Axiomatization Tableaux A tableau calculus for STIT imagination logic Heinrich Wansing Ruhr University Bochum Heinrich.Wansing@rub.de Workshop Doxastic Agency and Epistemic Logic 15 - 16 December 2017 1 / 30
Axiomatization Tableaux In (Olkhovikov and Wansing 2017) we provide two different kinds of proof systems for the logic of imagination semantically introduced in (Wansing 2015), STIT imagination logic: a Hilbert-style axiomatization and a tableau calculus. The language of propositional STIT imagination logic makes use of a countably infinite set Var of propositional variables, the connectives ¬ (negation), ∧ (conjunction), and the following set of modalities: 2 / 30
Axiomatization Tableaux SA understood as ‘ A is settled true’; the dual modality is PA understood ‘ A is possible’. [ c ] a A understood as ‘agent a cstit -realizes A ’; another action modality, namely, [ d ] a A to be read ‘agent a dstit -realizes A ’, is in this setting a defined one with the following definition: [ c ] a A ∧ ¬ SA . I a A understood as ‘agent a imagines that A ’. All the agent indices are assumed to stand for pairwise different agents. 3 / 30
Axiomatization Tableaux For these modalities we assume the following ‘stit-plus-neighbourhood’ semantics defined in (Wansing 2015). An imagination model is a tuple M = � Tree , ≤ , Ag , Choice , { N a | a ∈ Ag } , V � , where: Tree is a non-empty set of moments, and ≤ is a partial order on Tree such that ∀ m 1 , m 2 ∃ m ( m ≤ m 1 ∧ m ≤ m 2 ) , and ∀ m 1 , m 2 , m (( m 1 ≤ m ∧ m 2 ≤ m ) → ( m 1 ≤ m 2 ∨ m 2 ≤ m 1 )) . The set History of all histories of M is then just a set of all maximal ≤ -chains in Tree . A history h is said to pass through a moment m iff m ∈ h . The set of all histories passing through m ∈ Tree is denoted by H m . Ag is a finite set of all agents acting in Tree and is assumed to be disjoint from all the other items in M . The set Ag is fixed for the language of STIT imagination logic and does not vary from model to model. 4 / 30
Axiomatization Tableaux Choice is a function defined on the set Tree × Ag , such that for an arbitrary ( m , a ) ∈ Tree × Ag , the value of this function, that is to say Choice ( m , a ) (more commonly denoted Choice m a ) is a partition of H m . If h ∈ H m , then Choice m a ( h ) denotes the element of Choice m a , to which h belongs. Choice is assumed to satisfy the following two restrictions: “No choice between undivided histories”: for arbitrary a , and h , h ′ ∈ H m : m ∈ Tree , a ∈ Ag , e ∈ Choice m ( h ∈ e ∧ ∃ m ′ ( m < m ′ ∧ m ′ ∈ h ∩ h ′ )) → h ′ ∈ e . “Independence of agents”. If f is a function defined on Ag such that ∀ a ∈ Ag ( f ( a ) ∈ Choice m a ), then � a ∈ Ag f ( a ) � = ∅ . 5 / 30
Axiomatization Tableaux The set of moment/history-pairs in M , that is to say, the set MH ( M ) = { ( m , h ) | m ∈ Tree , h ∈ H m } is then to be used as a set of points, where formulas are evaluated. For every a ∈ Ag , we have N a : MH ( M ) → 2 (2 MH ( M ) ) . N a is thus a neighbourhood function, defining, for every moment history pair ( m , h ) the set of propositions imagined by the agent a at the moment m in history h . V is an evaluation function for atomic sentences, i.e., V : Var → 2 MH ( M ) . 6 / 30
Axiomatization Tableaux Definition The relation of satisfaction of sentences in the above defined language at moment/history-pairs in an imagination model M is then defined inductively as follows: M , ( m , h ) � p ⇔ ( m , h ) ∈ V ( p ) , for atomic p ; M , ( m , h ) � ( A ∧ B ) ⇔ M , ( m , h ) � A and M , ( m , h ) � B ; M , ( m , h ) � ¬ A ⇔ M , ( m , h ) � � A ; M , ( m , h ) � SA ⇔ ∀ h ′ ∈ H m ( M , ( m , h ′ ) � A ); M , ( m , h ) � [ c ] a A ⇔ ∀ h ′ ∈ Choice m a ( h )( M , ( m , h ′ ) � A ); M , ( m , h ) � I a A ⇔ ∀ h ′ ∈ Choice m a ( h ) ( { ( m ′ , h ′′′ ) ∈ MH ( M ) | M , ( m ′ , h ′′′ ) � A } ∈ N a (( m , h ′ ))) and ∃ h ′′ ∈ H m ( { ( m ′ , h ′′′ ) ∈ MH ( M ) | M , ( m ′ , h ′′′ ) � A } / ∈ N a (( m , h ′′ ))) . 7 / 30
Axiomatization Tableaux If by � A � M we denote the set { ( m , h ) ∈ MH ( M ) | M , ( m , h ) � A } , i.e., the truth set of A in model M , then the above satisfaction clause for formulas I a A can more compactly be rewritten as: M , ( m , h ) � I a A ⇔ ∀ h ′ ∈ Choice m a ( h ) ( � A � M ∈ N a (( m , h ′ ))) and ∃ h ′′ ∈ H m ( � A � M / ∈ N a (( m , h ′′ ))) 8 / 30
Axiomatization Tableaux Definition A formula is said to be valid in an imagination model M iff A is satisfied by every moment/history-pair in M , and A is said to be valid (simpliciter) iff A is valid in every imagination model. The language of STIT imagination logic, as presented here, does not contain temporal operators. Therefore, it may seem natural to use the alternative, atemporal Kripke STIT semantics developed in (Balbiani et al. 2008) for single agents (and in (Lorini and Schwarzentruber 2011) for collective agents) and to extend it by neighbourhood functions. This is a legitimate move, but we refrain from making it because we view the branching-time structures augmented by agent choice functions as the intended semantics of STIT theory. 9 / 30
Axiomatization Tableaux For the semantically presented logic G. Olkhovikov obtained the following strongly sound and complete axiomatization: (A0) Propositional tautologies. (A1) S is an S 5 modality. (A2) For every a ∈ Ag , [ c ] a is an S 5 modality. (A3) SA → [ c ] a A for every a ∈ Ag . (A4) ( P [ c ] a 1 A 1 ∧ . . . ∧ P [ c ] a n A n ) → P ([ c ] a 1 A 1 ∧ . . . ∧ [ c ] a n A n ), provided that all the a 1 . . . a n are pairwise different. (A5) I a A → ([ c ] a I a A ∧ ¬ SI a A ) for every a ∈ Ag . Rules of proof: (R1) Modus ponens. (R2) From A infer SA . (R3) From A ↔ B infer I a A ↔ I a B for every a ∈ Ag . 10 / 30
Axiomatization Tableaux Note . Observe that (A4) actually is a family of schemas parametrized by n . The above axiomatization is just the axiomatization of dstit logic proposed by Ming Xu plus the axiomatization of the logic of I a as a minimal neighbourhood modal system for the smallest classical (or congruential) modal logic E plus the special axiom (A5) stating the agentive character of the imagination operator. Note also that the converse of (A5) easily follows from (A2), so that we actually have a biconditional here. 11 / 30
Axiomatization Tableaux The tableau calculus for stit imagination logic is based on the tableau calculus for dstit logic in (Wansing 2006) and the labelled tableau calculus for E in (Indrzejczak 2007). The tableau rules are utilized to process semantic information about imagination models, and we will use (i) expressions h i ✁ m a h l to indicate that the histories h i and h l are choice-equivalent for agent a at moment m , (ii) statements m k ∈ h k to express that moment m k belongs to history h k , (iii) expressions m ≺ m k to state that moment m is earlier than moment m k , and (iv) statements � A � ∈ N a (( m , h )) ( � A � �∈ N a (( m , h ))) to express that the truth set of A belongs (does not belong) to N a (( m , h )). 12 / 30
Axiomatization Tableaux Moreover, it must be ensured that a model induced by a complete tableau satisfies the independence of agents condition and that ✁ m a indeed designates an equivalence relation. To guarantee the latter properties, the following structural tableau rules are assumed. REF SYM TRAN IND h i ✁ m h i ✁ m h l 1 ✁ a 1 m ∈ h i a h k a h k m h l 1 h k ✁ m . . . h l k ✁ m ↓ ↓ a h l a k h l k h i ✁ m h k ✁ m a h i a h i ↓ ↓ h i ✁ m a h l m ≺ m n m ∈ h n , m n ∈ h n , h l 1 ✁ a 1 m h n . . . h l k ✁ a k m h n where n is a new natural number not already occurring on the tableau. 13 / 30
Axiomatization Tableaux We annotate formulas by names of moment/history-pairs. If ∆ is a set of formulas, then ∆ 0 := { A , ( m , h 0 ) | A ∈ ∆ } . A tableau is a rooted tree; its nodes are sets of certain expressions. We shall use ’ ⊢ ’ to form derivability statements (i.e., sequents) in the tableau calculus we are about to define. If ∆ ⊢ A is a sequent, then the root of the tableau for ∆ ⊢ A is ∆ 0 ∪ { m ∈ h 0 , m ≺ m 0 , m 0 ∈ h 0 } ∪ {¬ A , ( m , h 0 ) } . To expressions from this root, decomposition rules and structural tableau rules can be applied to complete the tableau. 14 / 30
Axiomatization Tableaux A tableau is said to be complete iff each of its branches is complete. A branch is complete if there is no possibility to apply one more rule to expand this branch. A tableau branch is said to be closed iff there are expressions of the form A , ( m , h ) and ¬ A , ( m , h ) on the branch or expressions of the form � A � ∈ N a (( m , h )) and � A � �∈ N a (( m , h )). A closed branch is considered complete. A tableau is called closed iff all of its branches are closed, and it is called open if it is not closed. 15 / 30
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