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[[File:IceCream_Generic.pdf|500px|center]] | [[File:IceCream_Generic.pdf|500px|center]] | ||

By choosing "Show Source Code Coverage" in the Debug menu of ProB Tcl/Tk you can see which parts of the properties were translated to SAT (in green) and which were processed entirely by ProB (in red-orange): | |||

Here is the graphical rendering of a solution using the custom graph definition above: | |||

[[File:IceCreamCoverage.png|500px|center]] | |||

=== B2SAT in Jupyter Notebooks === | === B2SAT in Jupyter Notebooks === |

The current versions of ProB can make use of the new B2SAT backend as an alternate way of solving constraints. It translates a subset of B formulas to SAT and for solving by an external SAT solver. The new backend interleaves low-level SAT solving with high-level constraint solving performed by the default solver of ProB.

B2SAT solving consists of the following phases:

- the deterministic propagation phase(s) of ProB's solver: it performs deterministic propagations and can expand quantifiers and total functions.
- a compilation phase, whereby static values are inlined.
- the B to CNF conversion proper, which can translate a subset of B to propositional logic in conjunctive normal form. This phase currently supports:
- equalities and inequalities between boolean variables and constants,
- all logical connectives and
- some cardinality constraints.
- Subformulas that cannot be solved are sent to ProB's default solver and linked to the CNF via an auxiliary propositional variable.

- solving phase, where the generated CNF is sent to an external SAT solver.
- propagation of the SAT solution to B, by progressively grounding the B values and predicates linked to the propositional variables.
- complete constraint solving, solving the pending constraints in B by now performing the regular non-deterministic propagations and enumerations of ProB.

The following four commands are now available in REPL of probcli (start with `probcli --repl`):

- :sat PRED
- :sat-z3 PRED
- :sat-double-check PRED
- :sat-z3-double-check PRED

Here is a simple example:

>>> :sat f:1..n --> BOOL & n=3 & f(1)=TRUE & !i.(i:2..n => f(i) /= f(i-1)) PREDICATE is TRUE Solution: f = {(1|->TRUE),(2|->FALSE),(3|->TRUE)} & n = 3

The new preference SOLVER_FOR_PROPERTIES can be used to specify solver for PROPERTIES (axioms) when setting up constants. The valid settings are: prob (the default), kodkod, z3, z3cns, z3axm, cdclt, sat, sat-z3.

The last two will use B2SAT, {\tt sat} with Glucose as SAT solver and {\tt sat-z3} with Z3 as SAT solver.

Here is an example using this preference:

MACHINE IceCream_Generic // Dominating set example: // place ice cream vans so that every house (node) is at most one block away from a van // a version which can be solved by B2SAT DEFINITIONS N == 24; CUSTOM_GRAPH == rec(layout:"dot", rankdir:"TB", nodes: {j•j:NODES | rec(value:j, style:"filled", fillcolor:IF ice(j)=TRUE THEN "mistyrose" ELSE "white" END )}, edges:rec(color:"gray", arrowhead:"odot", arrowtail:"odot", dir:"both", label:"edge", edges: edge) ); bi_edge == (edge \/ edge~); SET_PREF_SMT == TRUE; SET_PREF_TIME_OUT == 50000; SET_PREF_SOLVER_STRENGTH == 300; SET_PREF_SOLVER_FOR_PROPERTIES == "sat"; SETS NODES = {n1,n2,n3,n4,n5,n6,n7,n8,n9,n10, n11,n12,n13,n14,n15,n16,n17,n18,n19,n20,n21,n22,n23,n24} CONSTANTS edge, ice, vans, neighbours PROPERTIES edge: NODES <-> NODES & edge = { n1|->n2, n1|->n4, n2|->n3, n3|->n4, n3|->n5, n3|->n7, n4|->n7, n5|->n6, n5|->n9, n6|->n7, n6|->n8, n7|->n8, n8|->n10, n8|->n13, n9|->n10, n9|->n11, n9|->n12, n11|->n12, n11|->n14, n12|->n13, n13|->n16, n14|->n15, n14|->n17, n15|->n16, n15|->n17, n15|->n18, n15|->n21, n16|->n18, n16|->n19, n17|->n19, n18|->n19, n18|->n20, n18|->n21, n19|->n20, n19|->n21, n20|->n21, n20|->n22, n21|->n22, n21|->n23, n21|->n24, n22|->n23, n21|->n24, n23|->n24 } & ice : NODES--> BOOL & neighbours = %x.(x:NODES|bi_edge[{x}]) & !x.(x:NODES => (ice(x)=TRUE or #neighbour.(neighbour: neighbours(x) & ice(neighbour)=TRUE) ) ) & vans = card(ice~[{TRUE}]) & card({x|x:NODES & ice(x)=TRUE})<=6 /* minimal solution requires 6 vans */ OPERATIONS v <-- NrVans = BEGIN v := vans END; xx <-- Get(yy) = PRE yy:NODES THEN xx:= ice(yy) END; v <-- Vans = BEGIN v:= ice~[{TRUE}] END END

Here is the graphical rendering of a solution using the custom graph definition above:

By choosing "Show Source Code Coverage" in the Debug menu of ProB Tcl/Tk you can see which parts of the properties were translated to SAT (in green) and which were processed entirely by ProB (in red-orange):

Here is the graphical rendering of a solution using the custom graph definition above:

The solve command in the ProB Jupyter kernel now supports `sat` and `satz3` as solvers.
They use B2SAT with Glucose and Z3 as SAT solvers respectively.

Here is a simple example to solve a constraint:

This shows how you can use cardinality constraints to iteratively find a minimal dominating set D of a graph g: