On the Conditional Logic of Simulation Models

Abstract

We propose analyzing conditional reasoning by appeal to a notion of intervention on a simulation program, formalizing and subsuming a number of approaches to conditional thinking in the recent AI literature. Our main results include a series of axiomatizations, allowing comparison between this framework and existing frameworks (normality-ordering models, causal structural equation models), and a complexity result establishing NP-completeness of the satisfiability problem. Perhaps surprisingly, some of the basic logical principles common to all existing approaches are invalidated in our causal simulation approach. We suggest that this additional flexibility is important in modeling some intuitive examples.

Probabilistic Reasoning Across the Causal Hierarchy

“…As in previous work, it is easy to see that none of the languages L 1 , L 2 , L 3 is compact (Perović et al 2008;Ibeling and Icard 2018;; consequently Thm. 5 shows weak completeness only.…”

“…Then there are polynomial terms 5 It is worth observing that this derivation would go through even if we considered the weaker logic for Lfull studied in (Ibeling and Icard 2018). That is, deriving (2)∧(3)∧(4)∧(5)∧( 6) → (1) does not depend on any of the causal axioms that characterize structural causal models (Halpern 2000;Pearl 2009).…”

“…Encoding causal knowledge in an implicit way via a generative probabilistic program has been explored recently by a number of research groups (Lake et al 2017;Bingham et al 2019;Tavares et al 2019). Deterministic conditionals over "simulation programs" have been axiomatized (Ibeling and Icard 2018), showing that in general such an interpretation validates strictly fewer principles than structural causal models. This weaker axiomatic system was also embedded in a probability logic with linear inequalities (Ibeling 2018).…”

Causal Modeling with Probabilistic Simulation Models

“…We work toward the definition of a language L for expressing probabilities involving probabilistic simulation models. Probabilistic simulation models extend the non-probabilistic 1 causal simulation models of [8,6]. Formally, a non-probabilistic simulation model is a Turing machine 2 , and a probabilistic simulation model is a probabilistic Turing machine, i.e., a deterministic Turing machine (that of course still has a read-write memory tape) given read access to a random bit tape whose squares represent the results of independent fair coin flips.…”

“…It is possible to define precisely this idea of programs as causal models and to generalize the idea of intervention from SEMs to programs [8]. It is also possible to give a sound and complete logic of conditionals in this setting [6]. However, these preliminary results have not fully explored the very important case-from, e.g., the Bayesian Logic modeling language [10] and implicit in the use of probabilistic programs as cognitive models [3]-of conditionals in a probabilistic setting, via using stochastic programs as the underlying causal model.…”