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## Syntax Matters

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**Syntax Matters**Tom Henzinger (IST Austria) with Barbara Jobstmann (Verimag) Maria Mateescu (EPFL) Verena Wolf (Saarbruecken)**Science**Experiment Theory**Mathematics**14**Mathematics**Semantics Syntax 14**Mathematics**Semantics Syntax 14 1110 XIV**Syntax Matters**• Expressiveness 0**Syntax Matters**• Expressiveness 0 • Succinctness IIII IIII IIII**Syntax Matters**• Expressiveness 0 • Succinctness IIII IIII IIII • Operations +1 addition multiplication 14 x 34 = 42 56 476**Syntax is More than Notation**• Expressiveness 0 • Succinctness IIII IIII IIII • Operations +1 addition multiplication 14 x 34 = 42 56 476 (* 14 34) =**Syntax is More than Notation**• Expressiveness 0 • Succinctness IIII IIII IIII • Operations +1 addition multiplication Two languages are equivalent if there is a linear translation from each to the other (e.g. prefix – infix, binary – decimal).**Computer Science**Semantics Syntax 1 1 2 6 24 ... 0 1 2 3 4 ... (defun f (n) (if (<= n 1) 1 (* n (f (- n 1)))))**Computer Science**Semantics Syntax 1 1 2 6 24 ... 0 1 2 3 4 ... (defun f (n) (if (<= n 1) 1 (* n (f (- n 1))))) finite infinite**Computer Science**Semantics Syntax 1 1 2 6 24 ... 0 1 2 3 4 ... (defun f (n) (if (<= n 1) 1 (* n (f (- n 1))))) finite recursively enumerable infinite**Computer Science**Semantics Syntax 1 1 2 6 24 ... 0 1 2 3 4 ... (defun f (n) (if (<= n 1) 1 (* n (f (- n 1))))) Composition: (defun h (n) (f (g n)))**Computer Science**Semantics Syntax x := 0; y := 0; while true do x := x + 1; y := y + 1 end**Computer Science**Semantics Syntax x := 0; y := 0; while true do x := x + 1; y := y + 1 end 2,2 3,2 1,1 2,1 0,0 1,0**Computer Science**Semantics Syntax (inc x)* 0,0 1,0 2,0**Computer Science**Semantics Syntax (inc x)* (inc y) * 0,2 0,1 0,0 1,0 2,0**Computer Science**Semantics Syntax (inc x)* (inc y) * 1,2 0,2 2,2 0,1 1,1 2,1 Composition: (inc x)* || (inc y)* 0,0 1,0 2,0**Markovian Population Models**State: ( 8 , 6 , 6 )**Markovian Population Models**State: ( 9 , 7 , 5 ) Transition: State: ( 8 , 6 , 6 )**Markovian Population Models**-discrete state -location unaware**Markovian Population Models**-discrete state -location unaware -stochastic transition -continuous time**Markovian Population Models**8, 6, 6 9, 7, 5 deterministic**Markovian Population Models**8, 6, 6 9, 7, 5 deterministic**Markovian Population Models**8, 6, 6 9, 7, 5 deterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time**Markovian Population Models**8, 6, 6 9, 7, 5 deterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time 0 0 1**Markovian Population Models**8, 6, 6 9, 7, 5 deterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time 1 0.4 0.6**Markovian Population Models**8, 6, 6 9, 7, 5 deterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time 2 0.64 0.36**Markovian Population Models**8, 6, 6 9, 7, 5 deterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time 3 0.784 0.216 (9,7,5): 0.36 (8,6,6): 0.64 (9,7,5): 0.216 (8,6,6): 0.784**Markovian Population Models**8, 6, 6 9, 7, 5 deterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time 0.5 8, 6, 6 9, 7, 5 continuous time exit rate 0.5 exp residence time 2**Markovian Population Models**0.5 8, 6, 6 9, 7, 5 continuous time 0exit rate 0.5 exp residence time 2 0 1**Markovian Population Models**0.5 8, 6, 6 9, 7, 5 continuous time 1exit rate 0.5 exp residence time 2 0.4 0.6**Markovian Population Models**0.5 8, 6, 6 9, 7, 5 continuous time 1.8exit rate 0.5 exp residence time 2 0.6 0.4**Markovian Population Models**8, 6, 6 9, 7, 5 nondeterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time 0.5 8, 6, 6 9, 7, 5 continuous time**Markovian Population Models**8, 6, 6 9, 7, 5 nondeterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time MDP 0.3 0.7 0.5 8, 6, 6 9, 7, 5 continuous time**Markovian Population Models**8, 6, 6 9, 7, 5 nondeterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time MDP 0.3 0.7 0.5 8, 6, 6 9, 7, 5 continuous time exit rate 2 exp residence time 0.5 1.5**Markovian Population Models**0.1 0.1 0.1 0.2 0.2 0.2 7, 5, 5 8, 6, 4 9, 7, 3 0.1 0.1 0.1 0.2 0.2 0.2 7, 5, 6 8, 6, 5 9, 7, 4 0.1 0.1 0.1 0.2 0.2 0.2 7, 5, 7 8, 6, 6 9, 7, 5 CTMC**Markovian Population Models**0.2 + 0.1 Syntax: set of transition classes (finite object)**Markovian Population Models**0.2 + 0.1 Syntax: set of transition classes (finite object) 0.1 0.1 0.2 0.2 8, 6, 6 9, 7, 5 Semantics: CTMC (infinite object)**Markovian Population Models**0.2 + 0.1 Syntax: set of transition classes (finite object) 0.6 0.5 8, 6, 6 9, 7, 5 Semantics: CTMC (infinite object)**Markovian Population Models**0.2 + 0.1 Syntax: set of transition classes (finite object) 0.6 0.5 9.6 12.6 8, 6, 6 9, 7, 5 Semantics: CTMC (infinite object)**Syntax: Transition Class Model (TCM)**Dimension n: state (x1, ..., xn) 2 S state space S = Nn Finite set of transition classes: each transition class consists of 1. guard G µ S 2. injective update function u: G ! S 3. rate function : G !R+**Syntax: Transition Class Model (TCM)**Dimension n: state (x1, ..., xn) 2 S state space S = Nn Finite set of transition classes: each transition class consists of 1. guard G µ S 2. injective update function u: G ! S 3. rate function : G !R+ n = 3 G1: x1¸ 1 Æ x2¸ 1 u1(x1,x2,x3) = (x1-1, x2-1, x3+1) 1(x1,x2,x3) = 0.2 ¢ x1 ¢ x2 G2: x3¸ 1 u2(x1,x2,x3) = (x1, x2, x3-1) 2(x1,x2,x3) = 0.1 ¢ x3**Semantics: Continuous-Time Markov Chain (CTMC)**For all times t 2R+, a random variable X(t) 2 S.**Semantics: Continuous-Time Markov Chain (CTMC)**For all times t 2R+, a random variable X(t) 2 S. Syntax ! Semantics: TCM ! CTMC For each transition class (Gi,ui,i) and all times t 2R+ and ! 0, Pr( X(t+) = ui(x) | X(t) = x ) = i(x) ¢. In addition, Pr( X(0) = x0 ) = 1 for some given initial state x02 S.

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