BOOLEAN CONSENSUS FOR SOCIETIES OF ROBOTS

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Workshop on New frontiers of Robotics - Interdep. Research Center E. Piaggio June 2-22, 22 - Pisa (Italy) BOOLEAN CONSENSUS FOR SOCIETIES OF ROBOTS Adriano Fagiolini DIEETCAM, College of Engineering, University of Palermo Interdep. Research Center E. Piaggio, University of Pisa lunedì 2 luglio 2

WHAT IS IT? A society is a collection of individuals with different levels of autonomy, hierarchical organization and interaction. lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

WHAT IS IT? A society is a collection of individuals with different levels of autonomy, hierarchical organization and interaction. A Society of Robots (SoR) is a complex system made of: Many, Fast/Slow, Different, Autonomous, Uncoordinated, (Fairly) Competing through rules, Joining in and quitting at will, Mostly well-behaved... robots. lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

IMPOSSIBLE? Societies in nature lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

IMPOSSIBLE? Societies in nature lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

IMPOSSIBLE? Societies in nature lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

IMPOSSIBLE? Societies in nature lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

IMPOSSIBLE? Societies in nature Societies of artificial systems A modern car is much more of a robot than a simple machine! Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

IMPOSSIBLE? Societies in nature Societies of artificial systems A modern car is much more of a robot than a simple machine! Streets with cars following driving rules are indeed SoR: Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

IMPOSSIBLE? Societies in nature Societies of artificial systems A modern car is much more of a robot than a simple machine! Streets with cars following driving rules are indeed SoR: *Many, *Fast/Slow, *Different, *Autonomous, *Uncoordinated, *Competing, *Joining in and quitting at will, *Mostly well-behaved...except for few jerks Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

WHY DO WE NEED A SOR? From industrial robots to service robots,... and finally to robots everywhere. lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

WHY DO WE NEED A SOR? From industrial robots to service robots,... and finally to robots everywhere. lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

WHY DO WE NEED A SOR? From industrial robots to service robots,... and finally to robots everywhere. low reconfigurability (e.g. the end-effector can be changed) lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

WHY DO WE NEED A SOR? From industrial robots to service robots,... and finally to robots everywhere. low reconfigurability (e.g. the end-effector can be changed) almost completely specified operating conditions (closed and obstacle-free spaces) limited Robot-Robot Interaction (ad-hoc solutions with few identical copies of robots) lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

WHY DO WE NEED A SOR? From industrial robots to service robots,... and finally to robots everywhere. low reconfigurability (e.g. the end-effector can be changed) almost completely specified operating conditions (closed and obstacle-free spaces) A robot must be rethought of as a physical entity embedded in a full-fledge society, interacting through social rules, not jeopardizing others efforts. limited Robot-Robot Interaction (ad-hoc solutions with few identical copies of robots) lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

MANY CHALLENGES How do we specify social robots behavior? Models / Formal languages Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

MANY CHALLENGES How do we specify social robots behavior? Models / Formal languages Localism implies partial/uncertain knowledge -> How do we reach consistent social behavior? Agreement mechanisms for global view/information reconstruction Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

MANY CHALLENGES How do we specify social robots behavior? Models / Formal languages Localism implies partial/uncertain knowledge -> How do we reach consistent social behavior? Agreement mechanisms for global view/information reconstruction Openness implies partial behavior knowledge -> How can robots learn others dynamics and social rules? Species classification [Martini, Egerstedt, Bicchi] Logical function learning by observations Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

MANY CHALLENGES How do we specify social robots behavior? Models / Formal languages Localism implies partial/uncertain knowledge -> How do we reach consistent social behavior? Agreement mechanisms for global view/information reconstruction Openness implies partial behavior knowledge -> How can robots learn others dynamics and social rules? Species classification [Martini, Egerstedt, Bicchi] Logical function learning by observations How do robots authenticate and establish trust relations? [Dini] Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

SOCIAL ROBOTS MODELING a social robot is a hybrid system i(t k ) q i (t) Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

SOCIAL ROBOTS MODELING a social robot is a hybrid system dynamics q i (t) =f i (q i (t),u i (t)) i(t k ) q i (t) Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

SOCIAL ROBOTS MODELING a social robot is a hybrid system automaton i(t k+ )= i ( i (t k ),e i (t k )) dynamics q i (t) =f i (q i (t),u i (t)) i(t k ) q i (t) Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

SOCIAL ROBOTS MODELING a social robot is a hybrid system controller u i (t) =u i (q i (t), i(t k )) u i (t) automaton i(t k+ )= i ( i (t k ),e i (t k )) dynamics q i (t) =f i (q i (t),u i (t)) i(t k ) q i (t) Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

SOCIAL ROBOTS MODELING a social robot is a hybrid system Comm. Sensor I i awareness e i (t k )=e i (s i (q i (t),i i (t))) e i (t k ) automaton i(t k+ )= i ( i (t k ),e i (t k )) controller u i (t) =u i (q i (t), i(t k )) u i (t) dynamics q i (t) =f i (q i (t),u i (t)) i(t k ) q i (t) Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

SOCIAL ROBOTS MODELING a social robot is a hybrid system Comm. Sensor natural-like specification language I i awareness e i (t k )=e i (s i (q i (t),i i (t))) e i (t k ) automaton i(t k+ )= i ( i (t k ),e i (t k )) controller u i (t) =u i (q i (t), i(t k )) u i (t) dynamics q i (t) =f i (q i (t),u i (t)) i(t k ) q i (t) Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

SOCIAL ROBOTS MODELING a social robot is a hybrid system Comm. Sensor natural-like specification language I i awareness e i (t k )=e i (s i (q i (t),i i (t))) e i (t k ) automaton i(t k+ )= i ( i (t k ),e i (t k )) controller u i (t) =u i (q i (t), i(t k )) u i (t) dynamics q i (t) =f i (q i (t),u i (t)) makers are only required to adhere to standard spec. protocols i(t k ) q i (t) Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

SOCIAL ROBOTS MODELING a social robot is a hybrid system Comm. Sensor natural-like specification language I i awareness e i (t k )=e i (s i (q i (t),i i (t))) e i (t k ) automaton i(t k+ )= i ( i (t k ),e i (t k )) i(t k ) controller u i (t) =u i (q i (t), i(t k )) u i (t) dynamics q i (t) =f i (q i (t),u i (t)) q i (t) makers are only required to adhere to standard spec. protocols an entire SoR can be described as a computer program: complex behaviors by few keywords and a grammar composability, reusability,... Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

SOCIETY AGREEMENT Global information reconstruction by consensus algorithms lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

SOCIETY AGREEMENT Global information reconstruction by consensus algorithms In the easiest case, the quantity to agree about is real and a linear iterative strategy can be used to update a robot s state: nx x i (t + ) = a i,j (x j (t) x i (t)) j= lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

SOCIETY AGREEMENT Global information reconstruction by consensus algorithms In the easiest case, the quantity to agree about is real and a linear iterative strategy can be used to update a robot s state: nx x i (t + ) = a i,j (x j (t) x i (t)) j= Optimal sensor coverage by nonlinear iterative strategy based on Voronoi tessellation q i = k(q i C Vi (q i )) lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

SOCIETY AGREEMENT Global information reconstruction by consensus algorithms In the easiest case, the quantity to agree about is real and a linear iterative strategy can be used to update a robot s state: nx x i (t + ) = a i,j (x j (t) x i (t)) j= Optimal sensor coverage by nonlinear iterative strategy based on Voronoi tessellation q i = k(q i C Vi (q i )) lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

SOCIETY AGREEMENT Global information reconstruction by consensus algorithms In the easiest case, the quantity to agree about is real and a linear iterative strategy can be used to update a robot s state: nx x i (t + ) = a i,j (x j (t) x i (t)) j= Optimal sensor coverage by nonlinear iterative strategy based on Voronoi tessellation q i = k(q i C Vi (q i )) Nonlinear set-valued iterative strategies can solve other important problems: Clock synchronization from confidence intervals [Marzullo, CASE 9], Geographical chart reconstruction from partial/uncertain aerial snapshots,... lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

INTRUSION / ATTACK DETECTION Effective intrusion reaction requires global, consistent reconstruction of suspicious people / objects presence. Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

INTRUSION / ATTACK DETECTION Effective intrusion reaction requires global, consistent reconstruction of suspicious people / objects presence. a complex environment binary input events uj 2 B Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

INTRUSION / ATTACK DETECTION Effective intrusion reaction requires global, consistent reconstruction of suspicious people / objects presence. a complex environment binary input events guards with limited visibility and communication binary alarm states uj 2 B Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

INTRUSION / ATTACK DETECTION Effective intrusion reaction requires global, consistent reconstruction of suspicious people / objects presence. a complex environment binary input events guards with limited visibility and communication binary alarm states only neighbor-to-neighbor interaction is possible! uj 2 B Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

A GENERAL SHARED FRAMEWORK (centralized) decisions depending on binary input events 8 < : y = f (u,...,u m ), y p = f p (u,...,u m ), u j,y i 2 B = {, } lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

A GENERAL SHARED FRAMEWORK (centralized) decisions depending on binary input events partial input visibility limited communication C i,j = a binary vector state V i,j = if agent i sees uj otherwise 8 < : i can receive a message from j otherwise X i =(x i,,,x i,p ) y = f (u,...,u m ), y p = f p (u,...,u m ), u j,y i 2 B = {, } lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

A GENERAL SHARED FRAMEWORK (centralized) decisions depending on binary input events partial input visibility limited communication C i,j = a binary vector state V i,j = if agent i sees uj otherwise 8 < : i can receive a message from j otherwise X i =(x i,,,x i,p ) y = f (u,...,u m ), y p = f p (u,...,u m ), u j,y i 2 B = {, } Logical Consensus Problem: Given visibility and adjaciency matrices, V = {V i,j } and C = {C i,j }, design a distributed protocol of the form X(t + ) = F (X(t),u(t)) converging to the consensus state n f(u,,u m ), from any X(). lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

LOGICAL ITERATIONS Traditional approaches do not apply to logical maps F : B n B m! B n, but tools for the convergence of automata are of use [Robert 74-83]. lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

LOGICAL ITERATIONS Traditional approaches do not apply to logical maps F : B n B m! B n, but tools for the convergence of automata are of use [Robert 74-83]. Iterations over finite domains either converge to an equilibrium or to a cycle lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

LOGICAL ITERATIONS Traditional approaches do not apply to logical maps F : B n B m! B n, but tools for the convergence of automata are of use [Robert 74-83]. Iterations over finite domains either converge to an equilibrium or to a cycle the incidence matrix of a logical map F is a binary matrix B(F )={b i,j } (b i,j = if the j-th element of F (x) depends on the i-th element of x) x + x 2 + x 3 F (x) = @ x 3 A! B(F )=@ A lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

LOGICAL ITERATIONS - GLOBAL CONVERGENCE the eigenvalues of a binary matrix A are all the scalars 2 B s.t. 9 x 2 B n s.t. Ax= x Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

LOGICAL ITERATIONS - GLOBAL CONVERGENCE the eigenvalues of a binary matrix A are all the scalars 2 B s.t. 9 x 2 B n s.t. Ax= x A = B @ C A A B @ A B @ C A = C A = B @ C A. Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

LOGICAL ITERATIONS - GLOBAL CONVERGENCE the eigenvalues of a binary matrix A are all the scalars its spectral radius 2 B s.t. 9 x 2 B n s.t. Ax= x A = (A) B @ C A is the biggest eigenvalue A B @ A B @ C A = C A = B @ C A. (A) = $ A is similar to a strictly lower/upper matrix lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

LOGICAL ITERATIONS - GLOBAL CONVERGENCE the eigenvalues of a binary matrix A are all the scalars its spectral radius 2 B s.t. 9 x 2 B n s.t. Ax= x A = (A) B @ C A is the biggest eigenvalue A B @ A B @ C A = C A = B @ C A. (A) = $ A is similar to a strictly lower/upper matrix Theorem: A logical iterative system x + = F (x) globally converges in finite time to its unique equilibrium if (B(F )) = lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

WHEN IS CONSENSUS FEASIBLE? Can every robot be reached by every input? C = B @ C A,V j = B @ C A A 5 A 4 A A 3 A 2 Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

WHEN IS CONSENSUS FEASIBLE? Can every robot be reached by every input? augment the graph with a node for each input determine the existence of paths from each input node to every robot C = B @ A 5 A 4 C A,V j = B @ A C A A 3 A 2 u j Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

WHEN IS CONSENSUS FEASIBLE? Can every robot be reached by every input? augment the graph with a node for each input determine the existence of paths from each input node to every robot introducing a binary reachability matrix C = B @ A 5 A 4 C A,V j = B @ A C A R j = V j CV j C 2 V j C n V j span(r j )={i I j (n ) = } I j = nx C k V j = k= nx R j (:,i) k= R j = B @ A 3 A 2 C A! I j(5) = u j B @ C A span(r j )={, 2, 3, 4} consensus is unfeasible! Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

DESIGN OF LOGICAL LINEAR CONSENSUS What is the best way to propagate every input? C = B @ Modified Example C A V j = B @ C A A 5 A 4 A A 3 A 2 u j Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

DESIGN OF LOGICAL LINEAR CONSENSUS What is the best way to propagate every input? C = B @ Modified Example C A V j = B @ C A A 5 A 4 A A 3 A 2 u j Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

DESIGN OF LOGICAL LINEAR CONSENSUS What is the best way to propagate every input? C = B @ Modified Example C A V j = B @ C A A A 5 u j A 3 A 2 A 4 Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

DESIGN OF LOGICAL LINEAR CONSENSUS What is the best way to propagate every input? C = B @ Modified Example C A V j = B @ C A A A 5 u j A 3 8 >< >: A 2 A 4 x,j (t + ) = u j (t) x 2,j (t + ) = u j (t) x 3,j (t + ) = x 2,j (t) x 4,j (t + ) = x 2,j (t) x 5,j (t + ) = x,j (t) Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

DESIGN OF LOGICAL LINEAR CONSENSUS What is the best way to propagate every input? Let Cj* be the adjacency matrix of the j-th Input Propagation Spanning Tree (IPST) C = B @ Modified Example C A V j = B @ C A A A 5 Theorem: A logical linear iterative system u j A 3 x + = C j x + V j u j is (C, V j ) compliant (B(F j (x, u j )) apple (C V j )), globally converges to the consensus equilibrium n u j, is optimal in terms of time and messages exchanged. Proof: F j is globally convergent as (B(F j )) =, and n u j is an equilibrium by construction. 8 >< >: A 2 A 4 x,j (t + ) = u j (t) x 2,j (t + ) = u j (t) x 3,j (t + ) = x 2,j (t) x 4,j (t + ) = x 2,j (t) x 5,j (t + ) = x,j (t) Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

CONSENSUS WITH ROBOT FAILURE spontaneous malfunctioning robots selfishness (unfairly trying to gain resource access) malicious reprogramming lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

CONSENSUS WITH ROBOT FAILURE spontaneous malfunctioning robots selfishness (unfairly trying to gain resource access) malicious reprogramming x i (t + ) = F i (x(t),u j (t)) d i Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

CONSENSUS WITH ROBOT FAILURE spontaneous malfunctioning robots selfishness (unfairly trying to gain resource access) malicious reprogramming nominal update function x i (t + ) = F i (x(t),u j (t)) d i operating condition d i correct agent inverted agent stuck on F i (x, u j ) stuck on F i (x, u j ) binary disturbance disjunctive or Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

CONSENSUS WITH ROBOT FAILURE spontaneous malfunctioning robots selfishness (unfairly trying to gain resource access) malicious reprogramming nominal update function x i (t + ) = F i (x(t),u j (t)) d i operating condition d i correct agent inverted agent stuck on F i (x, u j ) stuck on F i (x, u j ) binary disturbance disjunctive or linear consensus rule reaches a state such as F i = C j (i, :) x + V j(i) u j with e.g. x =(u j d,u j,u j d,u j,u j d ) T 6= n u j, d 6= lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

ROBUST LOGICAL CONSENSUS bounded number of faults among every robot s neighbors ( ) lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

ROBUST LOGICAL CONSENSUS bounded number of faults among every robot s neighbors ( ) Byzantine behaviors can be tolerated with 2 + neighbors if card(s (t)) >card(s x i (t + ) = (t)), if card(s (t)) <card(s (t)), S z (t) ={h C i,h =,x h (t) =z} lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

ROBUST LOGICAL CONSENSUS bounded number of faults among every robot s neighbors ( ) Byzantine behaviors can be tolerated with 2 + neighbors if card(s (t)) >card(s x i (t + ) = (t)), if card(s (t)) <card(s (t)), S z (t) ={h C i,h =,x h (t) =z} lunedì 2 luglio 2 a binary reachability matrix and IPST with multiplicity r I (k) j (i) = R r j (C, V j)= 8 < : K k i I () j I (2) j V j (i) k =, (k ) I j (i) k>, card(ki k) <r, k>, card(ki k) r, = {h C(i, h)i(k ) j (h) =} Workshop on New frontiers of Robotics - June 2-22, 22 I (n) j,

ROBUST LOGICAL CONSENSUS: AN EXAMPLE Consider the following system with C = B @ C A,V j = B @ = C A possible failures A 5 A 3 A A 4 A 2 u j Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

ROBUST LOGICAL CONSENSUS: AN EXAMPLE Consider the following system with C = B @ C A,V j = B @ = C A possible failures A 5 A 3 A The pair (C,Vj) is 3-reachable! A 2 A 4 u j Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

ROBUST LOGICAL CONSENSUS: AN EXAMPLE Consider the following system with C = B @ C A,V j = B @ = C A possible failures A 5 A 3 A The pair (C,Vj) is 3-reachable! A 2 A 4 u j Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

NONLINEAR LOGICAL CONSENSUS CONVERGENCE Majority rule lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

NONLINEAR LOGICAL CONSENSUS CONVERGENCE Theorem: Given a (2 + ) reachable pair (C, V j ) the nonlinear logical system x(t + ) = F (x(t),u j (t)), with Fi : B n B! B (x, u j ) 7! x() = x, P H2S i Majority rule u j if V j (i) =, h2h x h if V j (i) =, is (C, V j ) compliant, and globally converges in finite time to x i = u j for all correct i. Proof: F j is globally convergent as (B(F j )) =, and n u j is an equilibrium by construction. Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

DYNAMICS OF CONTINUOUS SETS X(t + ) = F (X(t)) F : P (X ) n! P (X ) n Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

DYNAMICS OF CONTINUOUS SETS X(t + ) = F (X(t)) F : P (X ) n! P (X ) n Can we have complex behaviors, such as accumulation points, chaotic behaviors, when the domain X is infinite? lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

DYNAMICS OF CONTINUOUS SETS X(t + ) = F (X(t)) F : P (X ) n! P (X ) n Can we have complex behaviors, such as accumulation points, chaotic behaviors, when the domain X is infinite? X X 2 X 3 X 4 X 5 X 6 6 6 6 6 6 6 2 2 2 2 2 2 8 8 6 28 8 6 28 8 6 28 8 6 28 8 6 28 6 28 Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

DYNAMICS OF CONTINUOUS SETS X(t + ) = F (X(t)) F : P (X ) n! P (X ) n Can we have complex behaviors, such as accumulation points, chaotic behaviors, when the domain X is infinite? X X 2 X 3 X 4 X 5 X 6 6 2 8 X 6 X 2 6 X 3 6 2 2 2 2 2 6 6 6 6 6 6 8 8 8 8 8 2 6 28 2 6 28 2 6 28 2 6 28 2 6 28 2 6 28 8 8 8 8 8 8 6 28 6 28 6 28 6 28 6 28 6 28 X 4 6 X 5 6 X 6 Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

DYNAMICS OF CONTINUOUS SETS X(t + ) = F (X(t)) F : P (X ) n! P (X ) n Can we have complex behaviors, such as accumulation points, chaotic behaviors, when the domain X is infinite? X X 2 X 3 X 4 X 5 X 6 6 2 8 X 6 X 2 6 2 2 2 2 2 6 6 6 6 6 8 X 8 8 8 8 2 6 28 2 6 X 28 2 6 2 X 28 2 6 3 X 28 2 6 4 28 X 3 6 X 4 6 X 5 6 6 X 5 X 6 2 6 28 8 8 6 28 6 8 6 28 6 8 6 28 6 8 6 28 6 8 6 28 6 6 28 6 2 2 2 2 2 2 8 8 6 28 8 6 28 8 6 28 8 6 28 8 6 28 6 28 X 6 Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

DYNAMICS OF CONTINUOUS SETS X(t + ) = F (X(t)) F : P (X ) n! P (X ) n Can we have complex behaviors, such as accumulation points, chaotic behaviors, when the domain X is infinite? X X 2 X 3 X 4 X 5 X 6 6 2 8 X 6 6 2 2 2 2 2 6 6 6 6 6 8 X 8 8 8 8 2 6 28 2 6 X 28 2 6 2 X 28 2 6 3 X 28 2 6 4 28 8 8 6 28 6 2 X 2 8 6 28 6 X 2 X 3 6 8 6 28 6 X 2 2 X 4 6 8 6 28 6 X 3 2 X 5 6 6 X 5 8 6 28 6 X 4 2 X 6 2 6 28 6 28 6 X 5 2 X 6 X 6 8 8 628 8 628 8 628 8 628 8 628 628 2 2 2 2 2 2 8 8 6 28 8 6 28 8 6 28 8 6 28 8 6 28 6 28 Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

DYNAMICS OF CONTINUOUS SETS X(t + ) = F (X(t)) F : P (X ) n! P (X ) n Can we have complex behaviors, such as accumulation points, chaotic behaviors, when the domain X is infinite? X X 2 X 3 X 4 X 5 X 6 6 2 8 X 6 6 2 2 2 2 2 6 6 6 6 6 8 X 8 8 8 8 2 6 28 2 6 X 28 2 6 2 X 28 2 6 3 X 28 2 6 4 28 8 8 6 28 6 2 X 2 8 8 628 8 6 28 6 X 2 X X 3 8 628 6 8 6 28 6 X 2 2 X 2 Establishing consensus on set-valued data may be much more difficult X 4 8 628 6 8 6 28 6 X 3 2 X 3 X 5 8 628 6 6 X 5 8 6 28 6 X 4 2 X 4 X 6 2 6 28 6 28 6 X 5 2 8 628 X 5 X 6 X 6 628 2 2 2 2 2 2 6 6 6 6 6 6 8 8 8 8 8 8 6 2 28 6 2 28 6 2 28 6 2 28 6 2 28 6 2 28 8 8 8 8 8 8 6 28 6 28 6 28 6 28 6 28 6 28 X 6 lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

SET-VALUED BOOLEAN DYNAMICS A Boolean Algebra (BA) is ( {z} B, {z} ^, {z} _, {z} domain meet join complement, {z} null, {z} unity ) a _ (b _ c) =(a _ b) _ c, a ^ (b ^ c) =(a ^ b) ^ c (associativity) a _ b = b _ a, a ^ b = b ^ a (commutativity). Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

SET-VALUED BOOLEAN DYNAMICS A Boolean Algebra (BA) is ( {z} B, {z} ^, {z} _, {z} domain meet join complement, {z} null, {z} unity ) a _ (b _ c) =(a _ b) _ c, a ^ (b ^ c) =(a ^ b) ^ c (associativity) a _ b = b _ a, a ^ b = b ^ a (commutativity). Every BA is isomorphic to an algebra of sets (Representation Theorem, Stones 36) lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

SET-VALUED BOOLEAN DYNAMICS A Boolean Algebra (BA) is ( {z} B, {z} ^, {z} _, {z} domain meet join complement, {z} null, {z} unity ) a _ (b _ c) =(a _ b) _ c, a ^ (b ^ c) =(a ^ b) ^ c (associativity) a _ b = b _ a, a ^ b = b ^ a (commutativity). Every BA is isomorphic to an algebra of sets (Representation Theorem, Stones 36) Theorem: A Boolean dynamic system X(t + ) = F (X(t)), defined over an n dimensional vector space P (X ) n can be simulated by a suitable 2 n dimensional logical iteration system x(t + ) = f( x(t)). lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

SET-VALUED BOOLEAN DYNAMICS A Boolean Algebra (BA) is ( {z} B, {z} ^, {z} _, {z} domain meet join complement, {z} null, {z} unity ) a _ (b _ c) =(a _ b) _ c, a ^ (b ^ c) =(a ^ b) ^ c (associativity) a _ b = b _ a, a ^ b = b ^ a (commutativity). Every BA is isomorphic to an algebra of sets (Representation Theorem, Stones 36) Theorem: A Boolean dynamic system X(t + ) = F (X(t)), defined over an n dimensional vector space P (X ) n can be simulated by a suitable 2 n dimensional logical iteration system x(t + ) = f( x(t)). Implications: F may possess only equilibria or cycles convergence study of f allows us to conclude also on F Workshop on New frontiers of Robotics - June 2-22, 22 lunedì 2 luglio 2

GEOGRAPHICAL CHART ESTIMATION A large landscape environment Many robots with partial and noisy snapshots X i () = (I i (),V i ()) lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

GEOGRAPHICAL CHART ESTIMATION A large landscape environment Many robots with partial and noisy snapshots X i () = (I i (),V i ()) lunedì 2 luglio 2 A global, consistent chart must be reconstructed only by local interaction ( Ii (t + ) = S T H 2 S +(C i ) h 2 H (V h(t) \ I h (t)), V i (t + ) = S T H 2 S + (C i ) h 2 H V h(t). Under suitable visibility and connectivity conditions, the image of the landscape is the unique equilibrium. Workshop on New frontiers of Robotics - June 2-22, 22

GEOGRAPHICAL CHART: SIMULATION Evolution of some robots states t = lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

GEOGRAPHICAL CHART: SIMULATION Evolution of some robots states t = t =2 lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

GEOGRAPHICAL CHART: SIMULATION Evolution of some robots states t = t =2 t =5 lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

GEOGRAPHICAL CHART: SIMULATION Evolution of some robots states t = t =2 t = t =5 lunedì 2 luglio 2 Workshop on New frontiers of Robotics - June 2-22, 22

Workshop on New frontiers of Robotics - Interdep. Research Center E. Piaggio June 2-22, 22 - Pisa (Italy) BOOLEAN CONSENSUS FOR SOCIETIES OF ROBOTS Adriano Fagiolini DIEETCAM, College of Engineering, University of Palermo Interdep. Research Center E. Piaggio, University of Pisa lunedì 2 luglio 2

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