An Efficient Formula Synthesis Method with Past Signal Temporal Logic

Abstract

In this work, we propose a novel method to find temporal properties that lead to the unexpected behaviors from labeled dataset. We express these properties in past time Signal Temporal Logic (ptSTL). First, we present a novel approach for finding parameters of a template ptSTL formula, which extends the results on monotonicity based parameter synthesis. The proposed method optimizes a given monotone criteria while bounding an error. Then, we employ the parameter synthesis method in an iterative unguided formula synthesis framework. In particular, we combine optimized formulas iteratively to describe the causes of the labeled events while bounding the error. We illustrate the proposed framework on two examples.

An automated system repair framework with signal temporal logic

“…Due to its expressivity and efficient algorithms for checking continuous signals against STL formulae, it is used in different areas including monitoring [10,11], fault detection via falsification [18] and formal control [33]. Synthe-sizing an STL formula from a dataset is studied in the literature in different forms such as finding a formula that is satisfied by all system traces to identify the system requirements [24,25], finding a formula that differentiates the given sets of good and bad signals [31], signal clustering [36] or finding a formula that would identify the bad events as they occur [10,17]. The proposed formula synthesis method as a part of the repair framework extends [17] by performing parameter synthesis in each iteration and eliminating formulae that cannot be part of the result for efficiency.…”

“…Synthe-sizing an STL formula from a dataset is studied in the literature in different forms such as finding a formula that is satisfied by all system traces to identify the system requirements [24,25], finding a formula that differentiates the given sets of good and bad signals [31], signal clustering [36] or finding a formula that would identify the bad events as they occur [10,17]. The proposed formula synthesis method as a part of the repair framework extends [17] by performing parameter synthesis in each iteration and eliminating formulae that cannot be part of the result for efficiency. In a recent work [12], STL formula synthesis is used as a part of a fault explanation framework for CPSs, where the authors synthesized a formula describing the "good" behaviors and checked it against the faulty behaviors to find a fault explanation.…”

“…Optimi ze(φ) finds the parameter valuation v that maximize T P(φ(v)) (13) while guaranteeing that F P(φ(v)) ( 14) is less than a predefined bound B. The diagonal parameter synthesis method based on monotonicity properties from [17] is used in this step. Then, starting from i = 0, Φ 0 = False and F 0 = F ; Φ i+1 and F i+1 are computed iteratively via Algorithm 2.…”

“…Definition 1 (Repaired system) Let S (16) be a control system, φ be a parametric ptSTL formula over the state and control variables of S as in (17), v be a valuation for φ, and g R (•) be a strategy as defined in (19) with respect to S and φ(v). Then, the repaired system S is defined as Proof By construction of g R (•), g R (x, x) ⊆ g(x, x) holds for each (x, x) ∈ (X × U) K × X for any K ≥ 0.…”

“…However, due to the complexity of the parameter optimization step, only 4% of the formulae from Φ ∈ {φ 1 ∨ φ 2 | φ 1 , φ 2 ∈ F 1 } are optimized over the dataset D(S) (congestion on link-1) in 10 hours on the same machine. In addition, we run the synthesis method from [17] on D(S). The computation takes 306s, the resulting formula Φ tn−c includes 18 operators, T P(Φ tn−c , D(S)) = 201 and F P(Φ tn−c , D(S)) = 101.…”