4 1 $_{ℏ}$S $^{(2)}$[ C ] = N − 2 N$_{1}$ + 4 N$_{2}$ − 2 N$_{3}$ (13) 1 $_{ℏ}$S $^{(4)}$[ C ] = N − N$_{1}$ + 9 N$_{2}$ − 16 N$_{3}$ + 8 N$_{4}$ , (14) L eff √ − g ℏ = b$_{0}$l − 4 k + b$_{1}$l − 2 k R + b$_{2}$R 2 + . . . (15) In each of 2D and 4D, we can now define a one pa- rameter family of candidate actions, S$_{k}$ [ C ], for a causal set, C , by summing B$_{k}$ ( − 1) over the elements of C , times ℏ l 2 to get the units right, times a number of order one which in 4D is the ratio of l 2 to l 2 $_{p}$, where l$_{p}$ = √$_{8}$$_{πG}$ ℏ is the rationalized Planck length. When the nonlocality length l$_{k}$ equals the discreteness length l , B$_{k}$ = B and the action, S [ C ] takes a particularly simple form as an alternating sum of numbers of small order intervals in C . Up to factors of order one, we have in 2D and 4D, respectively: and where N is the number of elements in C and N$_{i}$ is the number of ( i + 1) element inclusive order intervals in C . Because B is the most non-nonlocal of the operators in the family, the action S [ C ] is a sum of contributions each of which is not close to the value of the Ricci scalar at the corresponding point of the continuum approxima- tion. However, one might expect that if the curvature is slowly varying on some intermediate scale, which we might as well call l$_{k}$ , the averaging involved in the sum- mation might perform the same role of suppressing the fluctuations as the smearing out of the operator itself so that the whole action S [ C ] is a good approximation to the continuum action when l$_{k}$ is the appropriate size. There are many new avenues to explore. Can we use these results to define a quantum dynamics for causal sets? In 2D is there a relation with the Gauss-Bonnet theorem? Can we analytically continue the action in an appropriate way [19] to enable Monte-Carlo simulations of the path sum? What sort of phenomenology might emerge from such actions? To answer this latter question, we need to know how big l$_{k}$ must be so that the action S [ C ] is a good approximation to the Einstein-Hilbert ac- tion of the continuum S$_{EH}$ [ g ]. In [18], a rough estimate is reported that in dimension 4, l$_{k}$ >> ( l $^{2}$L ) 1 / $^{3}$. Tak- ing L to be the Hubble scale, that would mean that in the continuum regime, only spacetimes whose curvature was constant over a scale ( l $^{2}$L ) 1 / 3 would be able to have an approximately local fundamental action. One might expect therefore that the phenomenological IR theory of gravity that could emerge from such a fundamental the- ory would be governed by an effective Lagrangian where b$_{1}$ and b$_{2}$ are of order 1, b$_{0}$ is set to its observed value, and where l$_{k}$ varies with epoch and today is much larger than the Planck scale. The phenomenological im- plications of these ideas remain to be explored. We end by pointing out that these results have a rele- vance beyond causal set theory as they provide a “proof of concept” for the mutual compatibility of Lorentz in- variance, fundamental spacetime discreteness, and ap- proximate locality. We thank Rafael Sorkin for invaluable help and Michael Delph and Joe Henson for useful discussions. We also thank David Rideout for help with the simulations using his CausalSets toolkit in the Cactus framework (www.cactuscode.org). DMTB is supported by EPSRC. FD is supported by by EC Grant No. MRTN-CT-2004- 005616 and Royal Society Grant No. IJP 2006/R2. We thank the Perimeter Institute for Theoretical Physics, Waterloo, Canada where much of this work was done. [1] L. Bombelli, J.-H. Lee, D. Meyer, and R. Sorkin, Phys. Rev. Lett 59 , 521 (1987). [2] L. Bombelli, J. Henson, and R. D. Sorkin (2006), gr- qc/0605006. [3] C. Moore, Phys. Rev. Lett. 60 , 655 (1988). [4] L. Bombelli, J. Lee, D. Meyer, and R. D. Sorkin, Phys. Rev. Lett. 60 , 656 (1988). 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