal trap space containing  is the full space B 5 .Observe that there is no path from 11011 to the fixed point 10110.

Minimal trap spaces are good approximations for attractors
In this section we prove that, in asynchronous dynamics of linearly-cuttable networks, attractors and minimal trap spaces are in one-to-one correspondence.
Theorem 10.Suppose that (,  ) is -cuttable and  is an attractor for the asynchronous dynamics of  .Then [ ] is a trap space and, for every  ∈ [ ], there is a geodesic from  to .
Lemma 11.Suppose that (,  ) is -cuttable and  is an attractor for the asynchronous dynamics of  .Let  ∈ [ ] and  ∈ , and suppose that  is -canonical.Let  be the set of  ∈  with   () ≠   =   .Suppose that there is no -canonical configuration in [x  , [∩.Then there is a geodesic from  to .
(1) There is no  ∈ Δ(, ) \  such that ȳ ∈ .Suppose, for a contradiction, that  = ȳ is in  for some  ∈ Δ(, ) \ .Let  be the targets  of  such that  ∈  and   () ≠   .Since  is -canonical and  is an independent set, there is a geodesic from  to z , which is -canonical.Suppose, for a contradiction, that z ∉ [x (2)   () ≠   for some  ∈ Δ(, ).Suppose not, that is,   () =   for all  ∈ Δ(, ).Since  is -canonical, by Corollary 7 (), there is a geodesic from  to  = ȳΔ( ) , and a geodesic  from  to .Let  be the first component of the direction sequence of  with   ≠   .Since ,  ∈ [ ], we have Δ(, ) ⊆ Δ( ), thus this component  exists.Let  be the configuration of  with   () ≠   .Let us prove that  has at least two regulators.Suppose not.Since  ∈ Δ( ),  has only one regulator, and its regulator  is in Δ( ).Since  (, ) <  (, ), by induction, there is a geodesic from  to  and thus it has also a geodesic from  to .
Theorem 10 follows from Corollary 7 () and the next lemma.
Lemma 12. Suppose that (,  ) is -cuttable and  is an attractor for the asynchronous dynamics of  .Let  ∈ [ ] and  ∈ , and suppose that  is -canonical.Let  be the set of  ∈  with   () ≠   =   .Then there is a geodesic from  to some -canonical configuration  ∈ [x  , ] ∩ .
Proof.We proceed by induction on  (x  , ).Since   =   , we have  ∈ [x  , ], so if  (x  , ) = 0 then  =  and there is nothing to prove.So suppose that  (x  , ) > 0. If there there is no -canonical configuration in [x  , [∩, then, by Lemma 11, there is a geodesic from  to , so the lemma holds with  = .So suppose that there is an -canonical , ] we have Δ(x  ,  ) ⊆ Δ(x  , ) and we deduce that  (x  ,  ) <  (x  , ).
Consequently, by induction, there is an -canonical configuration  ∈ [x  ,  ] ∩  ⊆ [x  , ] ∩  such that there is a geodesic from  to .This completes the induction.
Remark 3. Theorem 10 shows that every linearly-cuttable network has the property that each minimal trap space contains only one attractor.While attractors of most permissive semantics coincide with minimal trap spaces (Paulevé et al, 2020), this is not always true for linearly-cuttable networks, as can be seen for instance by taking Boolean networks with interaction graph consisting of a negative cycle (see Remy et al, 2003, for a full characterisation of the dynamics associated to isolated circuits).

Cuttable extended semantics
Given a Boolean network, we obtain an extended network by replacing a subset of the interactions with linear components.We show that the trap spaces of the original network are also trap spaces of its extensions, which provide an over-approximation of the original asynchronous dynamics.We will focus on cuttable extended networks in which the additional linear components form a linear cut of the extended network.Cuttable extensions allow to define an execution semantics that takes advantage of the properties of cuttable networks for any Boolean network.
Biological Boolean networks are abstract models often used in absence of quantitative knowledge on precise concentrations and kinetic parameters.The non-determinism of the classical asynchronous semantics accounts for this lack of knowledge by enabling alternative trajectories corresponding to quantitative differences in initial conditions and kinetic parameters.However, it assumes that a change of the state of a component is reflected on all its targets at the same time.The introduction of intermediate linear components lets us eliminate this assumption.The alternative trajectories obtained in the asynchronous dynamics of an extended network then cover plausible behaviours that may be missing in the asynchronous dynamics of the original network.
Definition 4. Let  = (,  ) be a Boolean network with edges  and  ⊆  ⊆  2 a subset of its interactions.Consider the Boolean function E (  , ) : B  ∪ → B  ∪ defined as follows.For each  ∈  ∪ where   : B  ∪ → B  is defined for all  ∈  as: We call the Boolean network ( ∪ , E (  , )) an extended network and the -extension of  .Each row shows a Boolean network with its asynchronous dynamics (left), one of its multi-valued refinements (center) and linear extensions (right).White circles in the extended network denote intermediate linear variables, whereas numbered coloured circles in the refined network denote regulatory thresholds.Selected dynamical trajectories are depicted below each interaction graph.Groups of color-coded squares represent the states of all variables: white for level 0, blue for level 1 (or max), gray and red denote intermediate levels in refinements.Fixed points are marked with a dotted line on the right.The values of intermediate linear variables are represented with smaller squares on the right side of their regulators.a) An inconsistent feedforward loop: the first component has opposite (direct and undirect) effects on the last one.This competition can be relaxed by associating a higher threshold (center) or adding an intermediate component (left) to the direct interaction.b) A chain propagating an activation.In the most permissive semantics and some non-monotonic refinements (center), intermediate components can be disabled after propagating the signal.This behaviour can often be considered as an artefact and can not be reproduced in linear extensions.c) A chain with stabilizing feedback loops.This is an extension of the previous example where feedback loops are added to stabilize the unexpected (1, 0, 1) state.This state is still unreachable in the Boolean network, however it can now be reached in monotonic (single threshold) refinements and in linear extensions.d) A positive circuit showing that the reachability of the generalized asynchronous (where transitions can involve multiple components) can be reproduced in linear extensions, however it may not be faithfully reproduced in multi-valued refinements.
For an extended network ( ∪ , E (  , )), we call  the set of core variables and  the set of extender variables.We say that an -extension cuttable if it is -cuttable.We call the -extension of  its full extension.By construction, the -extension is cuttable.We will need the following additional notations.We write  : B  ∪ → B  for the projection onto B  , and define the map  : B  → B  ∪ that "copies" each regulator, once for each of its target variable: Note that  = ( ()) =   ( ()) for any  ∈ , and that if  contains no interaction with target , then   = .We call the states  ∈  ∪  that satisfy  (()) =  (that is, states for which the extender variables mirror their regulators) canonical states.Note that, by construction, all canonical states of an -extended network are -canonical.
Aside from the partition of their components into core and extender variables, extended networks are regular networks and the notations introduced above, such as  () and (), apply as usual.Depending on the context, extender variables will be referred to as regular variables (e.g. ∈ ( ∪ )) or as a pair of core variables (e.g.(, ) ∈  2 ).
Definition 5. Let  = (,  ) be a Boolean network,  and  two states of B  , and  a subset of its interactions.We say that  is -reachable from  if there is a trajectory from  () to  () in the asynchronous dynamics of the -extension of .
This definition of -reachability allows us to study reachability in any Boolean network using canonical initial states in an extended network.Note that the set of states that are reachable from a non-canonical state can differ significantly from the set of states that are reachable from the canonical state that projects to the same core variables.For instance, consider a Boolean network such that all components have at least one regulator, and take the full extension.Then all canonical states are reachable from any state in which all extender variables differ from their regulators.
It is worth observing that the elimination of the extender components from the extended network using the method described in Naldi et al (2011) allows to recover the original network.The asynchronous dynamics of an extended network is thus an over-approximation of the original asynchronous dynamics.As consequence, If  is -reachable from , then it is also -reachable for any  ⊃ .
In the following we compare in more detail the reachability properties of the original network and its cuttable extensions and relate the trap spaces of a Boolean network (,  ) to the trap spaces of its -extension.
Proposition 13.Consider a Boolean network (,  ) and its -extension ( ∪ ,   ).The proposition states that all trap spaces in extended networks project to trap spaces for the original network, and any trap space in the original network gives at least one trap space in any extension.In addition, if  is a canonical state in an extended network, that is  =  () for some , then the minimal trap space containing  is the canonical extension of the minimal trap space containing .
Clearly a Boolean network and its extensions do not necessarily have the same number of trap spaces.Multiple trap spaces in an extension can project to the same trap space in the original network.Take for instance the Boolean network  ( 1 ) =  1 and its extension   ( 1 ,  2 ) = ( 2 ,  1 ) with  = (1, 1).The trap spaces 00 and 0★ for   project on the same trap space (the fixed point 0).On the other hand, the mapping between trap spaces described in the proposition defines a one-to-one correspondence between minimal trap spaces of a Boolean network and any of its extensions.
Corollary 14.There is a one-to-one correspondence between the minimal minimal trap spaces of a Boolean network and the minimal trap spaces of any of its extensions.
Remark 4. Minimal trap spaces in extended networks are always canonical.Every trap space  in an extended network contains the canonical trap space  (()).

Canonical initial state
Unreachable fixed point Reachable fixed point

Minimal trap space
Reachable trap space

Reachable attractor
Unreachable space

Unreachable attractor
Figure 2: Summary of the reachability of trap spaces and attractors.Given an initial state, all states, and in particular all attractors, that are not contained in the minimal trap space containing the initial state are not reachable in any updating semantic.For -cuttable networks and -canonical initial states, all trap spaces and attractors included in the minimal trap space are reachable.
We now focus our study on cuttable extensions.As stated above, the full extension is always cuttable, but other cuttable extensions often exist in practice.Following the definition of cuttable networks, these more conservative cuttable extensions can be obtained by extending only interactions (, ) such that | ()| > 1 and |( )| > 1 as well as one interaction for each cycle which remains unextended.The following properties build on the previous results obtained on cuttable networks and can be applied to any cuttable extension.
Proposition 15.Let  be a Boolean network and  a subset of its interactions defining a cuttable extension.
(i) If there is a trajectory from  to  in the generalized asynchronous dynamics of , then  is -reachable from .
(ii) Given a state  and  the minimal trap space containing , all trap spaces contained in  are -reachable from .
(iii) There is a one-to-one correspondence between the minimal trap spaces of  and the attractors in the asynchronous dynamics of its -extension.
Proof.() It is sufficient to show that, if x is a successor of  in the generalized asynchronous dynamics of , then x is -reachable from .By definition of extended network we have, for all  ∈ , E (  , )  ( ()) =   () ≠   =   (), and  ()  is a successor of  () in the generalized asynchronous dynamics of the extended network.By Corollary 7 (ii),

𝜖 (𝑥)
is reachable from  () in the asynchronous dynamics of the extended network.Since  ()  and  (x  ) coincide on the core variables and  (x  ) is canonical,  (x  ) can be reached from  ()  .Combining the two paths we have that  (x  ) is reachable from  ().
() Consider a trap space  contained in .By Proposition 13,  ( ) is a trap space contained in  (), and  () is the minimal trap space cointaining  ().Theorem 9 then gives that  ( ) is reachable from  () in the extended network, that is, there exists  ∈  ( ) such that there is a path from  () to  in the asynchronous dynamics of the extended network.In addition, we can assume that  is canonical, that is,  (()) = .Then () is in  is -reachable from from .

Relation to single threshold refinements
Multi-valued networks are commonly used to refine the behaviour of some components of a Boolean network.They can account for some semi-quantitative knowledge, for instance by tracking different amounts of a component that are required to affect its different targets, or by encoding the existence of some specific condition leading to a higher production or a higher activity level for some target.To account for all these effects, multi-valued refinements can take many forms and involve complex modifications to the logical rules (Chaouiya et al, 2003).Here we introduce single threshold networks, a subset of multi-valued networks that adds different thresholds to the interactions but retains the same logical rules as the Boolean network.Such refinements are solely defined by a Boolean network and a mapping associating a single multi-valued threshold to each interaction of the network.We start by setting some notation and definitions.Given a Boolean network  = (,  ) with  = {1, . . ., }, we call any  :  2 → N * a threshold map for .For each  ∈ , we then define the value   and the mapping Ω  : N  → B  such that: We call ℵ =  ∈ [0,   ] the multi-valued space of (, ).For each component , we denote by e  the element of ℵ with component  equal to 1 and all other components equal to 0. In addition, we define the mapping  : B  → ℵ such that for each component  ∈ , ()  =   •   .Definition 6.Given Boolean network  = (,  ) and a threshold map  for , the function As is customary for multi-valued networks we consider dynamics that allow for asynchronous stepwise transitions that point in the direction defined by the multi-valued function.That is, we define the asynchronous dynamics of M as the graph with vertex set ℵ and edge set {(,  + e  ) |  ∈ ℵ,  ∈ Δ(, R (  , )()),  = sign(R (  , )  () −   )}.
Proposition 16.Let  be a Boolean network and  a threshold map for .If there exists a transition  → x in the asynchronous dynamics of  and there is no transition x → , then there is a trajectory from () to (x  ) in the asynchronous dynamics of the -refinement of .
The interaction graph  of a Boolean network  = (,  ) can be endowed with a label function  :  → P ({−1, 1}) that assigns signs to edges.For an edge ( , ) in  and  ∈ {−1, 1}, we have  ∈ (( , )) if there exists a state  ∈ B  such that (   (x  ) −   ())(x   −   ) = .Proposition 16 then gives the following corollary.Corollary 17.Let  = (,  ) be a Boolean network and suppose that the interaction graph of  has no loops with negative sign.If there is a path from  to  in the asynchronous dynamics, then there is a path from () to () in the asynchronous dynamics of all single threshold refinements of .
For some single threshold refinements of Boolean networks with negative loops in the interaction graph, the asynchronous dynamics can contain oscillations at intermediate levels and fail to capture the Boolean dynamics.
Proposition 18.Let  = (,  ) be a Boolean network, (ℵ, R (  , )) the single threshold refinement of  associated to a threshold map  and ( ∪ , E (  , )) the full extension of .If there is a transition  →  in the asynchronous dynamics of R (  , ), then for each state  ∈ Γ() there is a geodesic from  to at least one state  ∈ Γ() in the asynchronous dynamics of E (  , ).
If   =  then there is a geodesic from  to  that consists in updating all components of  (this is possible in any order).If   ≠  then since E (  , )  () =  there is a transition  → z , followed by a similar geodesic from z to .

Bloc-sequential
Most Permissive Synchronous Sequential Generalized Asynchronous Cuttable Extension Priorities STR Asynchronous Figure 3: Reachability properties across updating semantics.Boxes represent updating semantics and arrows between them indicate that the target semantics is an over-approximation of the source semantics.The gray area on the left groups classical deterministic semantics, while all others are non-deterministic.STR stands for single threshold refinement (Definition 6), and the blue area denotes the asynchronous semantics of all multi-valued refinements.
Corollary 19.Consider a Boolean network (,  ) and ,  Boolean states.If there exists a threshold map  such that () is reachable from () in the asynchronous dynamics of R (  , ), then  is -reachable from .
Note that in Proposition 18 and Corollary 19, we only considered the full extension.Whether the conclusions hold for any cuttable extension remains an open question.

Discussion
To reflect the lack of kinetic knowledge often associated with biological networks, the classical asynchronous semantics explores all possible alternative trajectories where a single component is updated in each transition.The generalized asynchronous semantics accounts for possible partial or total synchronism in updates.The binary nature of activity levels on the other hand implies that a change of the activity level of a single component simultaneously affects all its target components.In many networks, the effect of a component on different targets involves different mechanisms with their own kinetics and even sometimes different implicit intermediates.In case of competition (such as the inconsistent feedback loop in Fig. 1 a), the classical semantics then fail to capture some plausible behaviours.Multi-valued networks could be used to define separate thresholds for different targets, but would require either additional knowledge for all interactions or the identification of some key interactions that would benefit from a refinement.The most permissive semantics uses transitory states to address this issue and reproduce the behaviour of all multi-valued refinements, but also introduces undesired non monotonic behaviours.For example, a component in the increasing state can act in succession as inactive, then active, then inactive again for one of its targets as illustrated in Fig. 1 b).While such behaviours could be interpreted as stochastic effects in the neighbourhood of an activation threshold, they can often be considered as artefacts.Here, we focused on single threshold refinements, a small subset of multi-valued refinements that enable threshold separation while preserving the original Boolean functions (thus without introducing non monotonic behaviours).The extension of individual interactions with linear components can be used to emulate such refinements in absence of knowledge on the threshold values and within the established framework of asynchronous Boolean networks.
As a tool to study asynchronous trajectories we introduced implicant maps representing dependencies and conflicts controlling the possible change of value of the components compared to a specific initial state.These implicant maps correspond to classes of subgraphs in the implicant graph used for the identification of trap spaces (stable motifs, see Zañudo and Albert, 2013) or equivalently in the Petri net unfolding of the Boolean network (Chaouiya et al, 2011).We say that an implicant map is weakly consistent if it describes a set of satisfiable (complete and noncircular) dependencies.In absence of any weakly consistent map containing a given component, we know that there is no trajectory (in any semantics) in which the value of this component can be modified.This strong requirement is consistent with our observation that the maximal weakly consistent maps correspond to the smallest trap spaces containing the initial state.This weak consistency solely relies on dependencies and ignores the competition between components.In permissive trajectories this limitation is ignored and all components included in a weakly consistent map can be updated in a geodesic (following a partial order defined by the dependencies).However these competitions can play a role in asynchronous trajectories, where some of these components can only be updated after much longer trajectories, if ever.A weakly consistent implicant map is strongly consistent in absence of competition between its components.This stronger consistency property is both necessary and sufficient for the existence of asynchronous geodesics.
As the direct requirements and competitions described by implicant maps are associated to interactions in the regulatory graph, the consistency constraints correspond to undirected cycles in the interaction graph.We further observed that a linear component mirroring its unique regulator in the initial state can be used to relax such competitions.This led us to study the dynamical properties of cuttable networks, a structural class of Boolean networks in which a set of linear components cover all feedback loops and paths from any component with multiple targets to any component with multiple regulators.Our observations suggest that these two structural conditions correspond to different types of competitions.On one hand, the linear extension of feedback loops seems to be associated to synchronized update of multiple components, as illustrated in Fig. 1 d).It is thus required and could be sufficient to reproduce the generalized asynchronous trajectories.On the other hand, the linear extension of paths connecting a component with multiple targets to a component with multiple regulators could be related to threshold separation in feedforward loops.We observed strong similarities between the trajectories recovered through the extension of feedforward loops and in single threshold refinements as illustrated in Fig. 1 a,c).These two associations are consistent with the fact that the extended dynamics reproduces the reachability properties obtained in both the generalized asynchronous and all single threshold refinements.Further work is needed to clarify the role of feedback loops, feedforward loops, and other paths from components with multiple targets to components with multiple regulators in the dynamical properties of cuttable networks to elucidate whether the structural conditions for linear cuts could then be further generalized.
We have implemented the linear extension of Boolean networks in the bioLQM software (Naldi, 2018), enabling the use of the extended semantics in existing software tools supporting the classical asynchronous semantics.Note that efficient analysis based on trap spaces does not require this explicit extension and can be performed directly on the original Boolean networks using existing implementations of trap spaces identification in PyBoolNet (Klarner et al, 2017) or BioLQM.
As shown by Klarner et al (2014), prime implicants provide a compact and complete representation of the implicant graph enabling the identification of sets of implicants that cooperatively define a trap space as the solutions of a constraint solving problem.We plan to adapt this approach to the identification of implicant maps with the desired consistency level.The identification of strongly-consistent maps can be used as a proof of reachability in the asynchronous semantics, while the identification of weakly consistent maps can be used to pinpoint specific competitions that need to be relaxed to enable this reachability.Beyond the general question of reachability, this approach would provide valuable hints to assess the biological relevance of the corresponding extended trajectories.Note that this type of reasoning can only be used to formally validate a reachability property: if the competitions can not be realistically relaxed, then more complex trajectories to the target of interest may still exist.

Conclusion
In this paper we study the reachability properties of dynamical Boolean networks, and in particular the reachability of a subspace from a specific initial state.This question is known to be PSPACE-complete in the classical asynchronous semantics, however abstract interpretation approaches provide efficient solutions in some cases (Paulevé et al, 2012;Paulevé et al, 2020).Furthermore, this problem is polynomial for monotonic networks in the recently proposed most permissive semantics (Paulevé et al, 2020).This novel semantics extends the classical asynchronous semantics by adding intermediate activity levels explicitly accounting for the absence of information on the regulation thresholds.This approach enables the simulation of relevant behaviours missed by the standard asynchronous dynamics.The most permissive semantics can, on the other hand, also introduce some artefactual behaviours and should thus be considered as an over-approximation.This work starts with the characterisation of different structural conditions for individual transitions in asynchronous and permissive trajectories and leads to the identification of a class of Boolean networks and initial states for which these semantics have the same geodesics.These networks have a simple structural characterization: they are networks whose interaction graph admits a linear cut.We could show that trap spaces (also called stable motifs or symbolic steady states, see Zañudo and Albert, 2013;Klarner et al, 2014) always provide a precise characterization of all attractors in cuttable networks, and that their reachability solely depends on the minimal trap space containing the initial state.These results are strong improvements compared to the general case where trap spaces lack such formal guarantees, even if they are often considered as good estimators in practice.These results are similar to the properties of the most permissive dynamics but here they do not rely on intermediate activity levels that could induce known artefactual behaviours.
We then proposed an extended semantics based on linear extensions of Boolean networks.This type of extension can be interpreted as the explicit representation of hidden delays or threshold effects, and thus carries a natural biological justification.As trap spaces of the original network are also trap spaces of their extensions, the properties of cuttable networks (reachability of trap spaces and configuration of attractors) can then be applied directly to any Boolean network without explicitly constructing a cuttable extension.The reachability properties of this extended semantics provide an interesting middle ground between the asynchronous semantics and the most permissive semantics, as it recovers realistic trajectories missing in the former and excludes some artefactual behaviours of the latter (see Fig. 3).The reachability of trap spaces in the cuttable extension semantics has the same polynomial complexity as in the most permissive; however, the reachability of transient subspaces remains to be investigated.It is currently unclear if all permissive trajectories which are not captured by this new semantics are associated to non-monotonicity (and could be considered as artefacts) or if some relevant trajectories (to transient states) might also missing.Similarly, while the most permissive semantics capture all possible behaviours of multi-valued refinements, the ability of our extended semantics to reproduce behaviours emerging in multi-valued refinements has been only partially explored.We have shown that refinements that rely on a unique threshold per regulation can be captured by full extensions; however this condition does not fully characterized the emerging behaviours.
The strength of Boolean networks lies in their simple, parameter-free formulation.However, their ability to deal with lack of detailed kinetic information is also at the core of their intrinsic limitations.Although the parameter uncertainty can partially be encoded by resorting to non-deterministic semantics, many potential fine-grained behaviours that depend on specific parameter scenarios are inevitably inaccessible when relying to logical rules alone.The most permissive semantics provide an important step to ensure that all possible parameters are indeed captured, and can ) and  + (I, ) with vertex  and edge set {( , ) |  ∈ Δ(, I ())} and {( , ) |  ∈ Δ + (, I ())} respectively.For all  ∈ , define the sets − → Δ (I, , ) = {  ∈  | there is a path of length greater than zero from  to  in  (I, )}, − → Δ + (I, , ) = {  ∈  | there is a path of length greater than zero from  to  in  + (I, )}.We call − → Δ (I, , ) the set of full requirements of  and − → Δ + (I, , ) the set of strong full requirements of .An implicant map I is consistent if for each  ∈  we have − → Δ (I, , ) ⊆  \ {}.

Figure 1 :
Figure 1: Reachability properties in Boolean, refined and extended networks.Each row shows a Boolean network with its asynchronous dynamics (left), one of its multi-valued refinements (center) and linear extensions (right).White circles in the extended network denote intermediate linear variables, whereas numbered coloured circles in the refined network denote regulatory thresholds.Selected dynamical trajectories are depicted below each interaction graph.Groups of color-coded squares represent the states of all variables: white for level 0, blue for level 1 (or max), gray and red denote intermediate levels in refinements.Fixed points are marked with a dotted line on the right.The values of intermediate linear variables are represented with smaller squares on the right side of their regulators.a) An inconsistent feedforward loop: the first component has opposite (direct and undirect) effects on the last one.This competition can be relaxed by associating a higher threshold (center) or adding an intermediate component (left) to the direct interaction.b) A chain propagating an activation.In the most permissive semantics and some non-monotonic refinements (center), intermediate components can be disabled after propagating the signal.This behaviour can often be considered as an artefact and can not be reproduced in linear extensions.c) A chain with stabilizing feedback loops.This is an extension of the previous example where feedback loops are added to stabilize the unexpected (1, 0, 1) state.This state is still unreachable in the Boolean network, however it can now be reached in monotonic (single threshold) refinements and in linear extensions.d) A positive circuit showing that the reachability of the generalized asynchronous (where transitions can involve multiple components) can be reproduced in linear extensions, however it may not be faithfully reproduced in multi-valued refinements.
→ {0, 1, ★}  a strongly consistent implicant map of  for  and  + (I, ) the associated graph.Since , ) for all  ∈ , that is,  + (I, ) has no cycle.Hence  + (I, ) admits a topological ordering  1 , . . .,   .By definition, for each  ∈ {1, . . ., } the sub-ordering  1 , . . .,  −1 contains all components −1 }, we have − → Δ (I, ,   ) ⊆ { 0 , ...,  −1 }.Hence the map I is consistent.Consider I : There is no 0 ≤  ≤  ≤  such that  has an edge from   to   .Suppose, for a contradiction, that there is 1 ≤  ≤  ≤  such that  has an edge from   to   .Since  has no loop we have  < .Suppose first that   has only one regulator.Since   is in  and not in , we have    =    and since  a trap space and   is the unique regulator of   , we derive    () =    .Then   appears before   in the direction sequence of , a contradiction.So   has at least two regulators.Since  has a linear cut,   is the unique target of   , and we deduce from Lemma 8 that there is a geodesic from  to z  .Since   is in , this contradicts the minimality of . (I) , ].Then there is a component  such that z  ≠ x  =   .Since   =   we have  ∉ , thus z  ≠   =   .Since   ≠   we have  ≠ , thus x  ≠   =   .We deduce that  ∈ .Since   =   ≠   and since  is -canonical, we have   () =   () ≠   () =   =   thus  ∈ , a contradiction.This proves that z ∈ [x  , ], and since   ≠   we have  ∈ [x  , [.Since z is -canonical and reachable from , we have z ∈  and we obtain a contradiction.
and if  ∉ , then, since   ≠   , we have   () =   by (1).Since ,  ∈ Δ( ), we obtain   ( ) =   .Since   () ≠   =   , we have   ≠   and thus  appears before  in the direction sequence of .By the choice of , we have   =   and thus   () =   () =   ≠   , which contradicts our hypothesis.This proves that  has at least two regulators. to ȳ , and since ȳ is reachable from , we have ȳ ∈ .Since  has at least two regulators, this contradicts (1).By (2) there is a component  with   ≠   () =   .Then there is a transition from  to  = x .Let  be the set of  ∈  with   () ≠   =   (we have  ∉  since otherwise  has a negative loop).Let us prove that z ∈ [x  , ].Take a component  such that x  =   .We have to show that z  = x  =   .We have  ∉  by definition of , hence   =   .Since, by choice of ,   ≠   , we have  ≠ , so   = x  =   =   .Suppose that  is in , that is,  ∈  and   () ≠   .Since  is not in , we have   () =   , and  is therefore the unique regulator of .Since   =   , we have   () =   () ≠   , but then  is not -canonical, a contradiction.Hence  is not in  and z  = x  =   as wanted.This proves that z ∈ [x  , ] and thus [z  , ] ⊆ [x  , ].Hence, by hypothesis, there is no -canonical configuration in [z  , [∩.
Let  be the path from  = z to  contained in .Let  be the set of regulators  of  such that   ≠   .We have  ∉  ⊆ Δ(, ) ⊆ Δ( ).Hence, by the choice of ,  ∩ Δ( ,  ) = ∅, and since Δ( ,  ), Δ( , ) is a partition of Δ( ), we have  ⊆ Δ( , ).Hence ȳ ∈[ , ].By the definition of  and our hypothesis, we have   (ȳ  ) =   () =   ≠   = ȳ  .Since   =   , we deduce from Corollary 7 () that there is a geodesic from That is, given  such that x  =   , we have to show that x  = x  .Since, by definition of , we have  ∉ , we just need to show that  is not in  .Since  is in [x  , ], we have   =   ; as a consequence,  ∈  would imply  ∈ , a contradiction.We have Δ(,  ) \  ≠ ∅.Indeed, let  ∈ Δ(,  ).If  ∉  we are done.So suppose that  ∈  and let  be one of its regulators.Since  ,  are -canonical,