SAT Modulo Monotonic Theories
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1 SAT Modulo Monotonic Theories Sam Bayless, Noah Bayless, Holger H. Hoos, Alan J. Hu University of British Columbia Point Grey Secondary School Sam Bayless (UBC) SAT Modulo Monotonic Theories / 0
2 Procedural Content Generation Sam Bayless (UBC) SAT Modulo Monotonic Theories / 0
3 MonoSAT Applications Circuit Layout Sam Bayless (UBC) Data Center Allocation SAT Modulo Monotonic Theories /0
4 MonoSAT Applications Finite State Machine Synthesis b q0 a b q a b q a b q a a,b q4 CTL Controller Synthesis {p} {~p} {~p} {p} Sam Bayless (UBC) SAT Modulo Monotonic Theories 4 / 0
5 MonoSAT MonoSAT is a SAT Modulo Theory Solver for: Graph Predicates: Reachability Shortest paths Maximum s t flow Minimum Spanning Tree Acyclicity Sam Bayless (UBC) SAT Modulo Monotonic Theories 5 / 0
6 MonoSAT MonoSAT is a SAT Modulo Theory Solver for: Graph Predicates: Reachability Shortest paths Maximum s t flow Minimum Spanning Tree Acyclicity Collision Detection for Convex Hulls Finite State Machine String Acceptance L-Systems, Boolean Geometry, CTL checking (soon) These are all monotonic theories Sam Bayless (UBC) SAT Modulo Monotonic Theories 5 / 0
7 Boolean Monotonic Theories A function p is a Boolean monotonic predicate iff:. : p returns a Boolean. : All arguments are of p are Boolean. : p(..., F,...) = p(..., T,...) Sam Bayless (UBC) SAT Modulo Monotonic Theories 6 / 0
8 Boolean Monotonic Theories A function p is a Boolean monotonic predicate iff:. : p returns a Boolean. : All arguments are of p are Boolean. : p(..., F,...) = p(..., T,...). Definition (Boolean Monotonic Theory). A theory T with signature Σ is Boolean monotonic if and only if:. The only sort in Σ is Boolean;. all predicates in Σ are monotonic; and.. all functions in Σ are monotonic. Sam Bayless (UBC) SAT Modulo Monotonic Theories 6 / 0
9 Graph Constraints in SMMT A formula over Booleans, edges, and monotonic predicates: (a b) (b c) ( c d) (reaches, reaches,4 ) And or more symbolic graphs: a b c d 4 Sam Bayless (UBC) SAT Modulo Monotonic Theories 7 / 0
10 Graph Constraints in SMMT Reachability is Boolean monotonic: 4 reaches, False reaches,4 False Sam Bayless (UBC) SAT Modulo Monotonic Theories 8 / 0
11 Graph Constraints in SMMT Reachability is Boolean monotonic: 4 reaches, False reaches,4 False Sam Bayless (UBC) SAT Modulo Monotonic Theories 8 / 0
12 Graph Constraints in SMMT Reachability is Boolean monotonic: 4 reaches, True reaches,4 False Sam Bayless (UBC) SAT Modulo Monotonic Theories 8 / 0
13 Graph Constraints in SMMT Reachability is Boolean monotonic: 4 reaches, True reaches,4 False Sam Bayless (UBC) SAT Modulo Monotonic Theories 8 / 0
14 Graph Constraints in SMMT Reachability is Boolean monotonic: 4 reaches, True reaches,4 True Sam Bayless (UBC) SAT Modulo Monotonic Theories 8 / 0
15 Theory Propagation in SMMT Formula: (a b) (b c) ( c d) (reaches, reaches,4 ) Assignment: a 4 c b d 4 Underapproximation Overapproximation reaches, reaches,4 Sam Bayless (UBC) SAT Modulo Monotonic Theories 9 / 0
16 Theory Propagation in SMMT Formula: (a b) (b c) ( c d) (reaches, reaches,4 ) Assignment: b a 4 c d 4 Underapproximation Overapproximation reaches, reaches,4 Sam Bayless (UBC) SAT Modulo Monotonic Theories 9 / 0
17 Theory Propagation in SMMT Formula: (a b) (b c) ( c d) (reaches, reaches,4 ) Assignment: b, a a 4 a c d 4 Underapproximation Overapproximation reaches, reaches,4 Sam Bayless (UBC) SAT Modulo Monotonic Theories 9 / 0
18 Theory Propagation in SMMT Formula: (a b) (b c) ( c d) (reaches, reaches,4 ) Assignment: b, a, c a c 4 a c d 4 Underapproximation Overapproximation reaches, True reaches,4 Sam Bayless (UBC) SAT Modulo Monotonic Theories 9 / 0
19 Theory Propagation in SMMT Formula: (a b) (b c) ( c d) (reaches, reaches,4 ) Assignment: b, a, c, d a c 4 a c 4 Underapproximation Overapproximation reaches, True reaches,4 False Sam Bayless (UBC) SAT Modulo Monotonic Theories 9 / 0
20 Theory Propagation in SMMT Theory propagation in SMMT has useful properties: Easy to implement. Can use off-the-shelf algorithms. Improved worst-case clause learning. Sam Bayless (UBC) SAT Modulo Monotonic Theories 0 / 0
21 MonoSAT Applications: Shortest Path Constraints Shortest Path Runtime (s) MonoSAT clasp MiniSat 0 Dst.8-6 Dst.6- Dst.- 48 Dst.- 64 Dst.- 96 Dst.- 8 Dst Dst Dst.64-8 Sam Bayless (UBC) SAT Modulo Monotonic Theories / 0
22 MonoSAT Applications: Maximum Flow Constraints Maximum Flow Runtime (s) MonoSAT clasp MiniSat 0 8x8 6Flow 6x6 8Flow 6x6 6Flow 6x6 4Flow 4x4 6Flow x 6Flow Sam Bayless (UBC) SAT Modulo Monotonic Theories / 0
23 MonoSAT Applications: Convex Hull Containment Art Gallery Synthesis MonoSAT Z 0 points, hulls, cameras s 7s 0 points, 4 hulls, 4 cameras 6s 4s 0 points, 5 hulls, 5 cameras 87s > 600s 40 points, 6 hulls, 6 cameras 645s > 600s 50 points, 7 hulls, 7 cameras 5s > 600s Sam Bayless (UBC) SAT Modulo Monotonic Theories / 0
24 Conclusion Monotonic theories have many applications. Building SMT solvers for them is easy. MonoSAT supports many graph properties (and more!), and it is free & open-source: New! Bit vector support, New! Python support. Website: cs.ubc.ca/labs/isd/projects/monosat GitHub: github.com/sambayless/monosat Sam Bayless (UBC) SAT Modulo Monotonic Theories 4 / 0
25 MonoSAT + Python from monosat import a = Var ( ) b = Var ( ) c = Or ( a, Not ( b ) ) A s s e r t ( c ) r e s u l t = S o l v e ( ) Sam Bayless (UBC) SAT Modulo Monotonic Theories 5 / 0
26 MonoSAT + Python from monosat import g= Graph ( ) e = g. addedge (, ) e = g. addedge (, ) e = g. addedge (, ) A s s e r t ( Not ( And ( e, e ) ) ) A s s e r t ( g. r e a c h e s (, ) ) r e s u l t = S o l v e ( ) Sam Bayless (UBC) SAT Modulo Monotonic Theories 6 / 0
27 MonoSAT + Python from monosat import g= Graph ( ) e = g. addedge (, ) e = g. addedge (, ) e = g. addedge (, ) A s s e r t ( Or ( g. r e a c h e s (, ), g. d i s t a n c e l e q (,, ) ) ) r e s u l t = S o l v e ( ) Sam Bayless (UBC) SAT Modulo Monotonic Theories 7 / 0
28 MonoSAT + Python from monosat import g= Graph ( ) bv = B i t V e c t o r ( 4 ) bv = B i t V e c t o r ( 4 ) bv = B i t V e c t o r ( 4 ) e = g. addedge (,, bv ) e = g. addedge (,, bv ) A s s e r t ( g. d i s t a n c e l e q (,, bv ) ) A s s e r t ( Not ( g. d i s t a n c e l t (,, bv ) ) ) A s s e r t ( ( bv + bv ) == 9) r e s u l t = S o l v e ( ) Sam Bayless (UBC) SAT Modulo Monotonic Theories 8 / 0
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