An analysis of the Podelski–Rybalchenko termination theorem via bar recursion

Abstract

We present an effective proof (with explicit bounds) of the Podelski and Rybalchenko Termination Theorem. The sub-recursive bounds we obtain make use of bar recursion, in the form of the product of selection functions, as this is used to interpret the Weak Ramsey Theorem for pairs. The construction can be seen as calculating a modulus of well-foundedness for a given program given moduli of well-foundedness for the disjunctively well-founded finite set of covering relations. When the input moduli are in system T , this modulus is also definable in system T by a result of Schwichtenberg on bar recursion.

Bar recursion is not computable via iteration

“…Indeed, although Kleene's S1-S9 definition looks superficially very different from Plotkin's PCF, it turns out that in a certain sense, the two formalisms express exactly the same class of algorithms for higher-order computation (see [20, Sections 6.2 and 7.1]). 2 In the same paper, Kleene also considered another notion of computability in which S9 was replaced by a weaker scheme S10 for minimization (= unbounded search), giving rise to a substructure S min ⊂ S Kl of µ-computable functionals (Kleene's terminology was µ-recursive). Whereas we can regard S9 as giving us 'general recursion', it is natural to think of S10 as giving us a particularly simple kind of 'iteration': indeed, from a modern perspective, we may say that S1-S8 + S10 corresponds to a certain typed λ calculus W str 0 with strict ground-type iteration, or equivalently to a language T str 0 + min with strict ground-type primitive recursion and minimization.…”

A Direct Proof of Schwichtenberg’s Bar Recursion Closure Theorem

“…This seems to be the price we need to pay for having a more general construction that works uniformly in G and H. Remark 3.7. Our original motivation for this work started with our bar-recursive bound [1] for the Termination Theorem by Podelski and Rybalchenko [9]. The Termination Theorem characterizes the termination of transition-based programs as a properties of well-founded relations.…”

“…H(s)(λx τ .BR(G, H, Y )(s * x)) otherwise (1) where s : τ * , G : τ * → σ, H : τ * → (τ → σ) → σ and Y : (N → τ ) → N. As usualŝ denotes the infinite extension of the finite sequence s with 0's of appropriate type. For clarify of exposition we prefer to separate the arguments that stay fixed during the recursion, namely G, H and Y , from the mutable argument s.…”

A Combinatorial Bound for a Restricted Form of the Termination Theorem

“…In [1] bounds are extracted by using Spector's bar recursion, while in [17] the termination analysis is investigated from the point of view of Reverse Mathematics.…”