Runtime analysis of probabilistic programs with unbounded recursion

Abstract

We study the runtime in probabilistic programs with unbounded recursion. As underlying formal model for such programs we use probabilistic pushdown automata (pPDA) which exactly correspond to recursive Markov chains. We show that every pPDA can be transformed into a stateless pPDA (called "pBPA") whose runtime and further properties are closely related to those of the original pPDA. This result substantially simplifies the analysis of runtime and other pPDA properties. We prove that for every pPDA the probability of performing a long run decreases exponentially in the length of the run, if and only if the expected runtime in the pPDA is finite. If the expectation is infinite, then the probability decreases "polynomially". We show that these bounds are asymptotically tight. Our tail bounds on the runtime are generic, i.e., applicable to any probabilistic program with unbounded recursion. An intuitive interpretation is that in pPDA the runtime is exponentially unlikely to deviate from its expected value.A pPDA consists of a finite set of control states, a finite stack alphabet, and a finite set of rules of the form pX x ֒→ qα, where p, q are control states, X is a stack symbol, α is a finite sequence of stack symbols (possibly empty), and x ∈ (0, 1] is the (rational) probability of the rule. We require that for each pX, the sum of the probabilities of all rules of the form pX x ֒→ qα is equal to 1. Each pPDA ∆ induces an infinite-state Markov chain M ∆ , where the states are configurations of the form pα (p is the current control state and α is the current stack content), and pXβ x → qαβ is a transition of M ∆ iff pX x ֒→ qα is a rule of ∆. We also stipulate that pε 1 → pε for every control state p, where ε denotes the empty stack. For example, consider the pPDA∆ with two control states p, q, two stack symbols X, Y , and the rules pX ֒

Bounded expectations: resource analysis for probabilistic programs

“…Another body of research relies probabilistic pushdown automata and martingale theory to analyze the termination time [15] and the expected number of steps [33]. The use of martingale theory to automatically analyze probabilistic programs has been pioneered in [22].…”

“…As a result, it is very hard or impossible to automate. This example demonstrates how Absynth handles Boolean values (the variable flag ) to get the tightest expected bound 15 2 ·|[0, n]|. It assigns the potential 15·|[0, flag ]| to the variable flag for paying the cost of the inner loop, in which 15 2 ·|[0, n]|− 15 2 +15·|[0, flag ]| is the loop invariant.…”

“…For each iteration of the inner loop, a pair of adjacent elements in the input array can be swapped. However, the {q:= 15 2 · |[0, n] |− 15 2 } {. ; 15 2 · |[0, n] | } w h i l e ( n > 0 ) { {n >0; 15 2 +q } f l a g = 1 ;…”

Weakest Precondition Reasoning for Expected Run–Times of Probabilistic Programs

“…His analysis can be used to prove the soundness of experimental statistical methods to determine the average run-time of probabilistic programs. Brazdil et al [4] study the run-time of probabilistic programs with unbounded recursion by considering probabilistic pushdown automata (pPDAs). They show (using martingale theory) that for every pPDA the probability of performing a long run decreases exponentially (polynomially) in the length of the run, iff the pPDA has a finite (infinite) expected runtime.…”

“…They show (using martingale theory) that for every pPDA the probability of performing a long run decreases exponentially (polynomially) in the length of the run, iff the pPDA has a finite (infinite) expected runtime. As opposed to our program verification technique, [4] considers reasoning at the operational level. Fioriti and Hermanns [8] recently proposed a typing scheme for deciding almost-sure termination.…”