Probabilistic ACT (PACT)

• PACT is a formal logic integrating modal, temporal, and probabilistic operators to model actions and their uncertain effects in stochastic domains.
• It uses a structured syntax with time-point, fact, and event symbols to encode concurrent actions, conditional probabilities, and causality.
• Key inference principles such as Miller’s Principle and expected-value estimation provide robust methods for belief propagation and decision-making under uncertainty.

Probabilistic ACT (PACT) is a propositional, future-branching temporal probability logic designed to formalize reasoning about actions and their probabilistic effects over time in stochastic domains. It integrates modal and probabilistic operators to allow precise statements about inevitability, possibility, and objective chance regarding facts and events, including concurrent actions and their indeterminate outcomes. The syntax, semantics, model structure, and key inference principles provide a foundational framework for formal reasoning about actions under uncertainty, as developed by Haddawy (1304.1099).

1. Language and Syntax

PACT operates over a signature comprising three main types of symbols:

• Time-point symbols (TC): Labels for time, such as "now," "noon," or numeric timestamps.
• Fact symbols (FACTS): Fluents (e.g., $f, g, h$) whose truth may persist over intervals.
• Event symbols (EVENTS): Discrete occurrences, including agent actions (e.g., $e, e'$).

Well-formed formulas are constructed using Boolean connectives ($\land$, $\vee$, $\neg$); two dyadic predicates; indexed modal and probabilistic operators; and polynomial combinations of probability terms:

• $\mathrm{HOLDS}(t_1, t_2, f)$: Fact $f$ is true throughout $[t_1, t_2]$.
• $\mathrm{OCC}(t_1, t_2, e)$: Event/action $e$ occurs during $[t_1, t_2]$.
• $\Box_{t}\varphi$: $\varphi$ is inevitable at time $t$; dualized by $\Diamond_t \varphi := \neg\Box_t\neg\varphi$.
• $P_t(\varphi) \ge p$: The probability of $\varphi$ at $t$ is at least $p$.
• Linear inequalities of the form $$\alpha_1P_t(\varphi_1) + \cdots + \alpha_kP_t(\varphi_k) + \alpha_0 \geq 0$$, permitting representation of conditional probabilities via $$P_t(\varphi|\psi) = \alpha \iff P_t(\varphi\land\psi) = \alpha P_t(\psi)$$ (when $P_t(\psi) > 0$).

Formulas systematically encode temporal, modal, and probabilistic relationships among facts and events, including constraints on concurrent execution or conditional likelihoods.

2. Model-Theoretic Semantics

A PACT model is a tuple:

$M = (W, T, FA, EV, R, PR, F)$

where:

• $W$: Nonempty set of world-histories (maximal timelines).
• $T$: Linearly ordered time-points (commonly $\mathbb{R}$).
• $FA \subseteq T \times T \times W$: Denotation-set of facts.
• $EV \subseteq T \times T \times W$: Denotation-set of events.
• $R \subseteq T \times W \times W$: Accessibility relation; $R(t,w_1,w_2)$ iff $w_1$ and $w_2$ share the same history up to $t$.
• $PR$: Assigns each $(t, w)$ a probability measure on $\{w' \in W \mid R(t,w,w')\}$ (the possible continuations from history up to $t$).
• $F$: Interpretation function mapping time symbols, facts, and events to their denotation sets.

The semantic constraints ensure:

• Temporal inheritance: If $t_1 \leq t_2$ and $R(t_2, w_1, w_2)$, then $R(t_1, w_1, w_2)$. ($C1$)
• Branch equivalence: For fixed $t$, $R(t, \cdot, \cdot)$ is an equivalence relation. ($C2$)
• Fact persistence: If $(t_1, t_4, w) \in f \in FA$ and $t_1 \leq t_2 \leq t_3 \leq t_4$ then $(t_2, t_3, w) \in f$. ($C3$)
• Branch compatibility: $w_1$ and $w_2$ in a branch are indistinguishable regarding facts/events prior to $t$. ($C4$)
• Past-certainty: For any $(t, w)$, $PR(t, w)$ is 1 on all worlds that share the history up to $t$; past facts/events are settled. ($C5$)
• Branch homogeneity: Probability measures agree within a branch. ($C6$)

Truth, $M, w \models \varphi$, is defined recursively; modal operator $\Box_t$ corresponds to truth in all accessible worlds at $t$, and probabilistic operator $P_t(\varphi) \geq p$ is interpreted via the measure $PR(t,w)$ on possible future continuations.

3. Actions, Causality, and Probabilistic Dynamics

Actions are formally region-occurrence events $\mathrm{OCC}(t_a, t_a', A)$. Planning to execute an action is rendered by $\Box_{t_a}\,\mathrm{OCC}(t_a, t_a', A)$ (agent will surely try $A$ in $[t_a, t_a']$). PACT’s semantic structure supports Suppes-style causality:

• Potential Causation: $$P_{t_A}(\mathrm{OCC}(t_E, t'_E, E)\mid \mathrm{OCC}(t_A, t'_A, A)) > P_{t_A}(\mathrm{OCC}(t_E, t'_E, E))$$
• Actual Causation: If $E$ occurs and the above holds at $t_A$.

Concurrent actions are represented by conjunction, e.g., $$\mathrm{OCC}(t_1, t_2, A) \land \mathrm{OCC}(t_1, t_2, B)$$, and the model’s branching measure determines their joint outcome probabilities.

4. Expressiveness and Example Formulas

PACT accommodates conditional, temporal, and probabilistic assertions about states and transitions, as illustrated below:

• Concurrent exclusion:

$$\neg\,\Diamond_{t_0}\left(\mathrm{OCC}(t_1, t_2, \mathit{raise}) \land \mathrm{OCC}(t_1, t_2, \mathit{lower})\right)$$

• Probabilistic success conditioned on fluent:

$$P_{t_0}\bigl(\mathrm{OCC}(t_A, t'_A, \mathit{success})\,|\,\mathrm{OCC}(t_A, t'_A, A) \land \mathrm{HOLDS}(t_A, t'_A, g)\bigr) = 0.3$$

• Deterministic failure otherwise:

$$P_{t_0}\bigl(\mathrm{OCC}(t_A, t'_A, \mathit{success})\,|\,\mathrm{OCC}(t_A, t'_A, A) \land \neg\mathrm{HOLDS}(t_A, t'_A, g)\bigr) = 0$$

• Inevitable probability at a future time:

$$P_{t_0}(\Box_{t_2}\,\mathrm{OCC}(t_2, t_3, \mathit{crash})) = 0.5$$

These constructs enable specification of contingent outcomes, dependencies, and constraints on agent behavior and environmental stochasticity.

5. Inference Principles and Theoretical Results

The key principles underpinning inference in PACT include:

A. Past-Determination:

For any formula concerning intervals at or before $t$, $P_t(\varphi) \in \{0,1\}$.

B. Certainty from Inevitability:

$\Box_t \varphi \implies P_t(\varphi) = 1$.

C. Persistence of Inevitability:

If $t_1 \le t_2$, then $\Box_{t_1}\varphi \implies \Box_{t_2}\varphi$.

D. Weight Monotonicity (Probabilistic Detachment):

Given $\Box_t(\varphi \to \psi)$ and $P_t(\varphi) \geq p$, deduce $P_t(\psi) \geq p$.

E. Miller’s Principle:

For $t_1 \leq t_2$: $$P_{t_1}(\varphi \mid P_{t_2}(\varphi) \geq a) \geq a$$.

F. Expected-Value Principle:

$$P_{t_1}(\varphi) = \mathbb{E}_{w' \sim PR(t_1, w)} [P_{t_2}(\varphi)]$$ for all $t_1 \le t_2$.

These theorems underpin formal belief-update, optimal planning, and the logical distinction between possibility, inevitability, and probability across future branches.

6. Ontology, Applications, and Significance

PACT incorporates a two-sorted temporal ontology—facts and events—structured over branching time and objective chance. The explicit dualism between $\Box_t$ and $P_t$ facilitates precise distinctions between necessary, possible, and probable action consequences. This structure supports the formal encoding of:

• Enabling conditions and conditional effectivity of actions under uncertainty.
• Planning with concurrent and mutually exclusive actions.
• Disambiguation of actual and potential causes in the presence of indeterminacy.
• Evaluation of policies or plans via expectation over future probabilities.

The inference-theoretic foundations (e.g., Miller’s Principle) enable robust strategies for forward propagation of beliefs, expected-value estimation of outcomes, and reasoning about the temporal evolution of uncertainty. This makes PACT suitable for foundational studies in AI planning, temporal logic, causality, and reasoning about stochastic domains (1304.1099).