Good-Enough Synthesis

Abstract

We introduce and study good-enough synthesis (gesynthesis)-a variant of synthesis in which the system is required to satisfy a given specification ψ only when it interacts with an environments for which a satisfying interaction exists. Formally, an input sequence x is hopeful if there exists some output sequence y such that the induced computation x⊗y satisfies ψ, and a system ge-realizes ψ if it generates a computation that satisfies ψ on all hopeful input sequences. ge-synthesis is particularly relevant when the notion of correctness is multi-valued (rather than Boolean), and thus we seek systems of the highest possible quality, and when synthesizing autonomous systems, which interact with unexpected environments and are often only expected to do their best. We study ge-synthesis in Boolean and multi-valued settings. In both, we suggest and solve various definitions of ge-synthesis, corresponding to different ways a designer may want to take hopefulness into account. We show that in all variants, ge-synthesis is not computationally harder than traditional synthesis, and can be implemented on top of existing tools. Our algorithms are based on careful combinations of nondeterministic and universal automata. We augment systems that ge-realize their specifications by monitors that provide satisfaction information. In the multi-valued setting, we provide both a worst-case analysis and an expectation-based one, the latter corresponding to an interaction with a stochastic environment.

A Bit of Nondeterminism Makes Pushdown Automata Expressive and Succinct

“…We take the size of P to be |Q| + |Γ |. 1 A stack content is a finite word in ⊥Γ * (i.e. the top of the stack is at the end) and a configuration c = (q, γ) of P consists of a state q ∈ Q and a stack content γ.…”

“…The following problem is ExpTime-complete: Given a VPA P, is P GFG? Finally, we relate the GFGness problem to the good-enough synthesis problem [1], also known as the uniformization problem [9], which is similar to the Church synthesis problem, except that the system is only required to satisfy the specification on inputs in the projection of the specification on the first component.…”

“…In contrast, solving ω-VPA games is 2ExpTime-complete [27]. We also relate the problem of checking GFGness with the good-enough synthesis [1] or uniformization problem [9], which we show to be ExpTime-complete for DVPA and GFG-PDA.…”

History Determinism vs. Good for Gameness in Quantitative Automata

“….. More formally, given the automaton A = (Σ, Q, ι, δ) and a word w ∈ Σ * (resp. w ∈ Σ ω ), we define the arena G(A, w) with positions , w, δ(ι, w[0])), ε-labelled edges from (q, u, b) to (q, u, b ) when b is an immediate subformula of b, and x-labelled edges from (q, u, (x, q )) to (q , u[1..], δ(q , u [1])). Conjunctive positions belong to Adam while disjunctive ones belong to Eve.…”

“…Observe that a Val-automaton can be good for games, history deterministic, or determinizable by pruning when interpreted on infinite words, but not when interpreted on finite words, as demonstrated in Figure 1 . 1 2 over a unary alphabet that is determinizable by pruning, good for games, and history deterministic with respect to infinite words, but none of them with respect to finite words: For the single infinite word, the initial choice of going from q0 to q1 provides the optimal value of 1, making it all of the above. On finite words, on the other hand, it is not even threshold history deterministic (and by Theorem 4 neither of the rest), since in order to guarantee a value of at least 1, the first transition should be different for the word of length 1 and the word of length 2, going to q3 for the former and to q1 for the latter.…”