An Automata-Based View on Configurability and Uncertainty Martin - - PowerPoint PPT Presentation

An Automata-Based View on Configurability and Uncertainty

Martin Berglund1 and Ina Schaefer2

pmberglund@sun.ac.za, i.schaefer@tu-braunschweig.de

October 19, 2018

1Centre for AI Research (CSIR), Dept. of Information Science, Stellenbosch U.

• 2Inst. of Software Engineering and Automotive Informatics, TU Braunschweig

How did this come about?

A sprawling conversation: algorithms to handle configurability with unknowns and unpredictable actors, with a known configuration space

???

The process

Listen for and record instruction sequences of peer devices in network

The process

Listen for and record instruction sequences of peer devices in network Deduce probable configurations

• f peer devices

The process

Listen for and record instruction sequences of peer devices in network Deduce probable configurations

• f peer devices

Configuration space possible

Relating to the concept

• f configurability used in

software product line eng.

The process

Listen for and record instruction sequences of peer devices in network Deduce probable configurations

• f peer devices

Configuration space possible

Relating to the concept

• f configurability used in

software product line eng.

Feature selection probabilities

Novel here

The process

Listen for and record instruction sequences of peer devices in network Deduce probable configurations

• f peer devices

Attempt communication based on the features available in the deduced configuration Configuration space possible

Relating to the concept

• f configurability used in

software product line eng.

Feature selection probabilities

Novel here

Going for a formalism

Definition

An extensible automaton A is a tuple A = (B, ∆, wt) where

• B = (Q, Σ, q0, δ, F) is a DFA, the base automaton,
• ∆ ⊆ 2Q×Σ×Q are the features and,
• wt the weight function: dom(wt) = 2∆, range totally ordered

For each {δ1, · · · , δn} ⊆ ∆ A+{δ1,...,δn} = (Q, Σ, q0, δ ∪ δ1 ∪ · · · ∪ δn, F). is a realization of A. The weight of the realization is wt({δ1, . . . , δn). If A+δ1,...,δn is a DFA the realization is proper.

Going for a formalism

Definition

An extensible automaton A is a tuple A = (B, ∆, wt) where

• B = (Q, Σ, q0, δ, F) is a DFA, the base automaton,
• ∆ ⊆ 2Q×Σ×Q are the extension transition sets, and,
• wt the weight function: dom(wt) = 2∆, range totally ordered

For each {δ1, · · · , δn} ⊆ ∆ A+{δ1,...,δn} = (Q, Σ, q0, δ ∪ δ1 ∪ · · · ∪ δn, F). is a realization of A. The weight of the realization is wt({δ1, . . . , δn). If A+δ1,...,δn is a DFA the realization is proper.

Going for a formalism

Definition

An extensible automaton A is a tuple A = (B, ∆, wt) where

• B = (Q, Σ, q0, δ, F) is a DFA, the base automaton,
• ∆ ⊆ 2Q×Σ×Q are the extension transition sets, and,
• wt the weight function: dom(wt) = 2∆, range totally ordered

For each {δ1, · · · , δn} ⊆ ∆ A+{δ1,...,δn} = (Q, Σ, q0, δ ∪ δ1 ∪ · · · ∪ δn, F). is a realization of A. The weight of the realization is wt({δ1, . . . , δn). If A+δ1,...,δn is a DFA the realization is proper. Finite state? For tractability, makes sense for e.g. protocols wt may be a Boolean function (matching feature models), or costs/probabilities. Here mostly just monotonic

An example

Example feature model: the base functionality; “readX” to query the value X; can be configured with the features:

• F1. Monitor, 10% less likely, permits:

“monitorX · getX · getX · · · · getX · unmonitorX”

• F2. Logging, 50% less likely, permits: “log-readX · readX”, cannot

be combined with F1

• F3. Logging monitoring 30% less likely, permits:

“monitorX · getX · getX · · · · getX · unmonitorX” and “log-monX · monitorX · getX · getX · · · getX · unmonitorX · log-unmonX”, requires F2 Most likely for “log-mon·monitorX ·getX ·unmonitorX ·log-unmon” is F2 and F3 for 35% chance

An example extensible automaton

q1 q2 q3 q3,2 q3,1 q3,3 q0 qf readX

An example extensible automaton

q1 q2 q3 q3,2 q3,1 q3,3 q0 qf monitorX unmonitorX getX readX δF1

An example extensible automaton

q1 q2 q3 q3,2 q3,1 q3,3 q0 qf monitorX unmonitorX log-readX readX monitorX log-monX getX readX δF1 δF2

An example extensible automaton

q1 q2 q3 q3,2 q3,1 q3,3 q0 qf monitorX unmonitorX log-readX readX monitorX unmonitorX log-monX log-unmonX monitorX unmonitorX getX getX getX readX δF1 δF2 δF3

An aside: weighted automata relation

Weighted finite automata can (somewhat) describe extensible automata when

• wt describes a product in some semiring (e.g.

wt(∆′) =

δ∈∆′ wt(δ)); or;

• wt ranges over a group, preserving its identity

Then stratify the extensible automaton A into a WFA (but, exponential blowup, illustrative only for membership): A+∅ A+{δ1} A+{δ2} A+{δ3} A+{δ1,δ2} A+{δ2,δ3} A+{δ1,δ3}

wt({δ1}) wt({δ2}) wt({δ3})

wt({δ1,δ2}) wt({δ1}) wt({δ2,δ3}) wt({δ2}) wt({δ2,δ3}) wt({δ3}) wt({δ1,δ3}) wt({δ3})

Problems

Definition

An instance of the cost-constrained superset realization (CCSR) problem is a tuple (L, (A, ∆, wt), c) where

• L is a language given as a DFA,
• (A, ∆, wt) is an extensible automaton, and
• c ∈ range(wt), such that

there exists a proper realization A+∆′ of A with L ⊆ L(A+∆′) and a weight greater than or equal to c.

Problems

Definition

An instance of the cost-constrained superset realization (CCSR) problem is a tuple (L, (A, ∆, wt), c) where

• L is a language given as a DFA,
• (A, ∆, wt) is an extensible automaton, and
• c ∈ range(wt), such that

there exists a proper realization A+∆′ of A with L ⊆ L(A+∆′) and a weight greater than or equal to c.

Definition

An instance of CCSR where L is a singleton is an instance of the cost-constrained membership realization (CCMR) problem.

Hardness

Lemma

The CCRS and CCMR problems are NP-complete even for a constant wt. SAT reduction, variables x1, . . . , xn, clauses c1, . . . , cm, base automaton B: x1 x2 x3 xn c1 c2 c3 cm qf · · · · · · extensions s.t. a proper realization accepts an+m iff c1 ∧ · · · ∧ cm

• satisfiable. E.g. with literals x1 ∈ c2, x1 ∈ c4, ¬x2 ∈ c1 and xn ∈ c3

implies extensions δx1,{c2,c4}, δ¬x2,{c1} and δxn,{c3,cm} s.t. B+{δx1,{c2,c4},δ¬x2,{c1},δxn,{c3,cm}} is x1 x2 x3 xn c1 c2 c3 c4 c5 cm qf · · · · · · a a a a a a a a δx1,{c2,c4} δ¬x2,{c1} δ¬xn,{c3,cm}

With some node duplication and/or subset management taken into account.

Monotonic weights and bounded confusion depth

Unfortunate but unsurprising, the case seems somewhat unrealistic however

Definition

The extension confusion depth of an extensible automaton A = ((Q, Σ, q0, δ, F), ∆, wt) is the smallest k ∈ N such that for all α1, . . . , αn ∈ Σ (for n ∈ N) and q ∈ Q, we have |↓{∆′ ⊆ ∆ | q0

α1

− →A+∆′ · · · αn − →A+∆′ q}| ≤ k. where ↓S ⊆ S are the minimal incomparable sets of the set of sets S A mouthful, summary: bound the number of mutually exclusive realizations make some state reachable on some string More premissive than the prefix ambiguity of A+∆

CCMR w. monotonic wt and bounded confusion depth

Algorithm for CCMR with monotonic weights in terms of confusion depth:

❉❡✜♥✐t✐♦♥ ✾✳ ❚❤❡ ❡①t❡♥s✐♦♥ ❝♦♥❢✉s✐♦♥ ❞❡♣t❤ ♦❢ ❛♥ ❡①t❡♥s✐❜❧❡ ❛✉t♦♠❛t♦♥ ✇t ✐s t❤❡ s♠❛❧❧❡st s✉❝❤ t❤❛t ❢♦r ❛❧❧ ✭❢♦r ✮ ❛♥❞ ✱ ✇❡ ❤❛✈❡ ✳ ❘❡♠❛r❦ ✸✳ ❙♣❡r♥❡r✬s t❤❡♦r❡♠ ❬❙♣❡✷✽❪ ❞✐❝t❛t❡s t❤❛t t❤❡ ❡①t❡♥s✐♦♥ ❝♦♥❢✉s✐♦♥ ❞❡♣t❤ ♦❢ ❡①t❡♥s✐❜❧❡ ❛✉t♦♠❛t❛ ✐s ❜♦✉♥❞❡❞ ❜② ✭❛♥❞ t❤✐s ❜♦✉♥❞ ✐s t✐❣❤t✮✳ ❚❤❡ ❛ss✉♠♣t✐♦♥ t❤✐s s❡❝t✐♦♥ ♦♣❡r❛t❡s ✉♥❞❡r✱ ❤♦✇❡✈❡r✱ ✐s t❤❛t t❤❡ ❝♦♥❢✉s✐♦♥ ❞❡♣t❤ ✇✐❧❧ ✐♥ ❢❛❝t ❜❡ ❜♦✉♥❞❡❞ ❜② s♦♠❡ ♣♦❧②♥♦♠✐❛❧ ✐♥ ✳ ❊①❛♠♣❧❡ ✸✳ ❚❤❡ ❛✉t♦♠❛t♦♥ ✐♥ ❊①❛♠♣❧❡ ✷ ❤❛s ❡①t❡♥s✐♦♥ ❝♦♥❢✉s✐♦♥ ❞❡♣t❤ ✷✱ ❛s r❡❛❝❤✐♥❣ ♦♥ t❤❡ str✐♥❣ ♠♦♥✐t♦r ❣❡t ✉♥♠♦♥✐t♦r ❝❛♥ ❜❡ ❞♦♥❡ ✇✐t❤ ❡✐t❤❡r t❤❡ r❡❛❧✐③❛t✐♦♥

❋✶ ❋✷

♦r t❤❡ r❡❛❧✐③❛t✐♦♥

❋✸ ✱ ❜✉t ♥♦ s✉❜s❡t ♦❢ ❡✐t❤❡r✳

❋✉rt❤❡r ♥♦t❡ t❤❛t t❤❡ ❝♦♥str✉❝t✐♦♥ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ ✶ ✇✐❧❧ ♣r♦❞✉❝❡ ❛♥ ❡①t❡♥s✐❜❧❡ ❛✉t♦♠❛t♦♥ ✇✐t❤ ❝♦♥❢✉s✐♦♥ ❞❡♣t❤ ❛t ❧❡❛st ✭✇❤❡r❡ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s✱ ❛s ✐♥ t❤❡ ❝♦♥str✉❝t✐♦♥✮✱ ❛s ❧♦♥❣ ❛s ❡❛❝❤ ✈❛r✐❛❜❧❡ ♦❝❝✉rs ❜♦t❤ ♥❡❣❛t❡❞ ❛♥❞ ♥♦♥✲♥❡❣❛t❡❞ ✐♥ t❤❡ ❢♦r♠✉❧❛✳ ❚♦ s❡❡ t❤✐s✱ ♣✐❝❦ t❤❡ st❛t❡ ❛♥❞ t❤❡ str✐♥❣ ✱ t❤✐s str✐♥❣ r❡❛❝❤❡s t❤❡ st❛t❡ ❜② ♣✐❝❦✐♥❣ ❛♥② r❡❛❧✐③❛t✐♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ❡①t❡♥s✐♦♥s s❡tt✐♥❣ ❡❛❝❤ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s✱ ❢♦r ✐♥❝♦♠♣❛r❛❜❧❡ ♦♣t✐♦♥s✳ ❲✐t❤ t❤❡s❡ ❞❡✜♥✐t✐♦♥s ✐♥ ❤❛♥❞ ❆❧❣♦r✐t❤♠ ✶ s♦❧✈❡s ❈❈▼❘ ❢♦r ♠♦♥♦t♦♥✐❝ ✇❡✐❣❤t ❢✉♥❝t✐♦♥s✱ ❛♥❞ ❞♦❡s s♦ ❡✣❝✐❡♥t❧② ✐❢ t❤❡ ❝♦♥❢✉s✐♦♥ ❞❡♣t❤ ✐s ❜♦✉♥❞❡❞✳ ❆❧❣♦r✐t❤♠ ✶ ❙♦❧✈❡✲▼♦♥♦t♦♥✐❝✲❈❈▼❘

■♥♣✉t✿ ✭✐✮ ❛ str✐♥❣ α1 · · · αn❀ ✭✐✐✮ ❛♥ ❡①t❡♥s✐❜❧❡ ❛✉t♦♠❛t♦♥ A = (B, ∆, ✇t) ✇✐t❤ ❡①t❡♥✲ s✐♦♥ ❝♦♥❢✉s✐♦♥ ❞❡♣t❤ k ❛♥❞ ❛ ♠♦♥♦t♦♥✐❝ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ ✇t✱ ❧❡tt✐♥❣ B = (Q, Σ, q0, δ, F)❀ ❛♥❞❀ ✭✐✐✐✮ ❛ ♠✐♥✐♠✉♠ ✇❡✐❣❤t c✳ P❡r❢♦r♠ st❡♣s✿ ✶✳ ■♥✐t✐❛❧✐③❡ t❛❜❧❡s T, T ′ : Q → 22∆ t♦ ❜❡ ✉♥❞❡✜♥❡❞ ❡✈❡r②✇❤❡r❡✳ ✷✳ ❙❡t T(q0) := ∅✳ ✸✳ ❋♦r ❡❛❝❤ s②♠❜♦❧ α ✐♥ α1, . . . , αn✱ ✐♥ ♦r❞❡r✿ ✸✳✶ ❋♦r ❡❛❝❤ q ∈ ❞♦♠(T)✿ ✸✳✶❆ ■❢ q

α

− →B q′ s❡t T ′(q′) := T(q)✳ ✸✳✶❇ ❖t❤❡r✇✐s❡✱ ✐t❡r❛t✐✈❡❧②✱ ❢♦r ❡❛❝❤ δ′ ∈ ∆ ✇✐t❤ q

α

− →A+{δ′} q′ ❢♦r s♦♠❡ q′ ∈ Q✱ s❡t T ′(q′) := ↓(T ′(q′) ∪ {∆′ ∪ {δ′} | ∆′ ∈ T(q)) ✐❢ T ′(q′) ✐s ❞❡✜♥❡❞✱ ↓({∆′ ∪ {δ′} | ∆′ ∈ T(q)) ♦t❤❡r✇✐s❡✳ ✸✳✷ ❙❡t T := T ′ ❛♥❞ s❡t T ′ t♦ ❜❡ ✉♥❞❡✜♥❡❞ ❡✈❡r②✇❤❡r❡✳ ✹✳ ❋♦r ❡❛❝❤ qf ∈ F ❛♥❞ ❡❛❝❤ ∆′ ∈ T(qf)✿ ✹✳✶ ❝❤❡❝❦ ✐❢ A+∆′ ✐s ❛ ♣r♦♣❡r r❡❛❧✐③❛t✐♦♥✱ ✹✳✷ ✐❢ ✇t(∆′) ≥ c✱ ❛♥s✇❡r ✏tr✉❡✑✳ ✺✳ ❖t❤❡r✇✐s❡ ❛♥s✇❡r ✏❢❛❧s❡✑✳

◆❡①t t♦ ❞❡♠♦♥str❛t❡ t❤❡ ❛❧❣♦r✐t❤♠ ❝♦rr❡❝t✳ ✶✶

Runs in O(nmk2) (n input, m automaton, k confusion depth)

CCSR w. monotonic weights and bounded confusion depth

CCSR remains hard:

Theorem

CCSR is NP-complete even for extensible automata with extension confusion depth 1 and a constant wt. x1 xn . . . cm c1 . . . q0 qf x1 x3 c4 c1 · · · q4,1 q4,2 q4,3 · · · q1,3 q1,2 q1,1 a a a a a δ¬x1 δx3 δ1,3 δ4,3 a a

Unreachable parts cheating? Confusion depth 2 then.

Refining CCSR for monotonic weights

We can at least count carefully: starting with an exponential search procedure finding a proper realization with low enough weight

• Explore space of L × A+∆ building up mappings σ : L → A+∆
• Keep ensuring that the mapping of transitions under σ corresponds

to some proper realization ∆′

• Ensure the candidates do not exceed weight c

This is then successively refined by pruning the search space L × A+∆

Refining CCSR for monotonic weights

We can at least count carefully: starting with an exponential search procedure finding a proper realization with low enough weight

• Explore space of L × A+∆ building up mappings σ : L → A+∆
• Keep ensuring that the mapping of transitions under σ corresponds

to some proper realization ∆′

• Ensure the candidates do not exceed weight c

This is then successively refined by pruning the search space L × A+∆

Theorem

For a CCSR instance (L, A = (B, ∆, wt), c), letting D be the minimal DFA accepting L, evaluating Algorithm 4 using deterministic search runs in time O(nml2min(s,|∆|)) where n is the number of states in B, m the number of states in D, l the size of the input alphabet, and s is the degree of nondeterminism of b-prune(f-prune(D × A+∆)).

Conclusions

We’ve seen:

• A take on capturing an important class of questions
• A formalization approximating the questions
• Some results attempting to make the induced problems

tractable in a computationally constrained environment Left as future work:

• The formalization should be extended: transducers (with
• rigin information?) and more complex state obvious choices
• The modeling of feature models should be refined
• The specifics of the probabilistic case should be explored

Conclusions

We’ve seen:

• A take on capturing an important class of questions
• A formalization approximating the questions
• Some results attempting to make the induced problems

tractable in a computationally constrained environment Left as future work:

• The formalization should be extended: transducers (with
• rigin information?) and more complex state obvious choices
• The modeling of feature models should be refined
• The specifics of the probabilistic case should be explored

Thanks for listening!