δ-Decidability over the Reals
From SAT to SMT: The light at the end of a tunel
One Solver to rule them all,
One Solver to find them,
One Solver to bring them all,
and in the darkness bind them.
Goal:
A universal tool for solving problems.
A universal language for defining problems
??
?
From SAT to SMT: The light at the end of a tunel
SAT
(Boolean Satisfiability Problem)
NP-complete:
1-planarity, 3-dimensional matching, Bipartite dimension, Capacitated minimum spanning tree, Clique problem, Complete coloring, Domatic number, Dominating set, Bandwidth problem, Clique cover problem, Rank coloring, Degree-constrained spanning tree, Feedback vertex set, Feedback arc set, Graph homomorphism problem, Graph coloring, Hamiltonian completion, Longest path problem, Maximum independent set, Maximum Induced path, Graph intersection number, Metric dimension of a graph, Minimum k-cut, Pathwidth, Set splitting problem, Shortest total path length spanning tree, Slope number two testing, Treewidth, Vertex cover, Mathematical programming, 3-partition problem, Bin packing problem, Knapsack problem and several variants, Bottleneck traveling salesman, Numerical 3-dimensional matching, Partition problem, Quadratic assignment problem, Quadratic programming, Subset sum problem, Formal languages and string processing, Closest string, Longest common subsequence problem, The bounded variant of the Post correspondence problem, Shortest common supersequence, String-to-string correction problem, ...
From SAT to SMT: The light at the end of a tunel
SAT's hidden secret
is
From SAT to SMT: The light at the end of a tunel
QBF
PSPACE-complete:
Uhm...actually, it's
but never mind
Quantified boolean formulas, Stochastic satisfiability, Linear temporal logic satisfiability and model checking, Type inhabitation problem for simply typed lambda calculus, Integer circuit evaluation, Word problem for linear bounded automata, Word problem for quasi-realtime automata, Emptiness problem for a nondeterministic two-way finite state automaton, Equivalence problem for nondeterministic finite automata, Word problem and emptiness problem for non-erasing stack automata, Deterministic finite automata intersection emptiness, A generalized version of Langton's Ant, Minimizing nondeterministic finite automata, planarity of succinct graphs, acyclicity of succinct graphs, connectedness of succinct graphs, existence of Eulerian paths in a succinct graph, Canadian traveller problem, Dynamic graph reliability, Deterministic constraint logic (unbounded), Nondeterministic Constraint Logic (unbounded), Bounded two-player Constraint Logic, Word problem for context-sensitive language, Regular language intersection, Regular expression star freeness, Equivalence problem for regular expressions, Emptiness problem for regular expressions with intersection, Equivalence problem for star-free regular expressions with squaring, Covering for linear grammars, Structural equivalence for linear grammars, Equivalence problem for Regular grammars, Emptiness problem for ET0L grammars, Word problem for ET0L grammars
(Quantified Boolean Formulae)
From SAT to SMT: The light at the end of a tunel
SMT
(Satisfiability Modulo Theories)
UNDECIDABLE
(for real numbers)
1948: A. Tarski
A decision method for elementary algebra and geometry
(quantified polynomials over reals)
Complexity: EXPSPACE ... uff...
From SAT to SMT: The light at the end of a tunel
SMT
(Satisfiability Modulo Theories)
UNDECIDABLE
(for real numbers
transcendental functions)
A: You look so sad! What happened?
B: Universe is continuous and undecidable. We will never be able to understand it!
A: Are you sure?
B: Pretty sure, sine is undecidable.
A: No. Are you sure universe is continuous?
B: Well... I was...
The honest answer is "We don't know". Physical theories do not describe how the universe actually works, the only thing we know is that their predictions match experimental results.
δ-Decidability over the Reals
The universe does not care, so why should we?
Assuming x is computable with arbitrary precision, it can be mapped to f(x) with arbitrary precision.
Common functions are computable: addition, multiplication, min, max, exp, sin, Lipschitz-continuous ODE...
Important properties:
??!
We can compute necessary precision of inputs assuming we know the final desired precision.
The loss of precision is finite and quantifiable.
Reasoning about Type 2 functions
Where do the formulae come from?
- Bounded quantifiers - will be needed later
- Negation is pushed down to propositions
Bring in the δ!
δ-Decidability vs. δ-Robustness
δ-Decidability vs. δ-Robustness
δ-Decidability (Proof)
(Similar to δ-strengthening and δ-weakening)
δ-Decidability (Proof)
δ-Decidability (Proof)
Observations:
Complexity, revisited
- SAT: NP
- QBF: PSPACE
- SMT over Real Closed Fields: EXPSPACE
- SMT over Transcendental fun.: ?!
Complexity, revisited
- SAT: NP
- QBF: PSPACE
- SMT with δ-Decidability: PSPACE
- SMT over Real Closed Fields: EXPSPACE
- SMT over Transcendental fun.: ?!
delta-SMT
By Samuel Pastva
delta-SMT
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