Propositional Dynamic Logic for Planning

This paper presents an on going work on Propositional Dynamic Logic PDL in which atomic programs are STRIPS actions. We think that this new framework is appropriate to reasoning about actions and plans when dealing with planning problem. Unlike PDL atomic programs, STRIPS actions have pre-conditions and post-conditions. We propose a novel operator of action composition that takes into account the features of STRIPS actions. We propose an axiomatization and prove its soundness. Completeness, decidability and computational complexity are left as future work. Resumo. Este artigo apresenta um trabalho em andamento sobre Lógica Dinâmica Proposicional PDL em que programas atômicos são ações do STRIPS. Pensamos que esse novo arcaboço é apropriado para raciocinar sobre ações e planos ao lidar com problemas de planejamento. Ao contrário dos programas atômicos PDL, as ações do STRIPS têm pré-condições e pós-condições. Propomos um novo operador de composição de ações que leva em consideração os recursos das ações do STRIPS. Propomos uma axiomatização e provamos sua correção. Completude, decidibilidade e complexidade computacional são deixadas como trabalho futuro. 1. Background This section presents a brief overview of two topics on which the later development is based on. First, we make a brief review of the syntax and semantics of [Harel et al. 2000]. Second, we present the classical planning problem. Finally, we present a brief introduction to STRIPS. 1.1. Propositional Dynamic Logic In this section, we present the syntax and semantics of the most used dynamic logic called PDL for regular programs. Definition 1.1 The PDL language consists of a set Φ of countably many proposition symbols, a set Π of countably many basic programs, the boolean connectives ¬ and ∧, the program constructors ; (sequential composition), ∪ (non-deterministic choice) and ? (iteration) and a modality 〈π〉 for every program π. The formulas are defined as follows: φ ::= p | > | ¬φ | φ1 ∧ φ2 | 〈π〉φ, with π ::= a | π1; π2 | π1 ∪ π2 | π | φ? where p ∈ Φ and a ∈ Π. In all the logics that appear in this paper, we use the standard abbreviations ⊥ ≡ ¬>, φ ∨ φ ≡ ¬(¬φ ∧ ¬φ), φ→ φ ≡ ¬(φ ∧ ¬φ) and [π]φ ≡ ¬〈π〉¬φ. Each program π corresponds to a modality 〈π〉, where a formula 〈π〉φ means that there is a run of π after which φ is true, considering that π halts. There is also the possibility of using [π]φ (as an abbreviation for ¬〈π〉¬φ) indicating that the property denoted by φ holds after every possible run of π. The semantics of PDL is normally given using a transition diagram, which consists of a set of states and binary relations (one for each program) indicating the possible execution of each program at each state. In PDL literature a transition diagram is called a frame. Definition 1.2 A frame for PDL is a tuple F = 〈W,Rπ〉 where • W is a non-empty set of states; • Ra is a binary relation over W , for each basic program a ∈ Π; • We can inductively define a binary relation Rπ, for each non-basic program π, as follows – Rπ1;π2 = Rπ1 ◦Rπ2 , – Rπ1∪π2 = Rπ1 ∪Rπ2 , – Rφ? = {(w,w) | M, w φ}, – Rπ? = R π, where R π denotes the reflexive transitive closure of Rπ. Definition 1.3 A model for PDL is a pairM = 〈F ,V〉, where F is a PDL frame and V is a valuation function V : Φ→ 2 . The semantical notion of satisfaction for PDL is defined as follows: Definition 1.4 LetM = 〈F ,V〉 be a model. The notion of satisfaction of a formula φ in a modelM at a state w, notationM, w φ, can be inductively defined as follows: • M, w p iff w ∈ V(p); • M, w > always; • M, w ¬φ iffM, w 6 φ; • M, w φ1 ∧ φ2 iffM, w φ1 andM, w φ2; • M, w 〈π〉φ iff there is w′ ∈ W such that wRπw andM, w′ φ. For more details on PDL see [Harel et al. 2000]. 1.2. Planning Problem The planning problem consists of an automated process to check if a goal is achievable, given the starting state and the defined actions. With the information acquired in the process, it’s possible to construct the possibility graph [Luger 2008, Russell and Norvig 2003]. This is interesting for the field of artificial intelligence because it results in a tree of possibilities given a set of actions and an initial state, with which one can find a path to reach the intended state. This can be applied in robot and games programming, for performance analysis, and in some decision making processes. Definition 1.5 The classic planning problem is a tuple 〈S,Ac, s0, SG〉, where S is the set of all possible states, Ac a set of actions, s0 ∈ S is the initial state, and SG ⊆ S is a set containing the possible goals.