Sep 19, 2018 - b → f(a) f(x) → x ⇐ g(x) → c(y) we have. DPH(R) : G(a) → B. (1). B → F(a) (2) ... 2 Proving termination properties of CTRSs with 2D Dependency Pairs. 3 The Open ... The chain is minimal if for all i ≥ 1, terms obtained after a. Λ.
The 2D Dependency Pair Framework for conditional rewrite systems Salvador Lucas1 1 DSIC, 2 CS
Jos´e Meseguer2
Ra´ ul Guti´errez1
Universitat Polit` ecnica de Val` encia, Spain
Dept., University of Illinois at Urbana-Champaign, USA
´ n y Lenguajes XVIII Jornadas sobre Programacio PROLE 2018 Sevilla, Spain
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The 2D DP Framework for CTRSs
Introduction
For Conditional TRSs (CTRSs) which are not operationally terminating, two nonterminating behaviors can be observed [LM17]: 1 Infinite reduction sequences, a ‘horizontal’ dimension: nontermination 2 Infinitely many attempts to satisfy the conditions of the rules before issuing a reduction step, a ‘vertical’ dimension: non-V -termination The 2D DP Framework is used to automatically (dis)prove operational termination, termination and V -termination by using dependency pairs [LMG18]
Example (A terminating but non-V-terminating CTRS) For the CTRS R: g(a) → c(b)
b → f(a)
f(x) → x ⇐ g(x) → c(y )
we have DPH (R) :
G(a) → B
(1)
DPV (R) :
F(x) → G(x) (3)
B → F(a) (2) and DPVH (R) = DPH (R) 2
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The 2D DP Framework for CTRSs
Summary
Summary 1
2D Dependency Pairs and chains
2
Proving termination properties of CTRSs with 2D Dependency Pairs
3
The Open 2D DP Framework Some processors:
4
SCC Processor Narrowing Processors 3 Detection of Nontermination 1 2
5
Implementation
6
Conclusions and future work
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2D Dependency pairs and chains
Dependency pairs
2D Dependency Pairs: main idea Use new rules u → v ⇐ c to track function calls 1
Horizontal: ` = f (`1 , . . . , `m ) → C [g (v1 , . . . , vn )] = r ⇐ c
2
Vertical: ` = f (`1 , . . . , `m ) → r ⇐ · · · ∧ C [g (v1 , . . . , vn )] → ti ∧ · · ·
1
Rules in DPH (R) track horizontal computations that never go up Rules in DPV (R) track computations that go infinitely often up Rules in DPVH (R) are used for the horizontal connection of rules in DPV (R)
2 3
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2D Dependency pairs and chains
Dependency pairs
2D Dependency Pairs: main idea Use new rules u → v ⇐ c to track function calls 1
Horizontal: ` = f (`1 , . . . , `m ) → C [g (v1 , . . . , vn )] = r ⇐ c
2
Vertical: ` = f (`1 , . . . , `m ) → r ⇐ · · · ∧ C [g (v1 , . . . , vn )] → ti ∧ · · ·
1
Rules in DPH (R) track horizontal computations that never go up Rules in DPV (R) track computations that go infinitely often up Rules in DPVH (R) are used for the horizontal connection of rules in DPV (R)
2 3
Feasible pairs! Only R-feasible pairs are considered. For each of such pairs u → v ⇐ c there is a substitution σ such that for all s → t ∈ c, σ(s) →∗R σ(t) holds. Feasibility is undecidable. In practice, we ‘accept’ all pairs as feasible, except those which can be proved R-infeasible [LG18] 4
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2D Dependency pairs and chains
Chains of dependency pairs
Λ
Given CTRSs R and S we write s →S,R t if there is ` → r ⇐ c ∈ S and σ such that s = σ(`), t = σ(r ) and σ(u) →∗R σ(v ) for all u → v ∈ c.
Definition (Chain of pairs) Let P, Q, and R be CTRSs. A (P, Q, R)-chain is a sequence of pairs ui → vi ⇐ ci ∈ P with a substitution σ such that, for all i ≥ 1, 1
σ(s) →∗R σ(t) for all s → t ∈ ci and
2
σ(vi )(→R ∪ →Q,R )∗ σ(ui+1 ).
Λ
Λ
The chain is minimal if for all i ≥ 1, terms obtained after a →P∪Q,R -step are all R-operationally terminating
Definition (CTRS problem) A CTRS problem is a tuple τ = (P, Q, R, f ), where P, Q and R are CTRSs, and f is a flag ranging on F = {a, m} (referring arbitrary/minimal chains) 5
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Proving termination properties of CTRSs with 2D DPs
A CTRS problem τ = (P, Q, R, f ) is finite iff there is no infinite τ -chain (arbitrary or minimal, depending on f ); τ is infinite otherwise. Two kind of initial CTRS problems (for f ∈ F): τfH = (DPH (R), ∅, R, f ) and τfV = (DPV (R), DPVH (R), R, f ) Initial CTRS problems Requirements on R Fin/Inf τaH preserves terminating Fin substitutions Nontermination τfH , f ∈ F None Inf V -Termination τaV deterministic 3-CTRS Fin Non-V -Termination τfV , f ∈ F None Inf H Op. termination τf and τfV0 , f , f 0 ∈ F deterministic 3-CTRS Fin Op. nontermination τfH or τfV , f ∈ F None Inf Term. prop. Termination
Investigate all these termination properties using a single framework!
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Proving termination properties of CTRSs with 2D DPs
A CTRS problem τ = (P, Q, R, f ) is finite iff there is no infinite τ -chain (arbitrary or minimal, depending on f ); τ is infinite otherwise. Two kind of initial CTRS problems (for f ∈ F): τfH = (DPH (R), ∅, R, f ) and τfV = (DPV (R), DPVH (R), R, f ) Initial CTRS problems Requirements on R Fin/Inf τaH preserves terminating Fin substitutions Nontermination τfH , f ∈ F None Inf V -Termination τaV deterministic 3-CTRS Fin Non-V -Termination τfV , f ∈ F None Inf H Op. termination τf and τfV0 , f , f 0 ∈ F deterministic 3-CTRS Fin Op. nontermination τfH or τfV , f ∈ F None Inf Term. prop. Termination
Investigate all these termination properties using a single framework! How to avoid a fixed value for f ? 6
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The Open 2D DP Framework
Open CTRS problems and processors
Definition (Open CTRS problem) An Open CTRS problem is a tuple τˇ = (P, Q, R, ϕ), where P, Q and R are CTRSs, and ϕ is a constant flag in {a, m} or a variable flag f ∈ V
Definition (Open CTRS processor) An open CTRS processor P is a partial function from OCTRS problems τˇ = (P, Q, R, ϕ) into sets of OCTRS problems introducing no new variable flag; alternatively, it can return “no”.
Definition (Soundness and completeness) Let τˇ = (P, Q, R, ϕ), and f ∈ F be such that ςf (ϕ) = f . P is said to be: • (ˇ τ , f )-sound iff ςf (ˇ τ ) is finite whenever P(ˇ τ ) 6= no and for all
τˇ0 ∈ P(ˇ τ ), ςf (ˇ τ 0 ) is finite. • (ˇ τ , f )-complete iff ςf (ˇ τ ) is infinite whenever P(ˇ τ ) = no or there is
τˇ0 ∈ P(ˇ τ ) such that ςf (ˇ τ 0 ) is infinite. 7
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The Open 2D DP Framework
OCTRS Proof trees
Divide and conquer proof strategy using processors to build an Open CTRS Proof tree (OCTRSP-tree): ϕ1s [ τˇ1 ] ϕ1c
• Plugging schemes hϕs , ϕc i make
the soundness and completeness requirements of processors explicit when applied to an OCTRS
P1
ϕ2s [ τˇ2 ] ϕ2c P2
ϕ31 ˇ31 ] ϕ31 s [τ c
ϕ32 ˇ32 ] ϕ32 s [τ c
P31
L31
P32
ϕ41 ˇ41 ] ϕ41 s [τ c P41
L41 8
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The Open 2D DP Framework
OCTRS Proof trees
Divide and conquer proof strategy using processors to build an Open CTRS Proof tree (OCTRSP-tree): ϕ1s [ τˇ1 ] ϕ1c
• Plugging schemes hϕs , ϕc i make
the soundness and completeness requirements of processors explicit when applied to an OCTRS
P1
ϕ2s [ τˇ2 ] ϕ2c
• Leaves get labels ‘yes’ or ‘no’
P2
ϕ31 ˇ31 ] ϕ31 s [τ c
ϕ32 ˇ32 ] ϕ32 s [τ c
P31
L31
P32
ϕ41 ˇ41 ] ϕ41 s [τ c P41
L41 8
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The Open 2D DP Framework
OCTRS Proof trees
Divide and conquer proof strategy using processors to build an Open CTRS Proof tree (OCTRSP-tree): ϕ1s [ τˇ1 ] ϕ1c
• Plugging schemes hϕs , ϕc i make
the soundness and completeness requirements of processors explicit when applied to an OCTRS
P1
ϕ2s [ τˇ2 ] ϕ2c
• Leaves get labels ‘yes’ or ‘no’
P2
ϕ31 ˇ31 ] ϕ31 s [τ c
ϕ32 ˇ32 ] ϕ32 s [τ c
P31
L31
P32
• Soundness and completeness
equations Es or Ec are obtained from the tree
ϕ41 ˇ41 ] ϕ41 s [τ c P41
L41 8
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The Open 2D DP Framework
OCTRS Proof trees
Divide and conquer proof strategy using processors to build an Open CTRS Proof tree (OCTRSP-tree): ϕ1s [ τˇ1 ] ϕ1c
• Plugging schemes hϕs , ϕc i make
the soundness and completeness requirements of processors explicit when applied to an OCTRS
P1
ϕ2s [ τˇ2 ] ϕ2c
• Leaves get labels ‘yes’ or ‘no’
P2
ϕ31 ˇ31 ] ϕ31 s [τ c
ϕ32 ˇ32 ] ϕ32 s [τ c
P31
L31
P32
ϕ41 ˇ41 ] ϕ41 s [τ c
• Soundness and completeness
equations Es or Ec are obtained from the tree • A unifier ς of the equations is
eventually obtained
P41
L41 8
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The Open 2D DP Framework
OCTRS Proof trees
Divide and conquer proof strategy using processors to build an Open CTRS Proof tree (OCTRSP-tree): ϕ1s [ τˇ1 ] ϕ1c
• Plugging schemes hϕs , ϕc i make
the soundness and completeness requirements of processors explicit when applied to an OCTRS
P1
ϕ2s [ τˇ2 ] ϕ2c
• Leaves get labels ‘yes’ or ‘no’
P2
ϕ31 ˇ31 ] ϕ31 s [τ c
ϕ32 ˇ32 ] ϕ32 s [τ c
P31
L31
P32
ϕ41 ˇ41 ] ϕ41 s [τ c P41
L41
• Soundness and completeness
equations Es or Ec are obtained from the tree • A unifier ς of the equations is
eventually obtained • The instance ς(ϕ) of the flag
variable ϕ in τˇ1 is used to obtain a conclusion according to the table 8
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Some processors
The SCC processor
Given τˇ = (P, Q, R, ϕ), infinite τˇ-chains are represented as cycles in a Dependency Graph EG(ˇ τ ). The nodes are (R-feasible) pairs from P; arcs between nodes are established according to possible chains between them
Running example 1
2 EG(ˇ τH)
G(a) → B
1
F(x) → G(x)
3
2
B → F(a)
3 EG(ˇ τV )
Approximating the cycle in EG(ˇ τV )
SCC Processor Processor PSCC returns the set of CTRS problems obtained from the Strongly Connected Components of the graph associated to τˇ. It follows the plugging scheme hf , f i with all OCTRS problems 9
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Some processors
The SCC processor
With PSCC , we can simplify τˇH in our running example: PSCC (ˇ τH) = ∅ because there is no cycle in EG(ˇ τ H ).
Example (Termination of the running example) Since PSCC (ˇ τ H ) = ∅, the OCTRSP-tree Tˇ H for τˇH is: f0 [ˇ τ H ] f0 PSCC
yes
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Some processors
The SCC processor
With PSCC , we can simplify τˇH in our running example: PSCC (ˇ τH) = ∅ because there is no cycle in EG(ˇ τ H ).
Example (Termination of the running example) Since PSCC (ˇ τ H ) = ∅, the OCTRSP-tree Tˇ H for τˇH is: f0 [ˇ τ H ] f0 PSCC
yes Now, Es (Tˇ H ) = {f = f0 } is unified by the ground flag substitution ςa .
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Some processors
The SCC processor
With PSCC , we can simplify τˇH in our running example: PSCC (ˇ τH) = ∅ because there is no cycle in EG(ˇ τ H ).
Example (Termination of the running example) Since PSCC (ˇ τ H ) = ∅, the OCTRSP-tree Tˇ H for τˇH is: f0 [ˇ τ H ] f0 PSCC
yes Now, Es (Tˇ H ) = {f = f0 } is unified by the ground flag substitution ςa . Thus, ςa (ˇ τ H ) = (DPH (R), ∅, R, a) is proved finite.
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Some processors
The SCC processor
With PSCC , we can simplify τˇH in our running example: PSCC (ˇ τH) = ∅ because there is no cycle in EG(ˇ τ H ).
Example (Termination of the running example) Since PSCC (ˇ τ H ) = ∅, the OCTRSP-tree Tˇ H for τˇH is: f0 [ˇ τ H ] f0 PSCC
yes Now, Es (Tˇ H ) = {f = f0 } is unified by the ground flag substitution ςa . Thus, ςa (ˇ τ H ) = (DPH (R), ∅, R, a) is proved finite. Since R is a 2-CTRS (and preserves terminating substitutions), this proves R terminating 10
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Some processors
The SCC processor
With PSCC , we can simplify τˇH in our running example: PSCC (ˇ τH) = ∅ because there is no cycle in EG(ˇ τ H ).
Example (Termination of the running example) Since PSCC (ˇ τ H ) = ∅, the OCTRSP-tree Tˇ H for τˇH is: f0 [ˇ τ H ] f0 PSCC
yes Now, Es (Tˇ H ) = {f = f0 } is unified by the ground flag substitution ςa . Thus, ςa (ˇ τ H ) = (DPH (R), ∅, R, a) is proved finite. Since R is a 2-CTRS (and preserves terminating substitutions), this proves R terminating Note: ςm also unifies Es (Tˇ H ), but it does not lead to conclude termination! 10
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Some processors
Narrowing processors
Connections between dependency pairs u → v ⇐ c and u 0 → v 0 ⇐ c 0 , i.e., the existence of a substitution σ such that σ(v ) →∗R σ(u 0 ) can be investigated by using narrowing
Narrowing processors (sketch) Given τˇ = (P, Q, R, ϕ) and u → v ⇐ c ∈ P, the pair is replaced in P by a set N of pairs which is obtained by issuing a single narrowing step on v : • With PNR , the step is issued by using R • With PNQ , the step is issued (at the topmost position) by using Q;
furthermore, the output flag is fixed to a Both processors follow the plugging schemes hf , ai and h•, ai
Application of PNQ to τˇV (running example) For τˇV = ({F(x) → G(x)}, {G(a) → B, B → F(a)}, R, ϕ), we have: 1
PNQ (ˇ τ V ) = {ˇ τ1V }, where τˇ1V = ({F(a) → B}, {(1), (2)}, R, a)
2
PNQ (ˇ τ1V ) = {ˇ τ2V }, where τˇ2V = ({F(a) → F(a)}, {(1), (2)}, R, a) 11
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Some processors
Infiniteness processor
The following processor is used to prove CTRS problems infinite.
Infiniteness processor Let τˇ = (P, Q, R, ϕ), u → v ⇐ c ∈ P and θ, η be substitutions such that for all s → t ∈ c, η(s) →∗R η(t) and η(v ) = θ(η(u)). Then, PInf (P, Q, R, ϕ) = no It follows the plugging scheme h•, ai
Application of PInf to τˇ2V (running example) We apply PInf to τˇ2V = ({F(a) → F(a)}, {(1), (2)}, R, a) by using the (trivial or empty) substitutions θ = η = . Thus, PInf (ˇ τ2V ) = no
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Some processors
Infiniteness processor
Non-V-termination of the running example The OCTRSP-tree Tˇ V for τˇV is: f0 [ˇ τV ] a PNQ
f1 [ˇ τ1V ] a PNQ
• [ˇ τ2V ] a PInf
no We have Ec (Tˇ V ) = {f = a, a = a}, which unifies with ςa . Thus, ςa (ˇ τ V ) = (DPV (R), DPVH (R), R, a) is proved infinite. Since R is a deterministic 3-CTRS, this proves R non-V-terminating 13
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Some processors
Additional processors
Other processors developed so far are: • P removes pairs u → v ⇐ c from P and Q by comparing subterms
of u and v without paying attention to the rules in R • PRTN uses well-founded relations to remove pairs from P and Q.
Rules in R must be ‘compatible’ with appropriate relations • PIR and PRIR (re)move infeasible rules from R • PNC transforms the conditions si of conditional rules
` → r ⇐ s1 → t1 , . . . , sn → tn by using narrowing. • PSUC simplifies the conditional part of the rules by removing
conditions si → ti such that si and ti unify. • PSUNC combines PNC and PSUC .
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Implementation
We have developed the first implementation of the 2D DP Framework as part of the tool mu-term (http://zenon.dsic.upv.es/muterm/) The development consists of 20 modules and more than 4000 lines of code written in Haskell. On the 117 benchmark examples of the 2018 Termination Problems Data Base (TPDB), all processors are used during the proofs: Proc PSCC PSUC PNQ
# uses 399 8 2
Proc P PNR
# uses 48 5
Proc PRTN PNC
# uses 21 5
Proc PInf PSUNC
# uses 16 3
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Conclusions and future work
First framework for (dis)proving termination properties of CTRSs (termination, V-termination, operational termination) using ‘native’ dependency pairs (without transformations into TRSs)
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Conclusions and future work
First framework for (dis)proving termination properties of CTRSs (termination, V-termination, operational termination) using ‘native’ dependency pairs (without transformations into TRSs) Thanks to the 2D DP Framework, our tool is currently the only one that disproves operational termination of CTRSs
16
S. Lucas, J. Meseguer, and R. Guti´ errez
JCSS 2018 @ PROLE 2018
September 17-19, 2018
16 / 18
Conclusions and future work
First framework for (dis)proving termination properties of CTRSs (termination, V-termination, operational termination) using ‘native’ dependency pairs (without transformations into TRSs) Thanks to the 2D DP Framework, our tool is currently the only one that disproves operational termination of CTRSs Since 2014, we have participated in the yearly International Termination Competition, each year obtaining the first position among the tools in the TRS Conditional subcategory, see http://zenon.dsic.upv.es/muterm/?page_id=82 for a summary
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S. Lucas, J. Meseguer, and R. Guti´ errez
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September 17-19, 2018
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Conclusions and future work
First framework for (dis)proving termination properties of CTRSs (termination, V-termination, operational termination) using ‘native’ dependency pairs (without transformations into TRSs) Thanks to the 2D DP Framework, our tool is currently the only one that disproves operational termination of CTRSs Since 2014, we have participated in the yearly International Termination Competition, each year obtaining the first position among the tools in the TRS Conditional subcategory, see http://zenon.dsic.upv.es/muterm/?page_id=82 for a summary
Future work • Improve the integration of model generators (used in proofs of
infeasibility and also to generate well-founded models) • Use as part of external tools like the Maude Termination Tool (MTT) 16
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The 2D DP Framework for CTRSs
The 2D Dependency Pair Framework for conditional rewrite systems
Thanks!
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References
S. Lucas and R. Guti´errez. Use of logical models for proving infeasibility in term rewriting. Information Processing Letters, 136:90-95, 2018. S. Lucas and J. Meseguer. Dependency pairs for proving termination properties of conditional term rewriting systems. Journal of Logical and Algebraic Methods in Programming, 86:236-268, 2017. S. Lucas, J. Meseguer, and R. Guti´errez. The 2D Dependency Pair Framework for conditional rewrite systems. Part I: Definition and basic processors. Journal of Computer Systems Sciences 96:74–106, 2018.
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