Safety of Uncertain Session Types with Interval Probability
Abstract
:1. Introduction
2. Probabilistic Multiparty Session Calculus
2.1. Syntax
- Initial decision: uses the channel with role in order to decide how to respond to the poll: by phone, instant messaging or by deferring the response. Probabilities are assigned to each option, indicating ’s choice.
- Phone communication: If chooses the option, communication occurs through a unique session channel () representing ’s phone number.
- ’s interaction: initiates the interaction by invoking , where becomes after communication. Here, represents the first question for to answer using the following labels: , no, or . excludes the option for this question, as she considers it inapplicable to the question at hand.
- Question–answer sequence: After receiving an answer, proceeds to the next question in the list (represented by ).
- Exiting the poll: Both processes are able to get out of the poll by selecting either the or label. These options have low probabilities, indicating a low likelihood of being chosen during the survey poll.
2.2. Operational Semantics
1ex |
implies . |
3. Global and Local Types Using Interval Probability
- ;
- ;
- ;
- .
- if s.t. , ;
- if and .
- for all , the projection is defined;
- in all its subterms of the form , the set of intervals is proper and reachable.
4. Interval Probability for Multiparty Session Types
4.1. Typing System
- ,, for ;
- ;
- whenever .
- , implies ;
- implies ;
- and implies .
4.2. Behavioral Properties of Typed Processes
- (substitution) If and then .
- (type weakening) If and is only end, then .
- (type strengthening) If and , then .
- (sort weakening) If and , then .
- (sort strengthening) If and , then .
- Case .⇒ Assume . By inverting the rule (TConc), we obtain and , where . By inverting the rule (TEnd), is only end and is such that . Then, by type weakening, we obtain that , where .⇐ Assume . By rule (TEnd), it holds that , where is only end and . By applying the rule (TConc), we obtain and since , by type strengthening, we obtain , as required.
- Case .⇒ Assume . By inverting the rule (TRes), we obtain where . By inverting the rule (TEnd), is only end, namely is only end. Using the rule (TEnd), we obtain .⇐ Assume . By rule (TRes), it holds that where , and . Then, by type weakening, we obtain that , as required.
- Case⇒ Assume . By inverting the rule (TConc), we obtain and , where . Using the rule (TConc), we obtain .⇐ It follows in a similar way (actually, symmetric to ⇒).
- Case⇒ Assume . By inverting the rule (TConc), we obtain , and , where . Using the rule (TConc), we obtain .⇐ It follows in a similar way (actually, symmetric to ⇒).
- Case⇒ Assume . By inverting the rule (TRes) twice, we obtain where and . Using the rule (TRes), we obtain .⇐ Proof is similar (actually, symmetric to ⇒).
- Case⇒ Assume . Since and , if (by using type weakening), then such that , and is only end. By type strengthening, we obtain . By inverting the rule (TConc), we obtain and , where . By inverting the rule (TRes), we obtain where . Using the rule (TConc), we obtain . Since , it means that does not contain types for s; from with , by using the rule (TRes) we obtain , where . Since is only end, by using type weakening we obtain .⇐ Assume . By inverting rule (TRes), we obtain , where . Since , by inverting the rule (TConc), we obtain and , where . Using rule (TRes), we obtain . By using rule (TConc), we obtain , as required.
- Case⇒ Assume . By inverting rule (TDef), we obtain and , where . By inverting rule (TEnd) we obtain that is only end. Applying rule (TEnd), we obtain .⇐ Assume . By inverting rule (TEnd), we obtain that is only end. Applying rule (TEnd), we obtain . Consider a typable process P such that . By applying rule (TDef) we obtain that , namely , as required.
- Case⇒ Assume . By inverting rule (TDef), we obtain and , where . By inverting the rule (TRes), we obtain where . Using rule (TDef), we obtain , while using rule (TRes), we obtain .⇐ Assume . By inverting rule (TRes), we obtain where . By inverting rule (TDef), we obtain and , where . Using rule (TRes), we obtain . By applying rule (TDef) we obtain that , namely , as required.
- Case⇒ Assume . By inverting the rule (TDef), we obtain and , where . By inverting the rule (TConc), we obtain and , where . By applying the rule (TDef), we obtain that , namely . Since and , then and so by sort strengthening we obtain . By applying the rule (TConc), we obtain that .⇐ Assume . By inverting rule (TConc), we obtain and , where . By inverting rule (TDef), we obtain and , where and . Since , by sort weakening we obtain . By applying rule (TConc), we obtain that . By using rule (TDef), we obtain , namely , as required.
- Case⇒ Assume . By inverting rule (TDef), we obtain and , where . By inverting rule (TDef), we obtain and , where . Since , it means that X and are different. Applying rule (TDef) twice, we obtain .⇐ It follows in a similar way (actually, symmetric to ⇒).
- implies . Let and .The proof is by structural induction on .
- –
- Case . This case is trivial, as is in fact .
- –
- Case . By assumption, . By inverting the rule (TRes), we obtain that . By structural congruence, implies . By induction, . By structural congruence, implies . By using the rule (TRes), we obtain , as required.
- –
- Case . By assumption, . By inverting rule (TDef), we obtain and , where . By structural congruence, implies . By induction, . By structural congruence, implies . By using the rule (TDef), we obtain .
- –
- Case . By assumption, . By inverting rule (TConc), we obtain and , where . By structural congruence, implies . By induction, . By structural congruence, implies . By using the rule (TConc), we obtain , as required.
- Case (Com): .By assumption, . By inverting the rule (TConc), we obtain and with . Since these can be inferred only from (TSelect) and (TBranch), we know that and . By inverting the rules (TSelect) and (TBranch), we obtain that , , and . Since , from and , by applying the substitution part of Lemma 1, we obtain that . By applying the rule (TConc), we obtain . By using the type reduction relation, we obtain , where and .
- Case (Call):By assumption, . By inverting rule (TDef), we obtain and . By inverting the rule (TConc), we obtain and , where . By inverting the rule (TCall), we obtain and is end only. Applying substitution (Lemma 1), we obtain . Since is end only, then by type weakening (Lemma 1) we obtain that . Applying rule (TConc), we obtain . Applying rule (TDef), we obtain , as desired.
- Case (Ctxt): implies , where p = . Let , . Proof is by structural induction on .
- –
- Case . This case is trivial, as is in fact .
- –
- Case . By assumption, . By inverting rule (TRes), we obtain that . By applying rule (Ctxt), implies , where . By induction, , where or , with either or and . By applying rule (Ctxt), implies , where . By using the rule (TRes), we obtain , where or and , as required.
- –
- Case . By assumption, . By inverting rule (TDef), we obtain and , where . By applying rule (Ctxt), implies , where . By induction, , where or and . By applying rule (Ctxt), implies , where . By using rule (TDef), we obtain , where or and , as required.
- –
- Case . By assumption, . By inverting rule (TConc), we obtain and , where . By applying rule (Ctxt), implies , where . By induction, , with or and . By applying rule (Ctxt), implies , where . By using the rule (TConc), we obtain , with or and , as required.
- Case (Struct): and and implies . We use Theorem 1.
- Case (Com): .This means that and thus , as desired.
- Case (Call): . This means that and thus . Since , then , as desired.
- Case (Ctxt): implies , with . This means that and thus . From and , it holds from the rules of Table 5 that there exists and such that and . By induction, since , and , it holds that and thus , as desired.
- Case (Struct): and and implies . From and , by using Theorem 1, it holds that . By induction, since , and , it holds that . Also, from , it holds that and thus , as desired.
- Case (Err): (if and ). This means that . Since , and , by Corollary 1, has no error. This contradicts the assumption that and thus there is no P with , and such that rule (Err) is applicable.
- All typing rules are deterministically invertible—i.e., at most one typing rule can be applied for any arbitrary given well-typed process;
- At each invert, the typing contexts, sortings and type annotations in the process in conclusion determine how typing contexts, sortings and type annotations in the process in premises are generated; note that in the worst case, when splitting a typing context , one might need to try all possible typing contexts and such that ;
- Each rule invert guarantees that the premises contain smaller subterms of the term in the conclusion and so the recursive derivation eventually terminates.
- Case . Then there exist processes and and also, in , the terms and such that and . By inverting the rules (TSelect) and (TBranch), we obtain that and . According to rule (Com), there exist and and a probability p such that ; by using rule (Ctxt), it holds that , as desired.
4.3. Properties Preserved by Removing Probabilities
- Case (Com): . Using the definition of the erase function, we obtain that.By using rule (SCom), it holds that .Since , then it results that , as required.
- . Assume that for all it holds that there exist and such that . There exists such that and according to rule (TSelect) of Table 5 it holds that , where . Since by induction for all it holds that and , then . According to rule (TSelect), it holds that . Since and , it implies that , as desired.
4.4. Typed Probabilistic Equivalence
- . Assume that for all it holds that there exist and such that . There exist such that and according to rule (TSelect) of Table 5 it holds that , where . From the assumption that contains at least one probability interval of the form in which and , one can construct a Q such that by considering , where for all it holds that and for at least one we have . Thus, by construction, it holds that . Since by induction for all it holds that , it implies that also , as desired.
- Reflexivity: Consider . Since , from Definition 7 it follows that , as desired.
- Symmetry: Consider . According to Definition 7, it holds that , and and thus .
- Transitivity: Consider and . According to Definition 7, it holds that , , , and and thus .
- , for .By induction and by the associativity of interval intersections, we obtain that , as requested.
- , for .By induction, we obtain that , as requested.
- , for .By induction, we obtain that .
- . Assume that for all it holds that there exist , , and such that and . There exist and such that and . According to rule (TSelect) of Table 5, it holds that and , where and together with . Since for all it holds that and , then also holds. By induction for all , it holds that . By applying the rule (TSelect) of Table 5, it holds that . Since , then , as desired.
- . Assume that for all it holds that there exist and such that . There exist such that and according to rule (TSelect) of Table 5, it holds that , where . If there exists such that , then it implies that . Since for all it holds that , then also holds. By induction for all it holds that . By applying the rule (TSelect) of Table 5, it holds that . Since , then , as desired.
- A typed probabilistic relation over the set of processes, the set of sortings and the set of typings is any relation .
- The identity typed probabilistic relation is
- The inverse of a typed probabilistic relation is
- The composition of typed probabilistic relations and is
- 1.
- is a typed probabilistic bisimulation if and implies that exists and such that and .
- 2.
- The typed probabilistic bisimilarity is the union ≃ of all typed probabilistic bisimulations .
- 1.
- Identity, inverse, composition and union of typed probabilistic bisimulations are typed probabilistic bisimulations.
- 2.
- ≃ is the largest typed probabilistic bisimulation.
- 3.
- ≃ is an equivalence.
- We treat each relation separately, showing that it respects the conditions from Definition 8 for being a typed probabilistic bisimulation.
- (a)
- The identity relation is a typed probabilistic bisimulation. Assume ; then and . Consider ; according to Theorem 2, there exists and such that and or . Since , according to Definition 5, it also holds that . According to Proposition 7, by reflexivity, it holds that . Thus, it results that , as desired.
- (b)
- The inverse of a typed probabilistic bisimulation is a typed probabilistic bisimulation. Assume , namely . Consider ; then for some and we have and , namely . By similar reasoning, if then we can find and such that and .
- (c)
- The composition of typed probabilistic bisimulations is a typed probabilistic bisimulation. Assume . Then for some R we have and . Consider ; then for some and , since , we have and . Also, since we have for some that and . Thus, . By similar reasoning, if then we can find and such that and .
- (d)
- The union of typed probabilistic bisimulations is a typed probabilistic bisimulation. Assume . Then for some we have . Consider ; then for some and , since , we have and . Thus, . By similar reasoning, if then we can find such that and , namely .
- Since the union of typed probabilistic bisimulations is a typed probabilistic bisimulation, ≃ is a typed probabilistic bisimulation and it includes any other typed probabilistic bisimulation.
- Proving that relation ≃ is an equivalence requires to show that it satisfies reflexivity, symmetry and transitivity. We consider each of them in what follows:
- (a)
- Reflexivity: For any process P, results from the fact that the identity relation is a typed probabilistic bisimulation.
- (b)
- Symmetry: If , then exists and such that for some typed probabilistic bisimulation . Hence and so because the inverse relation is a typed probabilistic bisimulation.
- (c)
- Transitivity: If and then and for some typed probabilistic bisimulations and . Thus, and so due to the fact that the composition relation is a typed probabilistic bisimulation.
4.5. Probabilistic Properties of Typed Processes
- Case . The proof proceeds by cases depending on the value of .
- –
- . We have only one subcase:
- ∗
- . Since , then and the sum is not computed.
- ∗
- Since is well-typed, this means that , and also that .
- –
- . We have two subcases:
- ∗
- . Since , it means that R is unable to evolve and only could evolve by applying the rule (Call). Thus, by also using rule (Ctxt), it holds that evolves with probability 1 to . This means that (as desired).
- ∗
- , where and , meaning that the rules (Com) and (Ctxt) are applied and evolves with probability (with ) to . From the definition of , it follows that . Thus, (as desired).
- –
- , with . In this case, there is able to perform at least a (Call) or (Com) rule such that and , where . Since , then by removing from P, what remains (namely ) is such that . Since is well-typed, then . Thus, .The normalization appears due to the application of rule (Ctxt).
- Case . Since , this means that . By considering the inductive step, we obtain that . Let us consider that another evolution step is performed, namely k steps starting from process . This means that can be broken into two parts: one computing the probability for the first () steps and another one using the probabilities obtained in the additional step k. Thus, it holds that . Since we have , thenAlso, since we have , then
5. Conclusions and Related Work
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Processes | |||
| | (branching from role r, ) | ||
| | (process definition & process call) | ||
| | (parallel & restriction) | ||
| | (inaction & error) | ||
Declarations | |||
D | (process declaration) | ||
Contexts | |||
(evaluation context) | |||
Channels | |||
c | x| | (variable, channel with role r) | |
Values | |||
v | (channel, base value) |
(if ) | (Com) |
(if , ) | (Call) |
implies (where ) | (Ctxt) |
and and implies | (Struct) |
(if and ) | (Err) |
Global | G | | t | end | |
Sorts | S |
Local | T | | t | end | |
Sorts | S |
(TVar), (TVal) | |
(TSelect) | |
(TBranch) | |
(TEnd), (TConc) | |
(TRes) | |
(TDef) | |
(TCall) |
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Aman, B.; Ciobanu, G. Safety of Uncertain Session Types with Interval Probability. Symmetry 2025, 17, 218. https://doi.org/10.3390/sym17020218
Aman B, Ciobanu G. Safety of Uncertain Session Types with Interval Probability. Symmetry. 2025; 17(2):218. https://doi.org/10.3390/sym17020218
Chicago/Turabian StyleAman, Bogdan, and Gabriel Ciobanu. 2025. "Safety of Uncertain Session Types with Interval Probability" Symmetry 17, no. 2: 218. https://doi.org/10.3390/sym17020218
APA StyleAman, B., & Ciobanu, G. (2025). Safety of Uncertain Session Types with Interval Probability. Symmetry, 17(2), 218. https://doi.org/10.3390/sym17020218