Decision and optimization problems in the unreliable-circuit logic

Abstract

The ambition constrained validity and the model witness problems in the logic UCL, proposed in [10], for reasoning about circuits with unreliable gates are analyzed. Moreover, two additional problems, motivated by the applications, are studied. One consists of finding bounds on the reliability rate of the gates that ensure that a given circuit has an intended success rate. The other consists of finding a reliability rate of the gates that maximizes the success rate of a given circuit. Sound and complete algorithms are developed for these problems and their computational complexity is studied.Keywords probabilistic logic, unreliable circuits, algorithms in logic, computational complexity. An overview of UCLThe unreliable-circuit logic UCL is an extension of propositional logic and was introduced in [10] for reasoning about logic circuits with single-fan-out unreliable gates that can produce the wrong output by fortuitous misfiring. Herein, we provide an overview of its syntax and semantics.We start by presenting a modicum of the theory of real closed ordered fields, RCOF, and of propositional logic, PL, that we need later on for presenting UCL.The first-order signature Σ rcof of RCOF contains the constants 0 and 1, the unary function symbol −, the binary function symbols + and ×, and the binary predicate symbols = and <. As usual, we may write t 1 ≤ t 2 for (t 1 < t 2 ) ∨ (t 1 = t 2 ) and t 1 t 2 for t 1 × t 2 . In the sequel, we denote by [[t]] ρ the denotation of term t over the RCOF structure based on R and the assignment ρ. We use fo for denoting satisfaction in first-order logic. In the sequel, we use extensively the fact that the theory RCOF is decidable [12].We need an enriched signature of propositional logic that we denote by Σ. Let Σ be the signature for PL containing the propositional constants tt (verum) and ff (falsum) plus the propositional connectives ¬ (negation), ∧ (conjunction), ∨ (disjunction), ⊃ (implication), ≡ (equivalence) and M 3+2k (k-ary majority) for each k ∈ N, as well as their negated-output counterparts ¬ (identity), ∧ (negated conjunction), ∨ (negated disjunction), ⊃ (negated implication), ≡ (negated equivalence) and M 3+2k (k-ary negated majority) for each k ∈ N. We denote by L(X) the set of propositional formulas over Σ and a set X of propositional variables. Given a formula ϕ ∈ L(X) and a valuation v : X → {⊥, ⊤}, we write v ϕ, for saying that v satisfies formula ϕ.We now are ready to review the unreliable-circuit logic. The signature of UCL is the triple (Σ uc , ν, µ) where:• Σ uc contains Σ and the following additional connectives used for rep-

An unexpected Boolean connective

“…These structures can for example be used to deal with paraconsistent behaviour [14], to model how a processor deals with information from multiple-sources [5], or for reasoning about computation errors [3]. The fact that ∧ ∨ can be seen as a defective ∧ or ∨ might indicate that (P)Nmatrices can be of value when reasoning about unreliable logic circuits [31,26]. Also, recently a natural interpretation of quantum states as valuations over Nmatrices was introduced [20].…”