# Factored Models for Probabilistic Modal Logic

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2 2007 provides reasoning ethods for general probabilistic odal structures but does not scale up to large doains. Another related work is (Milch & Koller 2000 in which probabilistic episteic logic is used to reason about the ental states of an agent. This logic is a special case of probabilistic knowledge with the additional assuption of agents having a coon prior probability distribution over states. Adding probabilistic notions to odal logic is also considered in (Herzig 2003; Nie The forer adds a unary odal operator expressing that a proposition is ore probable than its negation, whereas the latter defines an extension of fuzzy odal logic to perfor probabilistic seanticbased approaches for finding docuents relevant to a query. Related works in the gae theory literature ainly focus on iperfect inforation gaes. For exaple, (Koller & Pfeffer 1995 provides an algorith for finding optial randoized strategies in two-player iperfect inforation copetitive gaes. The state of these gaes can be represented with our odel. Factored Probabilistic Modal Structures In this section, we provide a copact Bayesian Network (BN representation for probabilistic Kripke structures. A BN is a directed acyclic graph in which nodes represent rando variables, and the joint distribution of the node values can be written as P r(x 1,..., X n = Π n i=1 P r(x i parents(x i. According to (Shirazi & Air 2007, a probabilistic Kripke structure consists of a set of states and a probabilistic accessibility relation (we redefine it in the next section. The probabilistic accessibility relation is defined for each pair of states. The probabilistic Kripke structure is basically a graph of states with labeled edges. Therefore, the size of this structure is quadratic in the nuber of states and it is not scalable to large doains. In the following section we review the probabilistic Kripke structures. Then, in Section we provide coplete specifications of our representation. Probabilistic Kripke Structures For siplicity we assue that the agent wishes to reason about a world that can be described in ters of a nonepty set, Z, of state variables. Probabilistic odal forulas are built up fro a countable set of state variables Z using equality (=, propositional connectives (,, and the odal function K. We use and for truth and falsehood constants. First, we need to define non-odal forulas. The foration rules are: 1. For every state variable X and value x, X = x is a nonodal forula. 2. and are non-odal forulas. 3. If X is a non-odal forula so is X. 4. If X and Y are non-odal forulas so is X Y. 5. Every non-odal forula is a probabilistic odal forula. 6. If X is a probabilistic odal forula so is (K(X r when 0 < r 1 and {<, =}. Note that we have a different odal function K i for each agent i in the doain. We take, and to be abbreviations in the usual way. We use Texas Holde poker gae as an exaple of a gae with iperfect knowledge that can be odeled using our fraework. In Holde, players receive two downcards as their personal hand, after which there is a round of betting. Three boardcards are turned siultaneously (called the flop and another round of betting occurs. The next two boardcards are turned one at a tie, with a round of betting after each card. A player ay use any five-card cobination fro the board and personal cards. There are soe rules that applied to the cards to rank hands. For ore info refer to In the Holde exaple, suppose that we introduce two new propositional sybols, w 1 and w 2, to show whether player 1 or player 2 wins the hand, respectively. The value of these sybols is deterined based on the gae rules applied to players hands and boardcards (players hands and boardcards are state variables. In this exaple there are two players, therefore we have two odal functions, K 1 and K 2 corresponding to player 1 and player 2. K 1 (w 1 < 1/2 is an exaple of a probabilistic odal forula whose truth value can be evaluated on the current state of the world. K 1 (w 1 < 1/2 deonstrates that the probability of player 1 winning the hand is less than 1/2 fro her perspective. Now, we describe the seantics of our language. Our approach is in ters of possible worlds which is siilar to Kripke structures in odal logic. Definition 0.1 A probabilistic Kripke structure M is a tuple (S, P, V in which 1. S is a nonepty set of states or possible worlds. 2. P is a conditional probability function. P(s s denotes the probability of accessing state s given that we are in state s. P(s s > 0 indicates that s is accessible fro s. Therefore, it is siilar to the accessibility relation of odal logic. Since P is a probability function, we should ensure that the following constraints hold: 0 P(s s 1 For each state s S: s S P(s s = 1 3. V is an interpretation that associates with each state in S a value assignent to the state variables in Z. Probabilistic knowledge is a special case of probabilistic odal logic in which the accessibility relation is an equivalence relation. This odel captures the intuition that agent i considers state t accessible fro state s if in both s and t she has the sae knowledge about the world. The above definition is for one odal function. When we have j odal functions (K 1,... K j we need a probability function P i for each odal function K i. Intuitively, the probability function P i represents the probability of accessing a state fro the perspective of agent i. In probabilistic odal logic, the true state of the world is a state in S. An agent has a probability distribution over all the states that are possible to her given the true state of the world. Exaple. Let (KJ, K3, KKQ32 be the cards of players and the boardcards. Since player 1 does not know her opponent s hand, P 1((KJ, 65, KKQ32 (KJ, K3, KKQ32 should be > 0. P 1 is unifor on all the possible states since the player does not have any inforation about her opponents hand. 542

3 Figure 2: Left: GKM of War. Right: Kripke structure of War. Figure 1: GKM of Holde. The truth value of any forula in a given state is evaluated with the following definitions. Definition 0.2 Let M = (S, P, V be a probabilistic Kripke structure. Let s be a state in S. For any probabilistic odal forula ϕ, we define val(s, ϕ recursively as follows. 1. val(s, X = x = (V(s, X = x for a state variable X. 2. val(s, = true. 3. val(s, = false. 4. val(s, X = val(s, X. 5. val(s, X Y = val(s, X val(s, Y. 6. val(s, K(X r = true iff P(s s r. s S,val(s,X=true We use val s definition to define logical entailent, =. Definition 0.3 M, s = X iff val(s, X is true. Graphical Kripke Models A probabilistic Kripke structure as defined above, (S, P, V, has size O( S 2. This representation is ipractical for large state spaces. In this section we provide a ore copact representation for probabilistic Kripke structures. In our new odel, a state is represented by a set of state variables, Z. P is represented by a BN with 2 Z variables: Z(i a and Zh (i for each Z (i Z. Z(i a stands for a state variable for the actual state of the world, whereas Z(i h represents a variable for a hypothetical state of the world (an agent cannot distinguish with certainty between this state and the actual state. P r(z h Z a is represented by the BN which serves as P of probabilistic Kripke structures (e.g., see Figure 1. Definition 0.4 A graphical Kripke odel (GKM M on a set of rando variables Z is a BN which is defined as follows: 1. The nodes are: Z a (1,..., Za ( Z, Zh (1,..., Zh ( Z 2. P r(z h Z a is defined by the edges of M. 3. There are no edges between the nodes in Z a. The above definition is for one odal function, K. For cases with j odal functions, K 1,... K j, we need to define Z hi and P r(z hi Z a for each odal function K i. Figure 1 shows the GKM of our Holde exaple. The nodes in the first row represent the actual state of the world, whereas the second row represents a possible state of the world. Each node takes values fro {A,2,...,10,J,Q,K} {,,, }. The first and second nodes are observed by player 1 to have values K and J, respectively. In each row, the first two nodes correspond to player 1 s hand, the second two nodes correspond to player 2 s hand, and the last five are the boardcards. Fro the perspective of player 1, player 2 can have any cards except the boardcards and the cards in her hand. In the BN, this is shown by the edges to the third and forth node in Z h. The boardcards and player 1 s hand cards are the sae in the actual state of the world and the hypothetical state of the world. Let Z 1,..., Z 9 stand for the nodes in each row (Z 1 be the leftost node, the conditional probability functions are: ( P r(zi h Zi a 1 if Zi h = Zi a ; = for i {1, 2, 5,..., 9} 0 otherwise. P r(z h 3 Z a 1, Z a 2, Z a 5,..., Z a 9 = 1 α ( 1 if Z h 3 {Z a 1, Z a 2, Z a 5,..., Z a 9 }; 0 otherwise. As shown in the above equation, Z h 3 has a unifor distribution. α is the noralization factor. The conditional probability function for Z h 4 is the sae as Z h 3 except that Z h 4 is a child of Z h 3 and should not be equal to Z h 3 as well. Theore 0.5 Let Z be the set of state variables and k be the nuber of agents. GKM has O(k Z nodes and O(k Z 2 edges and each node has at ost 2 Z 1 parents. Note that this odel is ost useful when the size of the largest Conditional Probability Table (CPT is uch saller than S or when the CPTs can be represented copactly (e.g., unifor distribution. In those cases, the size of GKM is uch saller than the size of the corresponding probabilistic Kripke structure (O(2 Z nodes when state variables are binary. Exaple. We define a sipler 2-player gae, naed War, for the purpose of exposition with a saller set of states. In War, there is a deck of three cards, A, B, and C. One card is dealt to each player and the third card is face down. The player with the highest card wins the gae (A > B > C. The Kripke odel has six states, and so can be easily analyzed. The equivalence classes for player 1 are shown in the right part of Figure 2. The first rounded rectangle corresponds to the class in which player 1 has A and player 2 either has B or C (the actual state of the world is either AB or AC. In this equivalence class, player 1 knows that she is winning with probability 1. Player 1 has a probability distribution over each of these equivalence classes. The GKM representing the equivalence classes of player 1 is shown in the left part of Figure 2. Z a represent the actual 543

5 In the following sections we provide efficient algoriths for queries with inequalities (probabilistic odal queries. Ordered Variable Eliination In this section we provide an algorith to answer probabilistic odal queries. In the previous section we entioned that existing BN inference ethods cannot answer these queries. The algorith that we introduce is called Ordered Variable Eliination (OVE. The following exaple justifies that the original variable eliination (see (Pearl 1988 does not answer the following query because soe of the suations participate in inequalities. Therefore, the order of soe of the suations cannot be changed. Exaple. Assue the query K 1 (K 2 (x < 1 2 < 3 4 on s. This query is calculated as follows: K 1(K 2(x < 1 2 < 3 4 on s X = P r(z h 1 Z a = s K 2(x < 1 «2 on Zh 1 < 3 4 Z h 1 X = P r(z h 1 Z a = s Z h 1 `X P r(x Z h 2 P r(z h 2 Z h 1 < 1 «< Z h 2 In this forula we cannot ove Z h 1 inside Z h 2, since the latter participates in an inequality. OVE perfors variable eliination on this forula in two rounds. It eliinates variables Z h2 in the first round and variables Z h1 in the second round. Assue that Z which is a subset of Z h1 is the set of parents of Z h2 in the BN calculated by Function Q2BN of the previous section. After the first round of variable eliination ( Z h 2 P r(x Z h2 P r(z h2 Z h1 < 1 2 is replaced by f(z < 1 2. The result is a suation over Z h 1 which is coputed in the second round of variable eliination. The algorith is shown in figure 5. There are a few standard ways to speed up this function. For exaple, instead of suing over all Zs we can su over those in which P r(z Z is not zero. This will provide a faster approach when P r(z Z is sparse. Theore 0.6 Let q be a query, s be a state, v be the axiu size of the doain of rando variables, and t be the size of the largest factor. Function OVE calculates the value of q on s in O(v t tie. Eliination is deriven by an ordering on variables. OVE does not allow all the orders. Therefore, for soe graphs its running tie is worse than the variable eliination s. Typically, v t << S. However, the worst-case running tie of this algorith is the sae as the running tie of GBU in (Shirazi & Air 2007 which is the fastest exact ethod in that paper. Sapling with Confidence In this section we provide a sapling ethod to answer queries on GKMs. Our ethod is based on probabilistic logic sapling of (Henrion 1986 which is the siplest and first proposed sapling algorith for BNs. This ethod is optial for our query answering because our evidence nodes FUNCTION OVE(query q, state s q: the query of the for K (1 (... (K ( (x < n (... < n (1 in which K (i {K 1,..., K j} for j agents a s: list of actual-state values Z a : list of actual-state variables Y : query node b Z (i : list of hypothetical-state variables corresponding to K (i 1. Z a set of new nodes for all state variables 2. Y Q2BN (q, Z a 3. F list of conditional probabilities in the odel 4. for i to 1 5. while Z (i is not epty 6. reove node z fro Z (i 7. su-out(f, z 8. f(z the result of previous loop /*Z Z (i 1 */ 9. add f(z < n (i to F 10. h the ultiplication of all factors in F 11. return h(y P Y h(y a For siplicity we only treat <. The inequality can be > as well. b Q2BN is siilar to the one in the previous section except that these queries have inequalities. But it does not affect the BN. Figure 5: Ordered Variable Eliination (OVE algorith. are root nodes. The details of the ethod is provided in the rest of this section. First, we show that the estiated values of soe queries ay not converge to the true values by increasing the nuber of saples. Consequently, the only way to answer these queries is to use an exact ethod. The following theore states this result. Theore 0.7 Let K(x < n be a query, s be a state, and s 1, s 2,... be a sequence of independent and identically distributed states sapled fro P r(z h Z = s. Define ˆK = x(s1+...+x(s to be the observed value of K(x using saples. P r( li ( ˆK < n does not exist = 1 when n is equal to the value of K(x on s and 0 < n < 1. Proof Sketch. We show the proof for n = 1 2. We show that P r( li ( ˆK < 1 does not exist = 1 when the value of 2 K(x on s is equal to 1 2. The proof for other n s uses the solution of the ruin proble for generalized one-diensional rando walks (Feller Since the value of ˆK < n for a specific is either 0 or 1, li ( ˆK < 1 2 (if exists should be either 0 or 1. First, we show that P r(li ( ˆK < 1 2 = 1 = 0 (the proof for value 0 is siilar to the proof for value 1. By definition of liit of a function at infinity, li ( ˆK < 1 2 = 1 if and only if for each ɛ > 0 there exists an N such that ( ˆK < < ɛ whenever > N. Since ˆK < 1 2 is binary, our definition would be ˆK < 1 2 for > N. Each saple is drawn fro a Bernoulli distribution with ean 1 2. To copute the above probability we need to answer the following question. In a one-diensional rando 545

6 walk, what is the probability that no return to the origin occurs up to and including tie 2? A rando walk is odeled with X(t + 1 = X(t + Φ(t. In this notation, Φ(ts are independent and identically distributed Bernoulli rando variables that have value +1 and 1 with probability 1/2 at all ties. By lea 1 of chapter III.3 of (Feller 1968, the probability that no return to the origin occurs up to and including tie 2 is the sae as the probability that a return occurs at tie 2 (i.e., the nuber of +1s is equal to the nuber of -1s. This probability is: ( 2 P r(return at tie 2 = 2 2 We calculate the probability that in an infinite sequence no return to the origin occurs by coputing the liit of above probability at infinity. ( 2 P r( li ( ˆK < 1 = 1 = li 2 n 2 2 = 0 Siilarly we can show that P r(li ( ˆK < 1 2 = 0 = 0. Using these results, it holds that P r( li ( ˆK < n does not exist = 1. Based on this theore, the accuracy of a sapling ethod does not necessarily increase with the nuber of saples. To estiate the accuracy of the value of a query, our sapling ethod not only calculates the truth value of the query but also returns the confidence level of the ethod in that value. The confidence level is the probability of the query being true given the set of saples. Function CoSa shown in Figure 6 presents our sapling ethod. It returns the probability of the query being true. The function first transfors the query to its BN representation. Then, it calculates the probability of q = 1 recursively using Function RecCS (e.g., if RecCS returns 0 the truth value of the query is equal to 0. For queries with no odal function, RecCS calculates the value of q on s (the details of this calculation is not shown in the function and returns the probability of q = 1 based on this value. For queries with odal functions such as q = K i (q < n, RecCS repeats the following step ties. It saples a new state s accessible fro s and recursively coputes the probability of q on s. Then, RecCS calculates the probability of q = 1 using the probabilities of the values of these saples. Function RecCS calls Function CalculateProb to perfor this calculation. In the next few paragraphs we explain how CalulateProb coputes the probability of the query. Iagine the query q = K 1 (x < 0.4 on s. First, we saple states fro the probability distribution P r(z h Z = s and we calculate the value of x on each sapled state. Then, we copute the probability of q = 1 given these values. Let there be k saples with value 1. Each saple is a Bernoulli trial whose probability of success is p = K 1 (x. The saple proportion ˆp is the fraction of saples with value 1 so ˆp = k. When is large, ˆp has an approxiately noral distribution. The standard deviation of the saple pro- FUNCTION CoSa(query q, state s q: the query of the for K (1 (... (K ( (x < n (... < n (1 in which K (i {K 1,..., K j} for j agents s: list of actual-state values Z a : list of actual-state variables Y : query node Z (i : list of hypothetical-state variables corresponding to K (i 1. Z a set of new nodes for all state variables 2. Y Q2BN (q, Z a 3. return RecCS(q, s, Z a FUNCTION RecCS(query q, state s, set of nodes Z 1. if q = `K (i (q < n then 2. T 3. for j 1 to 4. s j saple according to P r(z (i Z = s 5. s value of non-leaf nodes in Z (i 6. if s / T then 7. T add s to T with weight 1 8. conf(s RecCS(q, s j, Z (i 9. else 10. weight(s weight(s return CalculateProb(q, T 12. else if q = x then 13. if value x on s is true then return else return 0 Figure 6: Confidence Sapling (CoSa algorith. p(1 p portion is σ =. Since the true population proportion (p is unknown, we use standard error instead of σ. The standard error provides an unbiased estiate of the standard deviation. It can be calculated fro the equation ˆp(1 ˆp SE =. Therefore, K 1(x ˆp SE N(0, 1and the probability of the query is calculated fro: n ˆp P r(k 1(x < n = Φ (1 q ˆp(1 ˆp where Φ is the cuulative distribution function of the standard noral distribution. For queries with nested odal functions, the exact values of the sub-query on the sapled states are unknown. We only have the probability of the sub-query on those states. To calculate the probability of K 1 (q < n where q has a odal function, we use the equation bellow: P r(k 1(q < 0.4 = X (v 1,...,v {0,1} P r(k 1(q < 0.4 q s 1 = v 1,..., q s = v P r(q s 1 = v 1... P r(q s 1 = v 1 where s 1,..., s are sapled states and q s i = v i eans the value of q on sapled state s i is equal to v i. P r(q s i = v i is calculated recursively. P r(k 1 (q < 0.4 q s 1 = v 1,..., q s = v is calculated using Forula 1 with ˆp equal to saple proportion. Theore 0.8 Let q be a query, s be a state, k be the nuber of nested odal functions, and be the nuber of saples at each stage. Function CoSa calculates the truth value of q on s in O( k+2 tie. 546

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