1. From Propositional SAT to SMT
Hossein M. Sheini and Karem A. Sakallah
SAT 2006
August 13, 2006
Seattle

2. Propositional Satisfiability
DPLL: systematic backtracking search
Branch
[Pure literal rule]
Propagate
Unit propagation  Boolean Constraint Propagation (BCP)
Backtrack
Modern implementations of DPLL
Conflict analysis
Clause learning
Non-chronological backtracking
Efficient propagation
Two watched literals per clause
Adaptive branching
VSIDS
Restarts

3. Quantifier-Free First-Order Logic
Boolean combination of atoms from background theories

4. Decidable Quantifier-Free FOL Theories
Empty
QF_UF: uninterpreted functions with equality (aka EUF)
Linear Real Arithmetic
QF_LRA
QF_RDL: real difference logic
UTVPI
Linear Integer Arithmetic
QF_LIA
QF_IDL: integer difference logic
Data Structures
QF_A: arrays
QF_BV: bit vectors
Lists, etc.

5. QF_UF Atoms

6. Linear Real Arithmetic Atoms
QF_LRA:
QF_RDL:
UTVPI:

7. Linear Integer Arithmetic Atoms
QF_LIA:
QF_IDL:
IUTVPI:

8. Chronology of SAT for Quantifier-Free FOL
Late 70s to mid 80s
Congruence closure algorithms
Combination strategies for disjoint theories
Mid 80s to mid 90s
Not much!
Mid to late 90s
Initial attempts at improved propositional reasoning
Early 21st Century
Significant activity across many fields

9. Reasoning About Equality
Logic of equality with uninterpreted functions and predicates (EUF)
To prove validity of EUF formula j
Construct disjunctive normal form (DNF) of ¬j
Prove unsatisfiability of each conjunct of ¬j using congruence closure
[Shostak 78]
[NelsonOppen 80]

10. Congruence Closure a b f
[NelsonOppen 80]

11. Reasoning About Integer Arithmetic
Quantifier-Free Presburger Arithmetic
To prove validity of QF Presburger formula j
Construct disjunctive normal form (DNF) of ¬j
Prove unsatisfiability of each conjunct of ¬j using integer linear programming (ILP)
[Shostak 79]

12. QF Presburger Arithmetic Example
Negate
“Normalize”
Convert to DNF
Solve ILPs
[Shostak 79]
Invalid!

13. Deciding Combinations of Theories
Quantifier-Free Presburger Arithmetic + EUF
Add all functional consistency axioms
(aka substitutivity axioms of equality)
Eliminate UFs and UPs using “Ackermann’s reduction” (adding consistency “axioms”) to get a pure equality formula
Convert complement to DNF
Solve each conjunct as an integer linear program (ILP)
Formula explodes
[Shostak 79]

14. QF Presburger Arithmetic + EUF Example
Add functional consistency axioms
Eliminate function symbols
[Shostak 79]

15. Alternatively … Add all functional consistency axioms
Eliminate UFs and UPs using “Ackermann’s reduction” (adding consistency “axioms”) to get a pure equality formula
Convert complement to DNF
Solve each conjunct as an integer linear program (ILP)
Formula explodes
Add functional consistency axioms as needed
Ignore functional consistency and solve as before
Check functional consistency for symbols with different values; if violated, add axiom and repeat
Has the flavor of “learning” on demand
ILPs can be solved incrementally
[Shostak 79]

16. Deciding Combinations of Theories
Convert negation to DNF
Add variables to purify different theory conjuncts
Solve separately
Propagate equalities
Split in case no equalities can be inferred
Supported theories:
Real numbers under + and leq: Simplex
Arrays under store and select
List structures with car, cdr, cons, and atom: congruence closure
Equality with UF: congruence closure
[NelsonOppen 79]

17. Example of Nelson-Oppen Procedure
Lists
EUF
LRA
[NelsonOppen 79]

18. Example of Nelson-Oppen Procedure

19. Deciding Combinations of Theories
Generalizes Nelson-Oppen method by eliminating the need for extra variables
Congruence closure is extended to handle different theories as long as they have “canonizers” and solvers
Found to be “buggy” and not as general as N-O
[Shostak 84]

20. Disadvantages of “Old” Combination Methods
Need to convert to DNF
Inefficient handling of Boolean structure
Can be viewed as lazy integration with an open feedback loop between a propositional enumerator and the theory solvers

21. Disadvantages of “Old” Combination Methods
SMT Instance
DNF
Converter
DNF Instance
Yes
More
Conjuncts?
Conjunction of
Theory Atoms No
UNSAT
Theory
Solvers
SAT

22. Improved Propositional Reasoning
Convert SMT instance to equi-satisfiable propositional form and apply Boolean reasoning techniques (BDDs or SAT)
Small-domain encoding
Per-constraint encoding
Add more sophisticated Boolean reasoning, but keep background theories intact
Re-write rules and better Boolean splits
Add full-fledged SAT or BDD solvers to handle the Boolean skeleton
Very Lazy: theory solver returns a conflict clause to SAT solver
Lazy: theory solver invoked incrementally but does not propagate
Eager: theory solver propagates and learns etc.
Variants
Layered

23. Reasoning About Equality: Revisited
EUF Formula
DNF
Converter
DNF of = and ≠ Conjuncts
More
Conjuncts? No
UNSAT
Yes
Congruence
Closure
SAT
Conjunction of
= and ≠

24. Reasoning About Equality: Revisited
SAT/BDD
Solver
SAT
UNSAT
EUF Formula
Reduction
E Formula
Range Analysis
&
Boolean Encoding

25. Reasoning About Equality: Revisited
Ackermann’s
Reduction
Congruence
Closure x y F1 F4 F3 F2
Bryant’s “ite” Reduction

26. Pros/Cons of SMT-to-SAT Conversion
Black-box use of modern SAT solvers ü
Leveraging of performance/capacity improvements
in SAT solvers ü û
Loss of theory semantics (e.g., arithmetic)
Increase of instance sizes û

27. Combine SAT and Theory Solvers
SMT Instance
Propositional
Abstraction
SAT Instance
Abstraction
Refinement
SAT
Solver
UNSAT
UNSAT
SAT
Theory Atoms
Theory
Solvers
SAT

28. Propositional Abstraction

29. Spectrum of Integration
How aggressively is refinement done?
Very lazily: theory solver does not feedback any info to SAT solver; no refinement
Lazily: theory solver returns a small explanation of infeasibility to SAT solver
Eagerly: theory solver participates in value propagation (implications) and in conflict analysis
Very eagerly: direct encoding of all theory semantics in propositional formula; no abstraction

30. Very Lazy Integration
Boolean Solver

31. Very Lazy Integration
Integer Solver
UNSAT

32. Spectrum of Integration
How aggressively is refinement done?
Very lazily: theory solver does not feedback any info to SAT solver; no refinement
Lazily: theory solver returns a small explanation of infeasibility to SAT solver
Eagerly: theory solver participates in value propagation (implications) and in conflict analysis
Very eagerly: direct encoding of all theory semantics in propositional formula; no abstraction

33. Lazy Integration
Boolean Solver

34. Lazy Integration Integer Solver Create conflict clause
and return to Boolean solver

35. Spectrum of Integration
How aggressively is refinement done?
Very lazily: theory solver does not feedback any info to SAT solver; no refinement
Lazily: theory solver returns a small explanation of infeasibility to SAT solver
Eagerly: theory solver participates in value propagation (implications) and in conflict analysis
Very eagerly: direct encoding of all theory semantics in propositional formula; no abstraction

36. Eager Integration: Incremental Propagation

37. Eager Integration: Incremental Propagation

38. Integration Trade-offs
Must balance cost of generating new “facts” against utility of such facts in pruning the search space
E.g., a priori generation of transitivity constraints for all possible equalities is overkill
Suggests an “on-demand” learning strategy analogous to conflict analysis in modern SAT solvers
On-demand learning requires incremental backtrackable theory solvers that maintain state
Cost of propagation for various theories and sub-theories:
EUF: congruence closure is O(n log n)
Difference constraints: negative cycle detection is O(nm)
UTVPI: transitive closure is O(n3)
Real arithmetic: incremental Simplex

39. Offline Integration of LRA Solver

40. Offline Integration of LRA Solver

41. Offline Integration of LRA Solver

42. Learning Strategies
CNF clauses: disjunctions of existing atoms (in terms of their indicator variables)
Introduction of new theory atoms: cutting planes

43. Introduction of New Atoms

44. Offline Integration of LRA Solver

45. DPLL(T) Framework
Declarative “calculus” for tight integration of a solver for theory T within a propositional DPLL solver
Inspired by CLP(X)
Defines SolverT as an abstract data type with the following methods:
Initialize(L: Literal set)
SetTrue(l: L-literal): L-literal set
IsTrue?(l: L-literal): Boolean
Backtrack(n: Natural)
Explanation(l: L-literal): L-literal set
[Tinelli 02]
[Nieuwenhuis-Oliveras 03]

46. Nelson-Oppen 1979 LPSAT 1999 Simplify 1998? CVC 2002/2004 Verifun 2003
SVC
1996
CVC Lite
2004
Sammy
2005
Shostak
1984
ICS
2001
Yices
2006
ICS+Chaff
2002
Simplics
2005
D. Cyrluk, P. Lincoln, and N. Shankar, “On Shostak’s Decision Procedure for Combinations of Theories,” CADE 1996
Compares the Nelson-Oppen and Shostak congruence closure procedures and identifies the three optimizations in Shostak’s procedure
Explains how so-called sigma-theories (i.e., theories that are canonizable and algebraically solvable) can be integrated much more closely than in the Nelson-Oppen framework, leading to much more efficient equality propagation (necessarily duplicated in the Neslon-Oppen approach)
Identifies the bugs in Shostak’s original algorithm
Suggests (for future) making Shostak a component in a Nelson-Oppen framework.
J.-C. Filliatre, S. Owre, H. Ruess, and N. Shankar, “ICS: Integrated Canonizer and Solver,” CAV 2001.
Implements (corrected) Shostak procedure extended with arithemtic inequalities
Has a flexible API for integration with other provers and simulators
Theories:
UF with equality and disequality
Rational linear arithmetic (with inequalities)
Sets
Tuples
Arrays
Bit vectors
L. de Moura and H. Ruess, “Lemmas of Demand for Satisfiability Solvers,” SAT 2002
Lazy integration of ICS and Chaff
Procedure for creating lemmas that “explain” the infeasibility of conjunctions of theory atoms
Mentions inefficiency of restarting the SAT solver after adding theory-induced lemmas
B. Dutertre and L. De Moura, “Simplics: Tools Description,” 2005
Successor to ICS that is limited to the theory of linear real arithmetic
Main application: BMC of infinite-state systems
Core: incremental Simplex that allows addition of equalities, inequalities, and disequalities
Modern SAT solver implemented in Ocaml
A. Armando and E. Giunchiglia, “Embedding Complex Decision Procedures inside an Interactive Theorem Prover,” Annals of Mathematics and Artificial Intelligence, 8(3-4): , 1993.
GETFOL
PTAUT: Basic DPPL-like procedure for deciding satisfiability of non-CNF formulas
PTAUTEQ: Equality propagation procedure (creates an equivalence partition on terms)
tautren, UE-dec: procedures to convert quantified FOL formulas to a quantifier-free FOL formulas
F. Giunchiglia and R. Sebastiani, “Building Decision Procedures for Modal Logics from Propositional Decision Procedures – The Case Study of Modal K(m),” CADE 1996.
KSAT
Theory: modal logics
Approach:
Propositional non-CNF solver based on DPLL using Bohm decision heuristic (fastest at the time)
Modal decider invoked when propositional decider return SAT
Enhancement: check modal decider before a split in propositional decider if partial assignment is likely to be unsatisfiable
R. Sebastiani, “Integrating SAT Solvers with Math Reasoners: Foundations and Basic Algorithms,” Tech report , ITC-IRST, November 2001.
MATH-SAT
G. Audemard, P. Bertoli, A. Cimatti, A. Kornilowicz and R. Sebastiani, “A SAT Based Approach for Solving Formulas over Boolean and Linear mathematical Propositions,” CADE 2002.
L0: propositional solver based on DPLL (SIM Library}
L1: equality solver and propagator (equivalence classes)
L2: difference inequality solver (Bellman-Ford)
L3: Simplex-based_LP solver
L4: negated equality checker and splitter
M. Bozzano, R. Bruttomesso, A. Cimatti, T. Junttila, P. van Rossum, S. Schulz, and R. Sebastiani, “An Incremental and Layered Procedure for the Satisfiability of Linear Arithmetic Logic,” TACAS 2005
Theory-driven backjumping (non-chronological backtracking)
Theory-driven learning
Theory-driven deduction (implication of theory atoms)
Theory solver is incremental and backtrackable (maintains state)
Clustering
Layering
Equational solver: congruence closure algorithm of Nieuwenhuis and Oliveras
Real Linear arithmetic solver: Cassowary
Disequalities checked against above
Integer linear arithmetic solver: Cassowary branch-and-cut
Finally, Omega (Fourier-Motzkin)
Boolean solver: Minisat
S. Wolfman, “The LPSAT Engine & its Application to Resourec Planning,” IJCAI 1999.
LPSAT
Inspired by Nelson-Oppen
Propositional solver: RELSAT (learns and backjumps)
Linear solver: Cassowary
Incremental
Allows for identifying a minimal infeasible subset of linear constraints by minimizing their “error”
C. Barrett, D. Dill, and J. Levitt, “Valdity Checking for Combinations of Theories with Equality,” FMCAD 1996.
SVC
Builds on Shostak’s congruence closure algorithm bust uses BDD-like datastructure to avoid conversion to DNF
Formula is simplified using splits and re-wrire rules
A first attempt at capturing Boolean structure
C. Barrett, D. Dill, and A. Stump, “Checking Satisfiability of First-Order Formulas by Incremental Translation to SAT,” CAV A Stump, C. barrett, and D. Dill, “CVC: A Cooperating Validity Checker,” CAV 2004.
CVC
Theories: uninterpreted functions, arithmetic, arrays, abstract data types using a variant of Nelson-Oppen
SAT solver (chaff) calls theory solver upon returning SAT
In case of a theory conflict, theory solver returns a (not necessarily minimal) conflict clause and restarts the SAT solver. This is called an abstract proof.
Notification policy:
Lazy: at end of execution of the SAT solver (theory solver is off-line)
Eager: in response to decisions made and revoked by SAT solver (theory solver is online)
C. Barrett and Sergey Berezin, “CVC Lite: A New Implementation of the Cooperating Validity Checker,” CAV 2004.
CVC Lite
An optimized implementation of CVC that adds an API, proof support, and quantifiers.
D. Detlefs, G. Nelson, and J. Saxe, “Simplify: A Theorem prover for Program Checking,” JACM 2005
Simplify
A very detailed description (109 pages)
Implements Nelson-Oppen procedure and supports quantification
Simple propositional search strategy (not a modern SAT algorithm)
C. Flanagan, R. Joshi, X. Ou, and J. Saxe, “Theorem Proving Using Lazy Proof Explication,” CAV 2003.
Verifun
Problems with Simplify:
Propositional SAT solver not competitive with more recent solvers
Theory conflicts did not guide propositional search (no theory learning or proofs of infeasibility)
Theory solvers generate “explicated” clauses which act similarly to conflict clauses in modern SAT solvers
Theories (they are non-backtracking, i.e., non-incremental, i.e. they don’t hold state)
EUF: based on Nelson-Oppen E-graph data structure for transitivity and congruence
Rational linear arithmetic: variation of Simplex
Arrays: pattern matching
Algorithmic Enhancements: (Use graphs in presentation)
Online (incremental) vs. Offline (non-incremental) SAT solving
Partial vs. complete truth assignments
Proxy reuse (use same prop variables for identical subformulas)
Eager transitivity: reduces # of refinement iterations, but not necessarily runtime (rreminicient of adding all consensus clauses to a SAT solver!)
Granularity of explication (size of infeasibility proof)
C. Tinelli, “A DPLL-based Calculus for Ground Satisfiability Modulo Theories,” JELIA (Europ. Conf. on Logic in AI), [Tin02]
Formalizes the integration of theories within a DPLL framework in a declarative fashion
Discusses implications of including theories within DPLL on search heuristics (e.g., branching heuristics, etc.)
DPLL(T) represents a tight integration of the propositional and theory solvers
R. Nieuwenhuis and A. Oliveras, “Congruence Closure with Integer Offsets,” LPAR (Conf. on Logic for Programming, Artificial Intelligence, and Reasoning) [NO03]
“Improved” implementation of congruence closure without conversion to graph
Incorporation of integer offsets (to handle succ/pred as in CLU)
Mentions DPLL(X) modeled after CLP(X)
H. Ganzinger, G. Hagen, R. Nieuwenhuis, A. Oliveras, and C. Tinelli, “DPLL(T): Fast Decision Procedures,” CAV 2004.
DPLL(T)
Builds on [Tin02] and [NO03]
Theories: EUF + succ/pred
DPLL(X) is modeled after Chaff; congruence closure with offsets as in [NO03]
Discusses propagtaion through the theory solver
Solvers with no papers:
Sammy
Uses SATO’s library for DPLL(X)
Uses CVCLite as theory solver
BarcelogicTools
Another implementation of DPLL(T)
Theories: EUF with succ/pred, integer and real DL
Yices
Succesor to ICS??
Ario
2005
DPLL(T)
2002
BarcelogicTools
2005
GETFOL
1993
KSAT
1996
MATH-SAT
2001
MATH-SAT
2002
MATH-SAT
2005

47. Nelson-Oppen 1979 LPSAT 1999 Simplify 1998? Verifun 2003 SVC 1996 CVC
2002/2004
CVC Lite
2004
Shostak
1984
ICS
2001
ICS+Chaff
2002
Simplics
2005
Yices
2006
D. Cyrluk, P. Lincoln, and N. Shankar, “On Shostak’s Decision Procedure for Combinations of Theories,” CADE 1996
Compares the Nelson-Oppen and Shostak congruence closure procedures and identifies the three optimizations in Shostak’s procedure
Explains how so-called sigma-theories (i.e., theories that are canonizable and algebraically solvable) can be integrated much more closely than in the Nelson-Oppen framework, leading to much more efficient equality propagation (necessarily duplicated in the Neslon-Oppen approach)
Identifies the bugs in Shostak’s original algorithm
Suggests (for future) making Shostak a component in a Nelson-Oppen framework.
J.-C. Filliatre, S. Owre, H. Ruess, and N. Shankar, “ICS: Integrated Canonizer and Solver,” CAV 2001.
Implements (corrected) Shostak procedure extended with arithemtic inequalities
Has a flexible API for integration with other provers and simulators
Theories:
UF with equality and disequality
Rational linear arithmetic (with inequalities)
Sets
Tuples
Arrays
Bit vectors
L. de Moura and H. Ruess, “Lemmas of Demand for Satisfiability Solvers,” SAT 2002
Lazy integration of ICS and Chaff
Procedure for creating lemmas that “explain” the infeasibility of conjunctions of theory atoms
Mentions inefficiency of restarting the SAT solver after adding theory-induced lemmas
B. Dutertre and L. De Moura, “Simplics: Tools Description,” 2005
Successor to ICS that is limited to the theory of linear real arithmetic
Main application: BMC of infinite-state systems
Core: incremental Simplex that allows addition of equalities, inequalities, and disequalities
Modern SAT solver implemented in Ocaml
A. Armando and E. Giunchiglia, “Embedding Complex Decision Procedures inside an Interactive Theorem Prover,” Annals of Mathematics and Artificial Intelligence, 8(3-4): , 1993.
GETFOL
PTAUT: Basic DPPL-like procedure for deciding satisfiability of non-CNF formulas
PTAUTEQ: Equality propagation procedure (creates an equivalence partition on terms)
tautren, UE-dec: procedures to convert quantified FOL formulas to a quantifier-free FOL formulas
F. Giunchiglia and R. Sebastiani, “Building Decision Procedures for Modal Logics from Propositional Decision Procedures – The Case Study of Modal K(m),” CADE 1996.
KSAT
Theory: modal logics
Approach:
Propositional non-CNF solver based on DPLL using Bohm decision heuristic (fastest at the time)
Modal decider invoked when propositional decider return SAT
Enhancement: check modal decider before a split in propositional decider if partial assignment is likely to be unsatisfiable
R. Sebastiani, “Integrating SAT Solvers with Math Reasoners: Foundations and Basic Algorithms,” Tech report , ITC-IRST, November 2001.
MATH-SAT
G. Audemard, P. Bertoli, A. Cimatti, A. Kornilowicz and R. Sebastiani, “A SAT Based Approach for Solving Formulas over Boolean and Linear mathematical Propositions,” CADE 2002.
L0: propositional solver based on DPLL (SIM Library}
L1: equality solver and propagator (equivalence classes)
L2: difference inequality solver (Bellman-Ford)
L3: Simplex-based_LP solver
L4: negated equality checker and splitter
M. Bozzano, R. Bruttomesso, A. Cimatti, T. Junttila, P. van Rossum, S. Schulz, and R. Sebastiani, “An Incremental and Layered Procedure for the Satisfiability of Linear Arithmetic Logic,” TACAS 2005
Theory-driven backjumping (non-chronological backtracking)
Theory-driven learning
Theory-driven deduction (implication of theory atoms)
Theory solver is incremental and backtrackable (maintains state)
Clustering
Layering
Equational solver: congruence closure algorithm of Nieuwenhuis and Oliveras
Real Linear arithmetic solver: Cassowary
Disequalities checked against above
Integer linear arithmetic solver: Cassowary branch-and-cut
Finally, Omega (Fourier-Motzkin)
Boolean solver: Minisat
S. Wolfman, “The LPSAT Engine & its Application to Resourec Planning,” IJCAI 1999.
LPSAT
Inspired by Nelson-Oppen
Propositional solver: RELSAT (learns and backjumps)
Linear solver: Cassowary
Incremental
Allows for identifying a minimal infeasible subset of linear constraints by minimizing their “error”
C. Barrett, D. Dill, and J. Levitt, “Valdity Checking for Combinations of Theories with Equality,” FMCAD 1996.
SVC
Builds on Shostak’s congruence closure algorithm bust uses BDD-like datastructure to avoid conversion to DNF
Formula is simplified using splits and re-wrire rules
A first attempt at capturing Boolean structure
C. Barrett, D. Dill, and A. Stump, “Checking Satisfiability of First-Order Formulas by Incremental Translation to SAT,” CAV A Stump, C. barrett, and D. Dill, “CVC: A Cooperating Validity Checker,” CAV 2004.
CVC
Theories: uninterpreted functions, arithmetic, arrays, abstract data types using a variant of Nelson-Oppen
SAT solver (chaff) calls theory solver upon returning SAT
In case of a theory conflict, theory solver returns a (not necessarily minimal) conflict clause and restarts the SAT solver. This is called an abstract proof.
Notification policy:
Lazy: at end of execution of the SAT solver (theory solver is off-line)
Eager: in response to decisions made and revoked by SAT solver (theory solver is online)
C. Barrett and Sergey Berezin, “CVC Lite: A New Implementation of the Cooperating Validity Checker,” CAV 2004.
CVC Lite
An optimized implementation of CVC that adds an API, proof support, and quantifiers.
D. Detlefs, G. Nelson, and J. Saxe, “Simplify: A Theorem prover for Program Checking,” JACM 2005
Simplify
A very detailed description (109 pages)
Implements Nelson-Oppen procedure and supports quantification
Simple propositional search strategy (not a modern SAT algorithm)
C. Flanagan, R. Joshi, X. Ou, and J. Saxe, “Theorem Proving Using Lazy Proof Explication,” CAV 2003.
Verifun
Problems with Simplify:
Propositional SAT solver not competitive with more recent solvers
Theory conflicts did not guide propositional search (no theory learning or proofs of infeasibility)
Theory solvers generate “explicated” clauses which act similarly to conflict clauses in modern SAT solvers
Theories (they are non-backtracking, i.e., non-incremental, i.e. they don’t hold state)
EUF: based on Nelson-Oppen E-graph data structure for transitivity and congruence
Rational linear arithmetic: variation of Simplex
Arrays: pattern matching
Algorithmic Enhancements: (Use graphs in presentation)
Online (incremental) vs. Offline (non-incremental) SAT solving
Partial vs. complete truth assignments
Proxy reuse (use same prop variables for identical subformulas)
Eager transitivity: reduces # of refinement iterations, but not necessarily runtime (rreminicient of adding all consensus clauses to a SAT solver!)
Granularity of explication (size of infeasibility proof)
C. Tinelli, “A DPLL-based Calculus for Ground Satisfiability Modulo Theories,” JELIA (Europ. Conf. on Logic in AI), [Tin02]
Formalizes the integration of theories within a DPLL framework in a declarative fashion
Discusses implications of including theories within DPLL on search heuristics (e.g., branching heuristics, etc.)
DPLL(T) represents a tight integration of the propositional and theory solvers
R. Nieuwenhuis and A. Oliveras, “Congruence Closure with Integer Offsets,” LPAR (Conf. on Logic for Programming, Artificial Intelligence, and Reasoning) [NO03]
“Improved” implementation of congruence closure without conversion to graph
Incorporation of integer offsets (to handle succ/pred as in CLU)
Mentions DPLL(X) modeled after CLP(X)
H. Ganzinger, G. Hagen, R. Nieuwenhuis, A. Oliveras, and C. Tinelli, “DPLL(T): Fast Decision Procedures,” CAV 2004.
DPLL(T)
Builds on [Tin02] and [NO03]
Theories: EUF + succ/pred
DPLL(X) is modeled after Chaff; congruence closure with offsets as in [NO03]
Discusses propagtaion through the theory solver
Solvers with no papers:
Sammy
Uses SATO’s library for DPLL(X)
Uses CVCLite as theory solver
BarcelogicTools
Another implementation of DPLL(T)
Theories: EUF with succ/pred, integer and real DL
Yices
Succesor to ICS??
Sammy
2005
Ario
2005
DPLL(T)
2002
BarcelogicTools
2005
GETFOL
1993
KSAT
1996
MATH-SAT
2001
MATH-SAT
2002
MATH-SAT
2005

48. 2005 Competition Results: QF_UF

49. 2005 Competition Results: QF_RDL

50. 2005 Competition Results: QF_IDL

51. 2005 Competition Results: QF_UFIDL

52. 2005 Competition Results: QF_LRA

53. 2005 Competition Results: QF_LIA

54. 2005 Competition Results: QF_AUFLIA

55. Conclusions SAT does it again!
Modern SAT technology critical enabler of SMT solvers
Clear winner: tight integration of SAT and Theory solvers
Incremental propagation
Incremental conflict analysis and learning
Careful tuning
SMT is bringing different communities together (SAT, CP, AI, OR)
Competition is good