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SMT Techniques and Solvers in Automated Termination Analysis Carsten Fuhs Birkbeck, University of London 2 nd July 2016 14 th Workshop on SAT Modulo Theories (SMT) Coimbra, Portugal Why analyze termination? 2/25 Why analyze termination? 1

  1. Example (Division)  minus ( x, 0 ) → x   minus ( s ( x ) , s ( y )) → minus ( x, y )  R = quot ( 0 , s ( y )) → 0   quot ( s ( x ) , s ( y )) → s ( quot ( minus ( x, y ) , s ( y )))  minus ♯ ( s ( x ) , s ( y )) minus ♯ ( x, y )  →  minus ♯ ( x, y ) quot ♯ ( s ( x ) , s ( y )) DP = → quot ♯ ( s ( x ) , s ( y )) quot ♯ ( minus ( x, y ) , s ( y )) →  Dependency Pairs [Arts, Giesl, TCS ’00 ] For TRS R build dependency pairs DP ( ∼ function calls) Show: No ∞ call sequence with DP (eval of DP ’s args via R ) Dependency Pair Framework [Giesl et al, JAR ’06 ] (simplified): while DP � = ∅ : 10/25

  2. Example (Division)  minus ( x, 0 ) � x   minus ( s ( x ) , s ( y )) � minus ( x, y )  R = quot ( 0 , s ( y )) � 0   quot ( s ( x ) , s ( y )) � s ( quot ( minus ( x, y ) , s ( y )))  minus ♯ ( s ( x ) , s ( y )) minus ♯ ( x, y )  �  minus ♯ ( x, y ) quot ♯ ( s ( x ) , s ( y )) DP = � quot ♯ ( s ( x ) , s ( y )) quot ♯ ( minus ( x, y ) , s ( y )) �  Dependency Pairs [Arts, Giesl, TCS ’00 ] For TRS R build dependency pairs DP ( ∼ function calls) Show: No ∞ call sequence with DP (eval of DP ’s args via R ) Dependency Pair Framework [Giesl et al, JAR ’06 ] (simplified): while DP � = ∅ : find well-founded order ≻ with DP ∪ R ⊆ � 10/25

  3. Example (Division)  minus ( x, 0 ) � x   minus ( s ( x ) , s ( y )) � minus ( x, y )  R = quot ( 0 , s ( y )) � 0   quot ( s ( x ) , s ( y )) � s ( quot ( minus ( x, y ) , s ( y )))   minus ♯ ( s ( x ) , s ( y )) minus ♯ ( x, y ) ( � )   quot ♯ ( s ( x ) , s ( y )) minus ♯ ( x, y ) DP = ( � ) quot ♯ ( s ( x ) , s ( y )) quot ♯ ( minus ( x, y ) , s ( y )) ( � )   Dependency Pairs [Arts, Giesl, TCS ’00 ] For TRS R build dependency pairs DP ( ∼ function calls) Show: No ∞ call sequence with DP (eval of DP ’s args via R ) Dependency Pair Framework [Giesl et al, JAR ’06 ] (simplified): while DP � = ∅ : find well-founded order ≻ with DP ∪ R ⊆ � delete s → t with s ≻ t from DP 10/25

  4. Example (Division)  minus ( x, 0 ) � x   minus ( s ( x ) , s ( y )) � minus ( x, y )  R = quot ( 0 , s ( y )) � 0   quot ( s ( x ) , s ( y )) � s ( quot ( minus ( x, y ) , s ( y )))   minus ♯ ( s ( x ) , s ( y )) minus ♯ ( x, y ) ( � )   quot ♯ ( s ( x ) , s ( y )) minus ♯ ( x, y ) DP = ( � ) quot ♯ ( s ( x ) , s ( y )) quot ♯ ( minus ( x, y ) , s ( y )) ( � )   Dependency Pairs [Arts, Giesl, TCS ’00 ] For TRS R build dependency pairs DP ( ∼ function calls) Show: No ∞ call sequence with DP (eval of DP ’s args via R ) Dependency Pair Framework [Giesl et al, JAR ’06 ] (simplified): while DP � = ∅ : find well-founded order ≻ with DP ∪ R ⊆ � delete s → t with s ≻ t from DP Find ≻ automatically and efficiently 10/25

  5. Polynomial interpretations Get ≻ via polynomial interpretations [ · ] over N [Lankford ’79] → ranking functions for rewriting Example minus ( s ( x ) , s ( y )) � minus ( x, y ) 11/25

  6. Polynomial interpretations Get ≻ via polynomial interpretations [ · ] over N [Lankford ’79] → ranking functions for rewriting Example minus ( s ( x ) , s ( y )) � minus ( x, y ) Use [ · ] with [ minus ] ( x 1 , x 2 ) = x 1 [ s ] ( x 1 ) = x 1 + 1 11/25

  7. Polynomial interpretations Get ≻ via polynomial interpretations [ · ] over N [Lankford ’79] → ranking functions for rewriting Example ∀ x, y. x + 1 = [ minus ( s ( x ) , s ( y )) ] ≥ [ minus ( x, y ) ] = x Use [ · ] with [ minus ] ( x 1 , x 2 ) = x 1 [ s ] ( x 1 ) = x 1 + 1 Extend to terms: [ x ] = x [ f ( t 1 , . . . , t n ) ] = [ f ] ( [ t 1 ] , . . . , [ t n ] ) ≻ boils down to > over N 11/25

  8. Example (Constraints for Division)  minus ( x, 0 ) � x   minus ( s ( x ) , s ( y )) minus ( x, y ) �  R = quot ( 0 , s ( y )) � 0   quot ( s ( x ) , s ( y )) � s ( quot ( minus ( x, y ) , s ( y )))   minus ♯ ( s ( x ) , s ( y )) minus ♯ ( x, y ) ( � )   quot ♯ ( s ( x ) , s ( y )) minus ♯ ( x, y ) DP = ( � ) quot ♯ ( s ( x ) , s ( y )) quot ♯ ( minus ( x, y ) , s ( y )) ( � )   12/25

  9. Example (Constraints for Division)  minus ( x, 0 ) � x   minus ( s ( x ) , s ( y )) � minus ( x, y )  R = quot ( 0 , s ( y )) � 0   quot ( s ( x ) , s ( y )) � s ( quot ( minus ( x, y ) , s ( y )))  minus ♯ ( s ( x ) , s ( y )) minus ♯ ( x, y )  ≻  quot ♯ ( s ( x ) , s ( y )) minus ♯ ( x, y ) DP = ≻ quot ♯ ( s ( x ) , s ( y )) quot ♯ ( minus ( x, y ) , s ( y )) ≻  Use interpretation [ · ] over N with [ quot ♯ ] ( x 1 , x 2 ) = x 1 + x 2 . [ quot ] ( x 1 , x 2 ) = x 1 + x 2 [ minus ♯ ] ( x 1 , x 2 ) = x 1 [ minus ] ( x 1 , x 2 ) = x 1 [ 0 ] = 0 [ s ] ( x 1 ) = x 1 + 1 � order solves all constraints 12/25

  10. Example (Constraints for Division)  minus ( x, 0 ) � x   minus ( s ( x ) , s ( y )) � minus ( x, y )  R = quot ( 0 , s ( y )) � 0   quot ( s ( x ) , s ( y )) � s ( quot ( minus ( x, y ) , s ( y )))    DP =  Use interpretation [ · ] over N with [ quot ♯ ] ( x 1 , x 2 ) = x 1 + x 2 . [ quot ] ( x 1 , x 2 ) = x 1 + x 2 [ minus ♯ ] ( x 1 , x 2 ) = x 1 [ minus ] ( x 1 , x 2 ) = x 1 [ 0 ] = 0 [ s ] ( x 1 ) = x 1 + 1 � order solves all constraints � DP = ∅ � termination of division algorithm proved � 12/25

  11. Automation Task: Solve minus ( s ( x ) , s ( y )) � minus ( x, y ) 13/25

  12. Automation Task: Solve minus ( s ( x ) , s ( y )) � minus ( x, y ) 1 Fix a degree, use pol. interpretation with parametric coefficients: [ minus ] ( x, y ) = a m + b m x + c m y, [ s ] ( x ) = a s + b s x 13/25

  13. Automation Task: Solve minus ( s ( x ) , s ( y )) � minus ( x, y ) 1 Fix a degree, use pol. interpretation with parametric coefficients: [ minus ] ( x, y ) = a m + b m x + c m y, [ s ] ( x ) = a s + b s x 2 From term constraint to polynomial constraint: s � t � [ s ] ≥ [ t ] Here: ∀ x, y. ( a s b m + a s c m ) + ( b s b m − b m ) x + ( b s c m − c m ) y ≥ 0 13/25

  14. Automation Task: Solve minus ( s ( x ) , s ( y )) � minus ( x, y ) 1 Fix a degree, use pol. interpretation with parametric coefficients: [ minus ] ( x, y ) = a m + b m x + c m y, [ s ] ( x ) = a s + b s x 2 From term constraint to polynomial constraint: s � t � [ s ] ≥ [ t ] Here: ∀ x, y. ( a s b m + a s c m ) + ( b s b m − b m ) x + ( b s c m − c m ) y ≥ 0 3 Eliminate ∀ x, y by absolute positiveness criterion [Hong, Jakuš, JAR ’98 ] : a s b m + a s c m ≥ 0 ∧ b s b m − b m ≥ 0 ∧ b s c m − c m ≥ 0 Here: 13/25

  15. Automation Task: Solve minus ( s ( x ) , s ( y )) � minus ( x, y ) 1 Fix a degree, use pol. interpretation with parametric coefficients: [ minus ] ( x, y ) = a m + b m x + c m y, [ s ] ( x ) = a s + b s x 2 From term constraint to polynomial constraint: s � t � [ s ] ≥ [ t ] Here: ∀ x, y. ( a s b m + a s c m ) + ( b s b m − b m ) x + ( b s c m − c m ) y ≥ 0 3 Eliminate ∀ x, y by absolute positiveness criterion [Hong, Jakuš, JAR ’98 ] : a s b m + a s c m ≥ 0 ∧ b s b m − b m ≥ 0 ∧ b s c m − c m ≥ 0 Here: 13/25

  16. Automation Task: Solve minus ( s ( x ) , s ( y )) � minus ( x, y ) 1 Fix a degree, use pol. interpretation with parametric coefficients: [ minus ] ( x, y ) = a m + b m x + c m y, [ s ] ( x ) = a s + b s x 2 From term constraint to polynomial constraint: s � t � [ s ] ≥ [ t ] Here: ∀ x, y. ( a s b m + a s c m ) + ( b s b m − b m ) x + ( b s c m − c m ) y ≥ 0 3 Eliminate ∀ x, y by absolute positiveness criterion [Hong, Jakuš, JAR ’98 ] : a s b m + a s c m ≥ 0 ∧ b s b m − b m ≥ 0 ∧ b s c m − c m ≥ 0 Here: 13/25

  17. Automation Task: Solve minus ( s ( x ) , s ( y )) � minus ( x, y ) 1 Fix a degree, use pol. interpretation with parametric coefficients: [ minus ] ( x, y ) = a m + b m x + c m y, [ s ] ( x ) = a s + b s x 2 From term constraint to polynomial constraint: s � t � [ s ] ≥ [ t ] Here: ∀ x, y. ( a s b m + a s c m ) + ( b s b m − b m ) x + ( b s c m − c m ) y ≥ 0 3 Eliminate ∀ x, y by absolute positiveness criterion [Hong, Jakuš, JAR ’98 ] : a s b m + a s c m ≥ 0 ∧ b s b m − b m ≥ 0 ∧ b s c m − c m ≥ 0 Here: Non-linear constraints (QF_NIA), even for linear interpretations 13/25

  18. Automation Task: Solve minus ( s ( x ) , s ( y )) � minus ( x, y ) 1 Fix a degree, use pol. interpretation with parametric coefficients: [ minus ] ( x, y ) = a m + b m x + c m y, [ s ] ( x ) = a s + b s x 2 From term constraint to polynomial constraint: s � t � [ s ] ≥ [ t ] Here: ∀ x, y. ( a s b m + a s c m ) + ( b s b m − b m ) x + ( b s c m − c m ) y ≥ 0 3 Eliminate ∀ x, y by absolute positiveness criterion [Hong, Jakuš, JAR ’98 ] : a s b m + a s c m ≥ 0 ∧ b s b m − b m ≥ 0 ∧ b s c m − c m ≥ 0 Here: Non-linear constraints (QF_NIA), even for linear interpretations Task: Show satisfiability of non-linear constraints over N � Prove termination of given term rewrite system 13/25

  19. Extensions Polynomials with negative coefficients and max-operator [Hirokawa, Middeldorp, IC ’07 ; Fuhs et al, SAT ’07, RTA ’08 ] models behavior of functions more closely automation via SMT for QF_NIA, more complex Boolean structure 14/25

  20. Extensions Polynomials with negative coefficients and max-operator [Hirokawa, Middeldorp, IC ’07 ; Fuhs et al, SAT ’07, RTA ’08 ] models behavior of functions more closely automation via SMT for QF_NIA, more complex Boolean structure Polynomials over Q + and R + [Lucas, RAIRO ’05 ] non-integer coefficients increase proving power SMT-based automation [Fuhs et al, AISC ’08 ; Zankl, Middeldorp, LPAR ’10 ; Borralleras et al, JAR ’12 ] 14/25

  21. Extensions Polynomials with negative coefficients and max-operator [Hirokawa, Middeldorp, IC ’07 ; Fuhs et al, SAT ’07, RTA ’08 ] models behavior of functions more closely automation via SMT for QF_NIA, more complex Boolean structure Polynomials over Q + and R + [Lucas, RAIRO ’05 ] non-integer coefficients increase proving power SMT-based automation [Fuhs et al, AISC ’08 ; Zankl, Middeldorp, LPAR ’10 ; Borralleras et al, JAR ’12 ] Matrix interpretations [Endrullis, Waldmann, Zantema, JAR ’08 ] interpretation to vectors over N k , coefficients are matrices useful for deeply nested terms QF_NIA instances with more complex atoms 14/25

  22. Extensions Polynomials with negative coefficients and max-operator [Hirokawa, Middeldorp, IC ’07 ; Fuhs et al, SAT ’07, RTA ’08 ] models behavior of functions more closely automation via SMT for QF_NIA, more complex Boolean structure Polynomials over Q + and R + [Lucas, RAIRO ’05 ] non-integer coefficients increase proving power SMT-based automation [Fuhs et al, AISC ’08 ; Zankl, Middeldorp, LPAR ’10 ; Borralleras et al, JAR ’12 ] Matrix interpretations [Endrullis, Waldmann, Zantema, JAR ’08 ] interpretation to vectors over N k , coefficients are matrices useful for deeply nested terms QF_NIA instances with more complex atoms “Arctic” matrices on the max-plus semiring on N or Z (instead of plus-times) [Koprowski, Waldmann, Acta Cyb. ’09 ] very useful for deeply nested terms can be encoded to QF_ L IA, but (unary!) bit-blasting seems to be faster in practice [Codish, Fekete, Fuhs, Giesl, Waldmann, SMT ’12 ] 14/25

  23. Extensions Polynomials with negative coefficients and max-operator [Hirokawa, Middeldorp, IC ’07 ; Fuhs et al, SAT ’07, RTA ’08 ] models behavior of functions more closely automation via SMT for QF_NIA, more complex Boolean structure Polynomials over Q + and R + [Lucas, RAIRO ’05 ] non-integer coefficients increase proving power SMT-based automation [Fuhs et al, AISC ’08 ; Zankl, Middeldorp, LPAR ’10 ; Borralleras et al, JAR ’12 ] Matrix interpretations [Endrullis, Waldmann, Zantema, JAR ’08 ] interpretation to vectors over N k , coefficients are matrices useful for deeply nested terms QF_NIA instances with more complex atoms “Arctic” matrices on the max-plus semiring on N or Z (instead of plus-times) [Koprowski, Waldmann, Acta Cyb. ’09 ] very useful for deeply nested terms can be encoded to QF_ L IA, but (unary!) bit-blasting seems to be faster in practice [Codish, Fekete, Fuhs, Giesl, Waldmann, SMT ’12 ] 14/25

  24. Example (bits)  half ( 0 ) → 0 bits ( 0 ) → 0  R = half ( s ( 0 )) → 0 bits ( s ( x )) → s ( bits ( half ( s ( x )))) half ( s ( s ( x ))) → s ( half ( x ))  15/25

  25. Example (bits)  half ( 0 ) → 0 bits ( 0 ) → 0  R = half ( s ( 0 )) → 0 bits ( s ( x )) → s ( bits ( half ( s ( x )))) half ( s ( s ( x ))) → s ( half ( x ))  half ♯ ( s ( s ( x ))) half ♯ ( x )  →  bits ♯ ( s ( x )) half ♯ ( s ( x )) DP = → bits ♯ ( s ( x )) bits ♯ ( half ( s ( x ))) →  15/25

  26. Example (bits)  half ( 0 ) � 0 bits ( 0 ) � 0  R = half ( s ( 0 )) � 0 bits ( s ( x )) � s ( bits ( half ( s ( x )))) half ( s ( s ( x ))) � s ( half ( x ))   half ♯ ( s ( s ( x ))) half ♯ ( x ) ( � )   bits ♯ ( s ( x )) half ♯ ( s ( x )) DP = ( � ) bits ♯ ( s ( x )) bits ♯ ( half ( s ( x ))) ( � )   15/25

  27. Example (bits)  half ( 0 ) � 0 bits ( 0 ) � 0  R = half ( s ( 0 )) � 0 bits ( s ( x )) � s ( bits ( half ( s ( x )))) half ( s ( s ( x ))) � s ( half ( x ))    DP = bits ♯ ( s ( x )) bits ♯ ( half ( s ( x ))) ≻  15/25

  28. Example (bits)  half ( 0 ) � 0 bits ( 0 ) � 0  R = half ( s ( 0 )) � 0 bits ( s ( x )) � s ( bits ( half ( s ( x )))) half ( s ( s ( x ))) � s ( half ( x ))    DP = bits ♯ ( s ( x )) bits ♯ ( half ( s ( x ))) ≻  Classic polynomials cannot solve bits ♯ ( s ( x )) ≻ bits ♯ ( half ( s ( x ))) 15/25

  29. Example (bits)  half ( 0 ) � 0 bits ( 0 ) � 0  R = half ( s ( 0 )) � 0 bits ( s ( x )) � s ( bits ( half ( s ( x )))) half ( s ( s ( x ))) � s ( half ( x ))    DP = bits ♯ ( s ( x )) bits ♯ ( half ( s ( x ))) ≻  Classic polynomials cannot solve bits ♯ ( s ( x )) ≻ bits ♯ ( half ( s ( x ))) [ bits ♯ ] ( x ) = x, [ half ] ( x ) = x − 1 Remedy: [ s ] ( x ) = x + 1 , 15/25

  30. Example (bits)  half ( 0 ) � 0 bits ( 0 ) � 0  R = half ( s ( 0 )) � 0 bits ( s ( x )) � s ( bits ( half ( s ( x )))) half ( s ( s ( x ))) � s ( half ( x ))    DP = bits ♯ ( s ( x )) bits ♯ ( half ( s ( x ))) ≻  Classic polynomials cannot solve bits ♯ ( s ( x )) ≻ bits ♯ ( half ( s ( x ))) [ bits ♯ ] ( x ) = x, [ half ] ( x ) = x − 1 Remedy: [ s ] ( x ) = x + 1 , But: Then ≻ not well founded any more: 0 ≻ half ( 0 ) ≻ half ( half ( 0 )) ≻ . . . 15/25

  31. Example (bits)  half ( 0 ) � 0 bits ( 0 ) � 0  R = half ( s ( 0 )) � 0 bits ( s ( x )) � s ( bits ( half ( s ( x )))) half ( s ( s ( x ))) � s ( half ( x ))    DP = bits ♯ ( s ( x )) bits ♯ ( half ( s ( x ))) ≻  Classic polynomials cannot solve bits ♯ ( s ( x )) ≻ bits ♯ ( half ( s ( x ))) [ bits ♯ ] ( x ) = x, [ half ] ( x ) = x − 1 Remedy: [ s ] ( x ) = x + 1 , But: Then ≻ not well founded any more: 0 ≻ half ( 0 ) ≻ half ( half ( 0 )) ≻ . . . ⇒ Solution [Hirokawa, Middeldorp, IC ’07 ] : [ half ] ( x 1 ) = max( x 1 − 1 , 0) ⇒ [ half ( s ( x )) ] = max(( x + 1) − 1 , 0) = x 15/25

  32. Example (bits)  half ( 0 ) � 0 bits ( 0 ) � 0  R = half ( s ( 0 )) � 0 bits ( s ( x )) � s ( bits ( half ( s ( x )))) half ( s ( s ( x ))) � s ( half ( x ))    DP =  Classic polynomials cannot solve bits ♯ ( s ( x )) ≻ bits ♯ ( half ( s ( x ))) [ bits ♯ ] ( x ) = x, [ half ] ( x ) = x − 1 Remedy: [ s ] ( x ) = x + 1 , But: Then ≻ not well founded any more: 0 ≻ half ( 0 ) ≻ half ( half ( 0 )) ≻ . . . ⇒ Solution [Hirokawa, Middeldorp, IC ’07 ] : [ half ] ( x 1 ) = max( x 1 − 1 , 0) ⇒ [ half ( s ( x )) ] = max(( x + 1) − 1 , 0) = x 15/25

  33. Problem: Expressions like max( x 1 − 1 , 0) are no polynomials For [ s ] > [ t ], show 16/25

  34. Problem: Expressions like max( x 1 − 1 , 0) are no polynomials For [ s ] > [ t ], show [ s ] left > [ t ] right [ s ] left under-approximation of [ s ] [ t ] right over-approximation of [ t ] [ s ] left , [ t ] right polynomials 16/25

  35. Problem: Expressions like max( x 1 − 1 , 0) are no polynomials For [ s ] > [ t ], show [ s ] left > [ t ] right [ s ] left under-approximation of [ s ] [ t ] right over-approximation of [ t ] [ s ] left , [ t ] right polynomials Automation initially: Generate-and-test Approx. for max( p, 0) depend on signum of constant addend of p [ s ( x ) ] right ⇒ [ s ( x ) ] = max( x + 1 , 0) = x + 1 [ half ( x ) ] right [ half ( x ) ] = max( x − 1 , 0) ⇒ = x 16/25

  36. Problem: Expressions like max( x 1 − 1 , 0) are no polynomials For [ s ] > [ t ], show [ s ] left > [ t ] right [ s ] left under-approximation of [ s ] [ t ] right over-approximation of [ t ] [ s ] left , [ t ] right polynomials Automation initially: Generate-and-test Approx. for max( p, 0) depend on signum of constant addend of p [ s ( x ) ] right ⇒ [ s ( x ) ] = max( x + 1 , 0) = x + 1 [ half ( x ) ] right [ half ( x ) ] = max( x − 1 , 0) ⇒ = x Solution [Fuhs et al, SAT ’07 ] : Encode case analysis . . . [ f ( x ) ] right = a f x 1 + c f ( x ) [ f ( x ) ] = max( a f x 1 + b f , 0) ⇒ . . . using side constraints ( b f ≥ 0 → c f ( x ) = b f ) ∧ ( b f < 0 → c f ( x ) = 0) Boolean structure in SMT quite handy! 16/25

  37. (SAT and) SMT solving for path orders Path orders: based on precedences of function symbols Recursive Path Order [Dershowitz, TCS ’82 ; Codish et al, JAR ’11 ] Weighted Path Order [Yamada, Kusakari, Sakabe, SCP ’15 ] 17/25

  38. (SAT and) SMT solving for path orders Path orders: based on precedences of function symbols Recursive Path Order [Dershowitz, TCS ’82 ; Codish et al, JAR ’11 ] Weighted Path Order [Yamada, Kusakari, Sakabe, SCP ’15 ] Knuth-Bendix Order [Knuth, Bendix, CPAA ’70 ] → SMT-Encoding to QF_LIA [Zankl, Hirokawa, Middeldorp, JAR ’09 ] outperformed polynomial time algorithm [Korovin, Voronkov, IC ’03 ] in experiments 17/25

  39. (SAT and) SMT solving for path orders Path orders: based on precedences of function symbols Recursive Path Order [Dershowitz, TCS ’82 ; Codish et al, JAR ’11 ] Weighted Path Order [Yamada, Kusakari, Sakabe, SCP ’15 ] Knuth-Bendix Order [Knuth, Bendix, CPAA ’70 ] → SMT-Encoding to QF_LIA [Zankl, Hirokawa, Middeldorp, JAR ’09 ] outperformed polynomial time algorithm [Korovin, Voronkov, IC ’03 ] in experiments Analogy: Exponential-time simplex vs. polynomial-time interior-point methods for QF_LRA? 17/25

  40. Further extensions Constrained term rewriting [Fuhs et al, RTA ’09 ; Kop, Nishida, FroCoS ’13 ; Rocha, Meseguer, Muñoz, WRLA ’14 ] term rewriting with predefined operations from SMT theories, e.g. integer arithmetic, . . . target language for translations from programming languages 18/25

  41. Further extensions Constrained term rewriting [Fuhs et al, RTA ’09 ; Kop, Nishida, FroCoS ’13 ; Rocha, Meseguer, Muñoz, WRLA ’14 ] term rewriting with predefined operations from SMT theories, e.g. integer arithmetic, . . . target language for translations from programming languages Complexity analysis [Hirokawa, Moser, IJCAR ’08 ; Noschinski, Emmes, Giesl, JAR ’13 ] Can re-use termination machinery to infer and prove statements like “runtime complexity of this TRS is in O ( n 3 ) ” 18/25

  42. SMT solvers from termination analysis Annual SMT-COMP, division QF_NIA Year Winner 2009 Barcelogic-QF_NIA 2010 MiniSmt 2011 AProVE 2012 no QF_NIA 2013 no SMT-COMP 2014 AProVE 2015 AProVE 2016 → today, 4 pm 19/25

  43. SMT solvers from termination analysis Annual SMT-COMP, division QF_NIA Year Winner 2009 Barcelogic-QF_NIA 2010 MiniSmt (spin-off of T T 2) T 2011 AProVE 2012 no QF_NIA 2013 no SMT-COMP 2014 AProVE 2015 AProVE 2016 → today, 4 pm ⇒ Termination provers can also be successful SMT solvers! 19/25

  44. SMT solvers from termination analysis Annual SMT-COMP, division QF_NIA Year Winner 2009 Barcelogic-QF_NIA 2010 MiniSmt (spin-off of T T 2) T 2011 AProVE 2012 no QF_NIA 2013 no SMT-COMP 2014 AProVE 2015 AProVE 2016 → today, 4 pm ⇒ Termination provers can also be successful SMT solvers! (disclaimer: Z3 participated only hors concours in the last years) 19/25

  45. Term Rewrite Systems (TRSs) 1 Imperative Programs 2 20/25

  46. Papers on termination of imperative programs often about integers as data 21/25

  47. Papers on termination of imperative programs often about integers as data Example (Imperative program) if x ≥ 0: while x � = 0: x = x − 1 Does this program terminate? 21/25

  48. Papers on termination of imperative programs often about integers as data Example (Imperative program) if x ≥ 0: ℓ 0 : ℓ 1 : while x � = 0: ℓ 2 : x = x − 1 Does this program terminate? Example (Equivalent translation to transition system) → [ x ≥ 0] ℓ 0 ( x ) ℓ 1 ( x ) ℓ 1 ( x ) → ℓ 2 ( x ) [ x � = 0] ℓ 2 ( x ) → ℓ 1 ( x − 1) ℓ 1 ( x ) → ℓ 3 ( x ) [ x == 0] 21/25

  49. Papers on termination of imperative programs often about integers as data Example (Imperative program) if x ≥ 0: ℓ 0 : ℓ 1 : while x � = 0: ℓ 2 : x = x − 1 Does this program terminate? Example (Equivalent translation to transition system) → [ x ≥ 0] ℓ 0 ( x ) ℓ 1 ( x ) ℓ 1 ( x ) → ℓ 2 ( x ) [ x � = 0] ℓ 2 ( x ) → ℓ 1 ( x − 1) ℓ 1 ( x ) → ℓ 3 ( x ) [ x == 0] Oh no! ℓ 1 ( − 1) → ℓ 2 ( − 1) → ℓ 1 ( − 2) → ℓ 2 ( − 2) → ℓ 1 ( − 3) → · · · 21/25

  50. Papers on termination of imperative programs often about integers as data Example (Imperative program) if x ≥ 0: ℓ 0 : ℓ 1 : while x � = 0: ℓ 2 : x = x − 1 Does this program terminate? Example (Equivalent translation to transition system) → [ x ≥ 0] ℓ 0 ( x ) ℓ 1 ( x ) ℓ 1 ( x ) → ℓ 2 ( x ) [ x � = 0] ℓ 2 ( x ) → ℓ 1 ( x − 1) ℓ 1 ( x ) → ℓ 3 ( x ) [ x == 0] Oh no! ℓ 1 ( − 1) → ℓ 2 ( − 1) → ℓ 1 ( − 2) → ℓ 2 ( − 2) → ℓ 1 ( − 3) → · · · ⇒ Restrict initial states to ℓ 0 ( z ) for z ∈ Z 21/25

  51. Papers on termination of imperative programs often about integers as data Example (Imperative program) if x ≥ 0: ℓ 0 : ℓ 1 : while x � = 0: ℓ 2 : x = x − 1 Does this program terminate? Example (Equivalent translation to transition system) → [ x ≥ 0] ℓ 0 ( x ) ℓ 1 ( x ) ℓ 1 ( x ) → ℓ 2 ( x ) [ x � = 0] ℓ 2 ( x ) → ℓ 1 ( x − 1) ℓ 1 ( x ) → ℓ 3 ( x ) [ x == 0] Oh no! ℓ 1 ( − 1) → ℓ 2 ( − 1) → ℓ 1 ( − 2) → ℓ 2 ( − 2) → ℓ 1 ( − 3) → · · · ⇒ Restrict initial states to ℓ 0 ( z ) for z ∈ Z ⇒ Find invariant x ≥ 0 at ℓ 1 , ℓ 2 21/25

  52. Papers on termination of imperative programs often about integers as data Example (Imperative program) if x ≥ 0: ℓ 0 : ℓ 1 : while x � = 0: ℓ 2 : x = x − 1 Does this program terminate? Example (Equivalent translation to transition system) → [ x ≥ 0] ℓ 0 ( x ) ℓ 1 ( x ) ℓ 1 ( x ) → ℓ 2 ( x ) [ x � = 0 ∧ x ≥ 0] ℓ 2 ( x ) → ℓ 1 ( x − 1) [ x ≥ 0] ℓ 1 ( x ) → ℓ 3 ( x ) [ x == 0 ∧ x ≥ 0] Oh no! ℓ 1 ( − 1) → ℓ 2 ( − 1) → ℓ 1 ( − 2) → ℓ 2 ( − 2) → ℓ 1 ( − 3) → · · · ⇒ Restrict initial states to ℓ 0 ( z ) for z ∈ Z ⇒ Find invariant x ≥ 0 at ℓ 1 , ℓ 2 21/25

  53. Proving termination with invariants Example (Transition system with invariants) ℓ 0 ( x ) → ℓ 1 ( x ) [ x ≥ 0] ℓ 1 ( x ) → ℓ 2 ( x ) [ x � = 0 ∧ x ≥ 0] ℓ 2 ( x ) → ℓ 1 ( x − 1) [ x ≥ 0] ℓ 1 ( x ) → ℓ 3 ( x ) [ x == 0 ∧ x ≥ 0] Prove termination by ranking function [ · ] with [ ℓ 0 ] ( x ) = [ ℓ 1 ] ( x ) = · · · = x 22/25

  54. Proving termination with invariants Example (Transition system with invariants) ℓ 0 ( x ) ℓ 1 ( x ) [ x ≥ 0] � ℓ 1 ( x ) � ℓ 2 ( x ) [ x � = 0 ∧ x ≥ 0] ℓ 2 ( x ) ≻ ℓ 1 ( x − 1) [ x ≥ 0] ℓ 1 ( x ) � ℓ 3 ( x ) [ x == 0 ∧ x ≥ 0] Prove termination by ranking function [ · ] with [ ℓ 0 ] ( x ) = [ ℓ 1 ] ( x ) = · · · = x 22/25

  55. Proving termination with invariants Example (Transition system with invariants) ℓ 0 ( x ) ℓ 1 ( x ) [ x ≥ 0] � ℓ 1 ( x ) � ℓ 2 ( x ) [ x � = 0 ∧ x ≥ 0] ℓ 2 ( x ) ≻ ℓ 1 ( x − 1) [ x ≥ 0] ℓ 1 ( x ) � ℓ 3 ( x ) [ x == 0 ∧ x ≥ 0] Prove termination by ranking function [ · ] with [ ℓ 0 ] ( x ) = [ ℓ 1 ] ( x ) = · · · = x Automate search using parametric ranking function: [ ℓ 0 ] ( x ) = a 0 + b 0 · x, [ ℓ 1 ] ( x ) = a 1 + b 1 · x, . . . 22/25

  56. Proving termination with invariants Example (Transition system with invariants) ℓ 0 ( x ) ℓ 1 ( x ) [ x ≥ 0] � ℓ 1 ( x ) � ℓ 2 ( x ) [ x � = 0 ∧ x ≥ 0] ℓ 2 ( x ) ≻ ℓ 1 ( x − 1) [ x ≥ 0] ℓ 1 ( x ) � ℓ 3 ( x ) [ x == 0 ∧ x ≥ 0] Prove termination by ranking function [ · ] with [ ℓ 0 ] ( x ) = [ ℓ 1 ] ( x ) = · · · = x Automate search using parametric ranking function: [ ℓ 0 ] ( x ) = a 0 + b 0 · x, [ ℓ 1 ] ( x ) = a 1 + b 1 · x, . . . Constraints e.g.: x ≥ 0 ⇒ a 2 + b 2 · x > a 1 + b 1 · ( x − 1) “decrease . . . ” x ≥ 0 ⇒ a 2 + b 2 · x ≥ 0 “. . . against a bound” 22/25

  57. Proving termination with invariants Example (Transition system with invariants) ℓ 0 ( x ) ℓ 1 ( x ) [ x ≥ 0] � ℓ 1 ( x ) � ℓ 2 ( x ) [ x � = 0 ∧ x ≥ 0] ℓ 2 ( x ) ≻ ℓ 1 ( x − 1) [ x ≥ 0] ℓ 1 ( x ) � ℓ 3 ( x ) [ x == 0 ∧ x ≥ 0] Prove termination by ranking function [ · ] with [ ℓ 0 ] ( x ) = [ ℓ 1 ] ( x ) = · · · = x Automate search using parametric ranking function: [ ℓ 0 ] ( x ) = a 0 + b 0 · x, [ ℓ 1 ] ( x ) = a 1 + b 1 · x, . . . Constraints e.g.: x ≥ 0 ⇒ a 2 + b 2 · x > a 1 + b 1 · ( x − 1) “decrease . . . ” x ≥ 0 ⇒ a 2 + b 2 · x ≥ 0 “. . . against a bound” Use Farkas’ Lemma to eliminate ∀ x , QF_LRA solver gives model for a i , b i . 22/25

  58. Proving termination with invariants Example (Transition system with invariants) ℓ 0 ( x ) ℓ 1 ( x ) [ x ≥ 0] � ℓ 1 ( x ) � ℓ 2 ( x ) [ x � = 0 ∧ x ≥ 0] ℓ 2 ( x ) ≻ ℓ 1 ( x − 1) [ x ≥ 0] ℓ 1 ( x ) � ℓ 3 ( x ) [ x == 0 ∧ x ≥ 0] Prove termination by ranking function [ · ] with [ ℓ 0 ] ( x ) = [ ℓ 1 ] ( x ) = · · · = x Automate search using parametric ranking function: [ ℓ 0 ] ( x ) = a 0 + b 0 · x, [ ℓ 1 ] ( x ) = a 1 + b 1 · x, . . . Constraints e.g.: x ≥ 0 ⇒ a 2 + b 2 · x > a 1 + b 1 · ( x − 1) “decrease . . . ” x ≥ 0 ⇒ a 2 + b 2 · x ≥ 0 “. . . against a bound” Use Farkas’ Lemma to eliminate ∀ x , QF_LRA solver gives model for a i , b i . More: [Podelski, Rybalchenko, VMCAI ’04 , Alias et al, SAS ’10 ] 22/25

  59. Searching for invariants using SMT Termination prover needs to find invariants for programs on integers Statically before the translation [Ströder et al, IJCAR ’14 ] In cooperation with a safety prover [Brockschmidt, Cook, Fuhs, CAV ’13 ] Using Max-SMT [Larraz, Oliveras, Rodríguez-Carbonell, Rubio, FMCAD ’13 ] Nowadays all SMT-based! 23/25

  60. Extensions Proving non -termination (infinite run from initial states is possible) [Gupta et al, POPL ’08 , Brockschmidt et al, FoVeOOS ’11 , Chen et al, TACAS ’14 , Larraz et al, CAV ’14 , Cook et al, FMCAD ’14 ] CTL ∗ model checking for infinite state systems based on termination and non-termination provers [Cook, Khlaaf, Piterman, CAV ’15 ] Complexity bounds [Alias et al, SAS ’10 , Hoffmann, Shao, JFP ’15 , Brockschmidt et al, TOPLAS ’16 ] 24/25

  61. Conclusion Automated termination analysis for term rewriting and for imperative programs developed in parallel over the last ∼ 15 years 25/25

  62. Conclusion Automated termination analysis for term rewriting and for imperative programs developed in parallel over the last ∼ 15 years Term rewriting: need to encode how to represent data structures 25/25

  63. Conclusion Automated termination analysis for term rewriting and for imperative programs developed in parallel over the last ∼ 15 years Term rewriting: need to encode how to represent data structures Imperative programs on integers: need to consider reachability and invariants 25/25

  64. Conclusion Automated termination analysis for term rewriting and for imperative programs developed in parallel over the last ∼ 15 years Term rewriting: need to encode how to represent data structures Imperative programs on integers: need to consider reachability and invariants Since a few years cross-fertilization 25/25

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