False-belief tasks and their formalisation Thomas Bolander, DTU Compute, Technical University of Denmark IRIT, Toulouse, 30 June 2014 Thomas Bolander, False-belief tasks, 30 June 2014 – p. 1/20
Social Intelligence and Theory of Mind Theory of Mind ( ToM ): The ability of attributing mental states—beliefs, intentions, desires, etc.—to other agents. Having a ToM is essential for successful social interaction in humans [Baron-Cohen, 1997] . The presence of a ToM in children is often tested through false-belief tasks , e.g. the Sally-Anne test . Thomas Bolander, False-belief tasks, 30 June 2014 – p. 2/20
Goal of the present work Goal : To formalise the Sally-Anne task in a suitable variant of dynamic epistemic logic. Thomas Bolander, False-belief tasks, 30 June 2014 – p. 3/20
Goal of the present work Goal : To formalise the Sally-Anne task in a suitable variant of dynamic epistemic logic. But why? Thomas Bolander, False-belief tasks, 30 June 2014 – p. 3/20
Goal of the present work Goal : To formalise the Sally-Anne task in a suitable variant of dynamic epistemic logic. But why? Three uses of logical formalisations of dynamical epistemic reasoning: 1. Specification , analysis and verification of agent systems (e.g. computer systems or security protocols). 2. Basis for reasoning engine of autonomous agents. 3. Providing formal models of human reasoning. My focus is on 2. My ultimate aim is to construct planning agents (e.g. robots) with ToM capabilities. Trying to find out what it takes for a computer or robot to pass the Sally-Anne test is a good test case for this research aim. Thomas Bolander, False-belief tasks, 30 June 2014 – p. 3/20
Comparison of false-belief task agents The Sally-Anne task requires second-order reasoning (the agent believes that Sally believes the cube to be in the large container). Some false-belief tasks require n -th order reasoning for n > 2. platform h-o other features reas. CRIBB Prolog ≤ 3 goal recognition, plan recognition [Wahl and Spada, 2000] Edd Hifeng event calc. ≤ 2 Second Life avatar [Arkoudas and Bringsjord, 2008] Leonardo C5 agent ≤ 2 goal recognition, arch. learning [Breazeal et al. , 2011] Epistemic planning DEL ∞ planning [Bolander and Andersen, 2011] ACT-R agent ACT-R ∞ learning [Arslan et al. , 2013] cogn. arch. Only [Bolander and Andersen, 2011] supports planning (but several of the other formalisms could possibly be extended to support it). Thomas Bolander, False-belief tasks, 30 June 2014 – p. 4/20
Dynamic Epistemic Logic (DEL) by example We use the event models of DEL [Baltag et al. , 1998] with added postconditions (ontic actions) as in [Ditmarsch et al. , 2008].
Dynamic Epistemic Logic (DEL) by example We use the event models of DEL [Baltag et al. , 1998] with added postconditions (ontic actions) as in [Ditmarsch et al. , 2008]. Example . The secret turn of a coin: i , u black epistemic model • Epistemic models : Multi-agent K models. Elements of domain called worlds . Actual world is colored green ( ). Thomas Bolander, False-belief tasks, 30 June 2014 – p. 5/20
Dynamic Epistemic Logic (DEL) by example We use the event models of DEL [Baltag et al. , 1998] with added postconditions (ontic actions) as in [Ditmarsch et al. , 2008]. Example . The secret turn of a coin: precond. postcond. event i , u � black , ¬ black � �⊤ , ⊤� black u i i , u epistemic model event model • Epistemic models : Multi-agent K models. Elements of domain called worlds . Actual world is colored green ( ). • Event model : Represent the action of secretly turning the coin. Thomas Bolander, False-belief tasks, 30 June 2014 – p. 5/20
Dynamic Epistemic Logic (DEL) by example We use the event models of DEL [Baltag et al. , 1998] with added postconditions (ontic actions) as in [Ditmarsch et al. , 2008]. Example . The secret turn of a coin: precond. postcond. event i , u � black , ¬ black � �⊤ , ⊤� ¬ black black u = ⊗ black u i i , u i i , u epistemic model epistemic model event model product update • Epistemic models : Multi-agent K models. Elements of domain called worlds . Actual world is colored green ( ). • Event model : Represent the action of secretly turning the coin. • Product update : The updated model represents the situation after the action has taken place. Thomas Bolander, False-belief tasks, 30 June 2014 – p. 5/20
Our version of Sally-Anne The Sally-Anne test exists in many variants [Wellman et al. , 2001] . We use the version where the observer (child) is asked: “Where does Sally think the cube is?”. Thomas Bolander, False-belief tasks, 30 June 2014 – p. 6/20
Our version of Sally-Anne The Sally-Anne test exists in many variants [Wellman et al. , 2001] . We use the version where the observer (child) is asked: “Where does Sally think the cube is?”. We will interpret this as meaning: “Where does Sally believe the cube to be?” Thomas Bolander, False-belief tasks, 30 June 2014 – p. 6/20
Constants of modelling language In the following we will use the following agent symbols: • O : The Observer (the child/agent taking the Sally-Anne test). • S : Sally. • A : Anne. We will use the following propositional symbols: • large : The cube is in the large container. • small : The cube is in the small container. • sally : Sally is present in the room with Anne and the observer. In epistemic models, we will use green nodes ( ) to denote the actual world. Thomas Bolander, False-belief tasks, 30 June 2014 – p. 7/20
Modelling Sally-Anne in DEL 1. Sally has placed cube in large container: O , S , A s 1 = large , sally A S O A S O Thomas Bolander, False-belief tasks, 30 June 2014 – p. 8/20
Modelling Sally-Anne in DEL 2. Sally leaves room: O , S , A s 1 = large , sally O , S , A a 2 = �⊤ , ¬ sally � A S O A O S Thomas Bolander, False-belief tasks, 30 June 2014 – p. 8/20
Modelling Sally-Anne in DEL 2. Sally leaves room: O , S , A s 1 = large , sally O , S , A a 2 = �⊤ , ¬ sally � O , S , A s 2 = s 1 ⊗ a 2 = large A S O A O S Thomas Bolander, False-belief tasks, 30 June 2014 – p. 8/20
Modelling Sally-Anne in DEL 3. Anne transfers cube to small container: O , S , A s 1 = large , sally O , S , A a 2 = �⊤ , ¬ sally � O , S , A s 2 = s 1 ⊗ a 2 = large A S O , A O , S , A O S a 3 = �⊤ , ¬ large ∧ small � �⊤ , ⊤� A O S Thomas Bolander, False-belief tasks, 30 June 2014 – p. 8/20
Modelling Sally-Anne in DEL 3. Anne transfers cube to small container: O , S , A s 1 = large , sally O , S , A a 2 = �⊤ , ¬ sally � O , S , A s 2 = s 1 ⊗ a 2 = large A S O , A O , S , A O S a 3 = �⊤ , ¬ large ∧ small � �⊤ , ⊤� O , A O , S , A S s 3 = s 2 ⊗ a 3 = small large A O S Thomas Bolander, False-belief tasks, 30 June 2014 – p. 8/20
Modelling Sally-Anne in DEL 4. Sally re-enters: O , S , A s 1 = large , sally O , S , A a 2 = �⊤ , ¬ sally � O , S , A s 2 = s 1 ⊗ a 2 = large A S O , A O , S , A O S a 3 = �⊤ , ¬ large ∧ small � �⊤ , ⊤� O , A O , S , A S s 3 = s 2 ⊗ a 3 = small large O , S , A a 4 = �⊤ , sally � A S O Thomas Bolander, False-belief tasks, 30 June 2014 – p. 8/20
Modelling Sally-Anne in DEL 4. Sally re-enters: O , S , A s 1 = large , sally O , S , A a 2 = �⊤ , ¬ sally � O , S , A s 2 = s 1 ⊗ a 2 = large A S O , A O , S , A O S a 3 = �⊤ , ¬ large ∧ small � �⊤ , ⊤� O , A O , S , A S s 3 = s 2 ⊗ a 3 = small large O , S , A a 4 = �⊤ , sally � s 4 = s 3 ⊗ a 4 = A S O O , A O , S , A S small , sally large , sally Thomas Bolander, False-belief tasks, 30 June 2014 – p. 8/20
Modelling Sally-Anne in DEL O , S , A 1. Sally has placed cube in large container: s 1 = large , sally O , S , A 2. Sally leaves the room: a 2 = �⊤ , ¬ sally � O , A O , S , A S 3. Anne transfers cube: a 3 = �⊤ , ¬ large ∧ small � �⊤ , ⊤� O , S , A 4. Sally re-enters: a 4 = �⊤ , sally � O , A O , S , A S s 4 = s 1 ⊗ a 2 ⊗ a 3 ⊗ a 4 = small , sally large , sally We have: s 4 | = B O B S large Thus the observer will answer the question “where does Sally believe the cube is” with “in the large container”, hence passing the Sally-Anne test! Thomas Bolander, False-belief tasks, 30 June 2014 – p. 9/20
Modelling Sally-Anne in DEL O , S , A 1. Sally has placed cube in large container: s 1 = large , sally O , S , A 2. Sally leaves the room: a 2 = �⊤ , ¬ sally � O , A O , S , A S 3. Anne transfers cube: a 3 = �⊤ , ¬ large ∧ small � �⊤ , ⊤� O , S , A 4. Sally re-enters: a 4 = �⊤ , sally � O , A O , S , A S s 4 = s 1 ⊗ a 2 ⊗ a 3 ⊗ a 4 = small , sally large , sally We have: s 4 | = B O B S large Thus the observer will answer the question “where does Sally believe the cube is” with “in the large container”, hence passing the Sally-Anne test! Now note that s 1 ⊗ a 3 = s 4 . Thus Sally leaving and re-entering doesn’t have any effect on the model the agent ends up with! Something is not right!... Thomas Bolander, False-belief tasks, 30 June 2014 – p. 9/20
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