### Table 2. The inference rules for PC. y0 are xed and all di erent. 3.3 Power to Simulate 2-Counter Machines The rst (and weakest) form of universality that we consider is that a process calculus has the expressive power of 2-counter machines (or, equivalently, Turing machines) in the sense that, for each n, we can exhibit a term U2CM n whose process graph simulates in lock step a universal 2-counter machine on input n. Calculi like CCS, CSP, ACP, and Meije are all universally expressive in this sense. Actu- ally, trying to code a 2-counter or Turing machine in each of these languages is a nice way to get familiar with them. Via a rather tricky encoding, we prove below that also PC has the power of 2-counter machines.1 Theorem 3.8 PC has the expressive power of 2-counter machines. Proof Suppose that a universal 2-counter machine has code of the form l1:

1993

"... In PAGE 11: ... It is also possible to view this operator as a special case of the action re nement operator as studied by Goltz and Van Glabbeek [17]: r re nes an action a into the nondeterministic sum of the actions in fb j r(a; b)g. The inference rules of PC are presented in Table2 . In the table a and b range over A,... ..."

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### Table 1. Comparison of the sizes of the different CSP encodings for k-SAT.

"... In PAGE 8: ...f the size of the resulting CSP w.r.t. the number of variables (#variables), the size of the variable domains, the number of constraints (#constraints), and the size of the constraints, using the encodings presented in the previous section is shown in Table1 . Notice that out of all the encodings, only the place encoding is linear in all of these parameters (number of variables, domain size, and constraint size).... In PAGE 9: ...shown in Table1 , variables associated with clauses have domains of exponential size in the hidden variable encoding, while these domains are of linear size in the place encoding. We begin by showing that enforcing FC consistency on the place encoding results in an arc consistent CSP.... ..."

### Table 1. Encodings for the different proof techniques CIRC. ELEM. SAT SWORD SMT CSP

"... In PAGE 3: ...1. Encoding Effort In Table1 an overview of the estimated effort for the en- coding of standard circuit elements is presented, including an n-bit variable declaration, n-bit logic operation and n-bit arithmetic operation. The focus is on the number of vari- ables and constraints that are necessary to declare a circuit element.... ..."

### Table 1: A summary of the CSPM notation, as compared to CSP.

2001

"... In PAGE 4: ... It is this functional language which gives speci cations written in CSPM the expressive power to represent Object-Z, drawing on the CSPM implementation of set theoretical and logic concepts to encode Object-Z schemas. A summary of the CSPM no- tation as compared to CSP is in Table1 , and similarly, a summary of the CSPM functional language notation as compared to Object-Z is found in Table 2. To encode Object-Z classes in CSPM by this approach, classes are decom- posed into their component aspects { state and operation de nitions.... In PAGE 8: ... The alteration also requires the use of the external choice operator for initial states, outputs and post-states. Given that not b amp; p abbreviates if b then STOP else p (see Table1 ), we have: OZSemantics(Ops; in; out; enable; e ect; init; event) = let OZ PART(s) = [ ] op : Ops @ enable(op)(s) amp; [ ]i : in(op) @ not empty(e ect(op)(s; i)) amp; ([ ](o; s0) : e ect(op)(s; i) @ event(op; i; o){ gt; OZ PART(s0)) OZ MAIN = [ ]s : init @ OZ PART(s) within OZ MAIN The encoding is completed by a process which calls the Semantics process with the derived class aspects: CreditCard = OZSemantics(Ops; in; out; enable; e ect; init; event) 3.2 A more e cient encoding? This initial adaption of the CSP-OZ encoding, while reducing the state-space of the compiled model, and hence some portion of the model checking time, is still somewhat time consuming as a model checking technique.... ..."

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### Table 4: The encoding of causal terms

"... In PAGE 22: ...e., CCS terms), Table4 gives us an encoding of the classical causal bisimulation of CCS [DD89, Kie91] into the observation equivalence of the monadic -calculus. It is straightforward to see that there is agreement between the de nitions of the encoding on processes and on sortings: Proposition 5.... ..."

### Table IV. Randomized rapid restarts (RRR) versus determinis- tic versions of backtrack search procedures (Satz solver used on SAT encodings; Ilog solver on CSP encodings).

2000

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### Table II. Estimates of the index of stability, ,with sample size k. The values within parentheses are the estimated asymptotic standard deviations. (Quasigroup instances encoded as CSP; other instances encoded as SAT).

2000

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### Table IV. Randomized rapid restarts (RRR) versus determin- istic versions of backtrack search procedures (Satz solver used on SAT encodings; Ilog solver on CSP encodings).

2000

Cited by 91